Lectures on Quantum Mechanics for mathematicians
Alexander Komech
TL;DR
This paper introduces quantum mechanics concepts to mathematicians, discusses the mathematical interpretation of quantum phenomena, and proposes a dynamical interpretation based on nonlinear PDE attractors, supported by various model results.
Contribution
It offers a new dynamical interpretation of quantum phenomena using attractors of nonlinear Hamiltonian PDEs, with partial confirmations and open problems.
Findings
Confirmed attractor conjectures for several nonlinear PDE models since 1990
Calculated diffraction amplitude for electron scattering using Kirchhoff approximation
Identified open problem for Maxwell–Schrödinger equations
Abstract
The main goal of these lectures -- introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation. We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of {\it model Hamiltonian nonlinear} PDEs by the results obtained since 1990 (we survey sketchy these results). However, for the Maxwell--Schr\"odinger equations this conjecture is still an {\it open problem}. We calculate the diffraction amplitude for the scattering of electron beams and Aharonov--Bohm shift via the Kirchhoff approximation.
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Quantum chaos and dynamical systems · advanced mathematical theories
Abstract
The main goal of these lectures – introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation.
We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of model Hamiltonian nonlinear PDEs by the results obtained since 1990 (we survey sketchy these results). However, for the Maxwell–Schrödinger equations this conjecture is still an open problem.
We calculate the diffraction amplitude for the scattering of electron beams and Aharonov–Bohm shift via the Kirchhoff approximation.
Abstract
We calculate the amplitude of diffraction for the electron beams in the framework of the Kirchhoff approximation applying the limiting amplitude and the limiting absorption principles, and the Sommerfeld radiation condition. The Aharonov–Bohm Ansatz and the corresponding shift of the diffraction pattern are justified for small magnetic field by a novel reduction to short-range magnetic potential and subsequent application of the Agmon–Jensen–Kato stationary scattering theory.
**Lectures on Quantum Mechanics
for mathematicians **
A.I. Komech 111 The research supported by the Austrian Science Fund (FWF) under Grant No. P28152-N35
Faculty of Mathematics of Vienna University
Institute for Information Transmission Problems of RAS, Moscow
Department Mechanics and Mathematics of Moscow State University (Lomonosov)
Keywords: Schrödinger equation; Maxwell equations; Maxwell–Schrödinger equations; semiclassical asymptotics; Hamilton–Jacobi equation; energy; charge; momentum; angular momentum; quantum transitions; wave-particle duality; probabilistic interpretation; magnetic moment; normal Zeeman effect; spin; rotation group; Pauli equation; electron diffraction; Kirchhoff approximation; Aharonov–Bohm shift; attractors; Hamiltonian equations; nonlinear partial differential equations; Lie group; symmetry group.
Contents
-
2 Planck’s law, Einstein’s photons and de Broglie wave-particle duality
-
5.2 Bohr’ postulates and asymptotics (5.6) by perturbation theory
-
8.3 Stern–Gerlach experiment and Goldsmith–Uhlenbeck hypothesis
-
9.2 Covariance of the Pauli equation with respect to rotations
-
13 Conjecture on attractors of -invariant Hamilton nonlinear PDEs
-
A.1 The Thompson electron and the Lorentz classical theory of the atom
1 Introduction
The first goal of these lectures is to give an introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of main quantum postulates:
I. Transitions between quantum stationary orbits (Bohr 1913).
II. Wave-particle duality (de Broglie 1923).
III. Probabilistic interpretation (Born 1927).
These postulates were inspired by empirical observations. However, after the discovery of the Schrödinger–Heisenberg Quantum Mechanics in the 1925–1926s, problems arose on the validity of these postulates in the dynamical theory. A rigorous dynamical description of these postulates is still unknown. This lack of theoretical clarity hinders the progress in the theory (e.g., in superconductivity and in nuclear reactions), and in numerical simulation of many engineering processes (e.g., of laser radiation and quantum amplifiers) since a computer can solve dynamical equations but cannot take into account the postulates.
It is obvious that Postulates I and II are not compatible with the linear Schrödinger equation. We suggest a novel mathematical conjecture on global attractors of the coupled (“self-consistent”) nonlinear Maxwell–Schrödinger equations which clarifies these postulates. This conjecture is confirmed for a list of model Hamilonian nonlinear PDEs by the results obtained since 1990, we survey sketchy these results in Section 14 (the details can be found in the survey [79]). However, for the Maxwell–Schrödinger equations this conjecture is still an open problem. Let us note that these coupled equations in particular underly modern theory of laser radiation [24]–[50].
On the other hand, the probabilistic interpretation, postulated by Born in 1927, agrees satisfactory with the linear Schrödinger theory in the case of electron beams with high intensity. Newertheless, for very low intensity, the diffraction becomes a random process with discrete registration of the diffracted electrons, and the diffraction pattern is obtained by long-time averaging. This was discovered for the first time by Biberman, Sushkin and Fabrikant [33] and confirmed later by Chambers, Tonomura, Frabboni, Bach & al. [37, 34, 36, 35]. This discrete registration is a genuine nonlinear effect, and we suggest its interpretation in the framework of the coupled nonlinear Maxwell–Schrödinger equations.
We discuss these questions and their connections with the theory of attractors of Hamilton nonlinear partial differential equations. The nonlinearity is inevitable due to the wave-matter interaction. For example, the electronic charge density generates the corresponding “own” scalar potential, which should be added to the Coulomb potential of molecular nucleus. Then the Schrödinger operator changes with the wave function that means a nonlinear self-action.
We start with a presentation of the Schrödinger non-relativistic quantum mechanics and of the Pauli spin theory for mathematically oriented readers. This is necessary for the introduction of nonlinear self-consistent Maxwell–Schrödinger equations and Maxwell–Pauli.
The Schrödinger equation is introduced as a wave equation with short-wave solutions propagating asymptotically along classical trajectories. The definition of quantum observables is also justified by an asymptotic correspondence with classic observables.
The electron spin is introduced to describe the results of the Stern–Gerlach experiment through the projective representation of the rotation group . A simple proof of the covariance of the Pauli equation with respect to the action of the group is given, as well as relevant conservation laws.
Further we introduce the self-consistent nonlinear Maxwell–Schrödinger equations. We formulate a conjecture which concerns attractors of “generic” -invariant nonlinear Hamiltonian partial differential equations. We survey the results obtained in the 1990–2019 which confirm this conjecture for the model equations. Then we apply this conjecture to the Maxwell–Schrödinger equations for dynamic interpretation of the basic postulates I III of quantum mechanics.
Finally we calculate the diffraction amplitude of the electron beam in Kirchhoff’ approximation. The amplitude satisfactory agrees with recent experimental data for the Young two-slit experiment. The Aharonov–Bohm effect is explained using the Agmon–Jensen–Kato stationary scattering theory.
In the Appendix we present the old quantum theory (1890–1924), which is necessary for understanding the emergence of the Schrödinger theory and for the introduction of electron spin.
Acknowledgements. The author thanks Alexander Shnirelman and Herbert Shpon for useful discussions and long-term collaboration.
2 Planck’s law, Einstein’s photons and de Broglie
wave-particle duality
The emergence of the Schrödinger theory and the Pauli spin theory was prepared by all the developments of “Old Quantum Mechanics” in the 1890–1925s, which is described in the Appendix. A decisive impetus for the emergence of Schrödinger’s theory came from the Planck-Einstein theory of the photons and de Broglie conjecture on wave-particle duality.
2.1 Planck’s law and Einstein’s photon
In 1900, Planck found a fundamental formula for spectral intensity of equilibrium radiation at absolute temperature (Nobel Prize for 1918):
[TABLE]
where is the Planck constant. Values of physical constants (the electron charge and mass, the Planck constant and the speed of light in the vacuum) in unrealized Gaussian units (Heaviside–Lorentz units ([18, p. 221])) are approximately equal
[TABLE]
In 1905, Einstein proposed a new derivation of the Planck formula (2.1) by a discretisation with step
[TABLE]
for allowed energies of waves with the frequency , see [8, Section 1.3.4]. At the same time Einstein proposed to identify this portion of energy with the energy of a photon (hypothetical “particle of light”) to explain the photo effect (Nobel Prize for 1921). Namely, the light of frequency , when falling onto a metallic surface, it knocks electrons out of the metal. Experimental observations give the maximal kinetic energy of the “photoelectrons”
[TABLE]
where is the work function [8, Section 8.4]. Einstein treated this empirical formula as the energy balance in the collision of the photons with an electron in the metal:
[TABLE]
The photon energy is partially transmitted to an electron of the metal; this electron leaves the metal losing an energy to overcome the attraction to the metal.
2.2 De Broglie wave-particle duality
The energy and momentum for free non-relativistic and relativistic particles are related by the equations
[TABLE]
respectively, where is the speed of light in the vacuum, and is the rest mass of particles (for photons ).
In 1923 de Broglie suggested the possibility of a wave description of matter as an antithesis of the Einstein corpuscular theory of light (photons) (2.5). The main postulate of de Broglie’ PhD was the correspondence
[TABLE]
where the vector is some function of the vector . The key de Broglie’s idea was that i) this correspondence must be relativistically covariant, and ii) is a Lorentz-invariant scalar product. This easily implies that these vectors are proportional: (see [8, pp 36–37]). Finally, the empirical Planck–Einstein law (2.5) gives which suggests the ratio
[TABLE]
which was experimentally confirmed by Davisson and Germer in the1924-1927s for the diffraction of electrons (see section 17 below).
In particular, the de Broglie wavelength is as follows:
[TABLE]
This great discovery of de Broglie still plays a key role in nuclear physics in calculating the energy and momentum of neutrons and other elementary particles in terms of the wavelength, which is measured in their diffraction.
Now formulas (2.6) give respectively
[TABLE]
3 The Schrödinger quantum mechanics
The next step was made by Schrödinger in the 1925–1926s by introduction of the Schrödinger equation and of stationary orbits.
3.1 Canonical Quantisation
Free particles. For the wave function from (15.1) it follows that
[TABLE]
Hence, the first formula (2.10) for free non-relativistic particles together with (2.8) implies the free Schrödinger equation
[TABLE]
Similarly, the second formula (2.10) for relativistic particles together with (2.8) implies the free Klein–Gordon equation
[TABLE]
Mnemonically, equations (15.2) and (17.2) are derived from expressions for energies (2.6) by replacements
[TABLE]
which are called the Canonical Quantization.
Bound particles. Now let us consider particles in the external Maxwell field
[TABLE]
with magnetic potential and scalar potential .
For nonrelativistic particles, the energy (Hamiltonian) is expressed by the formula [8, (12.90)]
[TABLE]
where is the charge of an electron. In this case, the canonical quantization (3.4) leads to the Schrödinger equation
[TABLE]
For relativistic particles the energy is expressed by the formula [8, (12.93)],
[TABLE]
which leads to the Klein–Gordon equation
[TABLE]
3.2 Wentzel–Kramers–Brillouin quasiclassical asymptotics
The derivation of equations (3.7) and (3.9) relies only on de Broglie’s relativistic arguments and formulas (2.6) of classical electrodynamics. However, a deeper basis for these equations lies in short-wave asymptotics. For example, the Hamiltonian equations corresponding to the nonrelativistic Hamiltonian (3.6) are equivalent to the Lorentz equation (A.1) (see [8, Section 12.6]), where the Maxwell fields are expressed in terms of potentials by formulas (3.5). The key argument for the Schrödinger equation (3.7) was the short-wave Wentzel–Kramers–Brillouin asymptotics (1926) for solutions to the Cauchy problem with fast-oscillating initial data :
[TABLE]
The main fact is that if outside the ball with small , then
[TABLE]
see [8, Sections 3.2 and 13.2]. In other words, short-wave solutions to the Schrödinger equation (3.7) propagate along the trajectories of the non-relativistic classical Lorentz equation (A.1). This is due to the fact that
a) The phase function satisfies the Hamilton–Jacobi equation
[TABLE]
where is the Hamiltonian (3.6), and
b) Solutions of the Hamilton-Jacobi equation are obtained by integrating along the trajectories of the corresponding Hamiltonian system [1, Section 46], which coincides in this case with the Lorentz equation (A.1), see [8, Section 12.6].
The Hamilton–Jacobi equation (3.12) is obtained by substituting the asymptotics (3.10) into the Schrödinger equation (3.7) and setting .
Remarks 3.1**.**
i) The structure of the Hamilton-Jacobi equation (3.12) dictates uniquely the rules of canonical quantization (3.4).
ii) Asymptotics (3.10) elucidate a new light into Thompson’s experiments with electrons in the 1893–1897s. It appears that he observed exactly short-wave asymptotics of cathode rays, which makes the concept of elementary particles in quantum theory problematic.
iii) These quasiclassical asymptotics are examples of the Bohrcorrespondence principle [8, Section 3.3.3]. We will call solutions of the type (3.10), (3.11) quasiclassical. Similar asymptotics hold for solutions of reativistic equations: for the Klein–Gordon equations and the Dirac equations (Pauli 1932), and for general hyperbolic systems (Lax 1957, Maslov 1965, Hörmander, and others). **
4 Quantum Observables
Quantum observables are invariants of the Schrödinger dynamics. The invariance is caused by the symmetry of external potentials: the electron energy is conserved if the external potentials are independent of time, the charge is conserved at any real potentials, the projection of the impulse on a certain direction is preserved if the potentials are invariant with respect to shifts in this direction, and the projection of angular momentum onto a certain direction is preserved if the potentials are invariant (in a certain sense) on rotations around this direction.
The Correspondence Principle is that the quantum and corresponding classical observables asymptotically coincide as , see (4.10).
4.1 Hamiltonian structure and energy
The linear Schrödinger equation (3.7) can be written in the Hamiltonian form as
[TABLE]
where is a variational derivative, and the Hamilton functional (= quantum energy ) reads as
[TABLE]
Therefore, the energy is conserved if the external potentials are independent of time.
4.2 Charges and currents
A comparison (4.2) with energy in electrostatics shows that the electric charge density should be defined as
[TABLE]
Now equation (4.1) implies the conservation of the total charge
[TABLE]
This follows from the symmetry of the Schrödinger operator :
[TABLE]
where the brackets mean anti-Hermitian scalar product.
For one electron , whence according to (4.3) the normalization condition holds
[TABLE]
Further, the current density should be defined as [7, (21.12)]):
[TABLE]
since this density together with (4.3) satisfies the continuity equation
[TABLE]
This can be either verified by direct differentiation [8, Section 3.4] or derived from Noether’s general theorem on invariants [8, Section 13.4.3] using the -invariance of Hamiltonian (4.2) with respect to the action , where
4.3 Quantum momentum and angular momentum
The quantum momentum and the angular momentum in Schrödinger’s theory are defined for any state as mean values
[TABLE]
where , are self-adjoint operators of the momentum and the angular momentum, is the multiplication operator by .
4.4 Correspondence Principle
The name of the momentum, angular momentum, as well as energy (4.2), are justified by the Correspondence Principle: for the quasiclassical solutions (3.10) with small and ,
[TABLE]
see [79, Section 3.3.3].
4.5 Conservation Laws
The formal mathematical motivation for these definitions of observables for the Hamiltonian system (4.1) consists in conservation laws [79, Section 3.3.2]: for solutions to the Schrödinger equation (3.7)
i) the energy if the Maxwell potentials are independent of time,
ii) the “mean momentum” if the potentials are independent of , and
iii) the “mean angular momentum”
[TABLE]
if the potentials are invariant with respect to rotations around the axis; i.e.,
[TABLE]
where is the rotation of the space through the angle around the axis , and the direction of rotation is determined by the gimlet rule.
Example. For a uniform magnetic field the vector potential has the form
[TABLE]
and satisfies the condition (4.12) with .
4.6 Proof of conservation laws
The energy conservation was proved above. The conservation of momentum and angular momentum follows from commutation relations. Namely, if the potentials are independent of , then the commutator and differentiation yields
[TABLE]
where we used the Schrödinger equation (4.1). Similarly, (4.12) implies the commutation
[TABLE]
because in cylindrical coordinates with the axis, while is a differential operator with coefficients independent of (Exercice). Therefore,
[TABLE]
Definition. A quantum observable is a quadratic form
[TABLE]
where is some self-adjoint operator in .
As above, the observable is conserved for solutions of the Schrödinger equation (4.1) if
[TABLE]
5 Bohr’s postulates, stationary orbits and attractors
In 1913, Bohr formulated two fundamental postulates of quantum theory of atoms:
I. An atom is always in one of quantum stationary orbits, and sometimes it jumps from one stationary orbit to another: in the Dirac notation
[TABLE]
II. The atom does not radiate in stationary orbits. Every jump is followed by a radiation of an electromagnetic wave with the frequency
[TABLE]
With the discovery of the Schrödinger theory in 1926, the question arose about the validility of these Bohr’s axioms in the new theory.
5.1 Schrödinger theory of stationary orbits
Besides the equation for the wave function, the Schrödinger theory contains quite a nontrivial definition of stationary orbits in the case when the Maxwell external potentials do not depend on time:
[TABLE]
In this case .
Definition 5.1**.**
Stationary orbits* are solutions of the form*
[TABLE]
to the Schrödinder equation (3.7).
Substitution into the Schrödinger equation (5.7) leads to the famous eigenvalue problem
[TABLE]
Definition of stationary orbits (5.4) is rather natural, since then does not depend on time. This definition probably was suggested by the de Broglie wave function forfree particles , which factorizes as . Namely, in the case of bound particles it is natural to change the spatial factor , since the spatial properties have changed and ceased to be homogeneous. On the other hand, the homogeneous time factor must be preserved, since the external potentials are independent of time. However, these “algebraic arguments” do not withdraw the question on agreement of the Schrödinger definition with the Bohr postulate (5.1)!
Thus, a problem arises on the mathematical interpretation of the Bohr postulate (5.1) in the Schrödinger theory. One of the simplest interpretation of the jump (5.1) is the long-time asymptotics
[TABLE]
for each finite energy solution, where and . However, for linear Schrödinger equation (5.7) such asymptotics are obviously wrong due to the superposition principle: for example, for solutions of the form with . It is exactly this contradiction which shows that the linear Schrödinger equation alone cannot serve as a basis for the theory compatible with the Bohr postulates.
Our main conjecture is that these postulates are inherent properties of the nonlinear Maxwell–Schrödinger equations, see (10.3) below. This conjecture is suggested by the following perturbative arguments.
5.2 Bohr’ postulates and asymptotics (5.6) by perturbation theory
The remarkable success of the Schrödinger theory was the explanation of the Bohr’ postulates and asymptotics (5.6) by perturbation theory in the case of static external potentials (5.3). Namely, in this case the Schrödinger equation (3.7) reads
[TABLE]
For “sufficiently good” external potentials and initial conditions, any finite energy solution can be expanded in eigenfunctions
[TABLE]
where the integration is performed over the continuous spectrum of the Schrödinger operator , and
[TABLE]
see, for example, [9, Theorem 21.1]. The substitution of this expansion into the expression for currents (4.7) gives
[TABLE]
where contains the continuous frequency spectrum. These currents enter into the Maxwell equations in the entire space of : in the Heaviside–Lorentz units ([19, p. 781])
[TABLE]
The second and third equations imply the Maxwell representations and Then in the Coulomb gauge , the Maxwell equations (5.11) are equivalent to the system
[TABLE]
where is the orthogonal projection in the Hilbert space onto free-divergent vector fields.
Thus, the currents (5.10) on the right of the Maxwell equations (5.12) contain, besides the continuous spectrum, only discrete frequencies . Hence, the discrete spectrum of the corresponding Maxwell radiation field also contains only these frequencies . This proves the Bohr rule (5.2) in the first order of perturbation theory, since this calculation ignores the back reaction of radiation onto the atom.
Moreover, these arguments also justify the jumps (5.1) as the long-time asymptotics (5.6). Namely, the currents (5.10) on the right of the Maxwell equation from (10.3) produce the radiation when nonzero frequencies are present. However, this radiation cannot last forever since the total energy is finite. Hence, in the long-time limit only should remain, which means exactly one-frequency asymptotics (5.6) by (5.9).
6 Stationary orbits of electron in atom
In the case of one electron in an atom with nucleous charge the electrostatic Coulomb potential of the nucleus reads as , while the magnetic potential can be neglected. Now the Schrödinger equation (4.1) becomes
[TABLE]
Stationary orbits of such an electron are defined as solutions of the form (5.4). Substitution into equation (6.1) results in the eigenvalue problem
[TABLE]
The “mean energy” of a stationary orbit is equal to the eigenvalue due to normalization (4.6),
[TABLE]
These eigenvalues were first calculated by Schrödinger [12, I] using separation of variables in spherical coordinates. Later, these calculations were simplified using irreducible representations for the Lie algebra of the rotation group to find the spectrum of the spherical Laplacian [8, Chapter 6]. The result is the Balmer formula.
[TABLE]
with the Rydberg constant
[TABLE]
which completely coincides with the formula of “Old Quantum Mechanics” (A.12). This confirmed both the Schrödinger theory, and the “Old Quantum Mechanics”. The eigenfunctions and eigenvalues are numbered with the same quantum numbers as in formulas (A.24), (A.42). However now the value for the angular momentum is excluded, and the formulas (A.24), (A.25) are slightly modified (see, e.g., [8, Ch. 6]):
[TABLE]
Here . The indices are called, respectively, the main, azimuth and magnetic quantum numbers. Let us note that the operators , , have common eigenfunctions due to their commutation
[TABLE]
which holds by the spherical symmetry of the Coulomb potential of the nucleus. These commutations imply that the observables and are conserved along the solutions to the Schrödinger equation (6.1).
7 Atom in magnetic field
In 1895, Zeeman observed a splitting of the spectral lines (5.2) in a magnetic field.
7.1 Normal Zeeman effect
The “normal Zeeman effect” is the splitting of one line into three separate lines
[TABLE]
where
[TABLE]
is the Larmor frequency. This normal effect was successfully explained in the classical Lorentz model (Section A.1) as well as in the “Old Quantum Theory” (Section A.6).
The Schrödinger theory leads to the same result. Namely, for an atom in a magnetic field with a potential , the Schrödinger equation (4.1) reads
[TABLE]
In particular, for a uniform external magnetic field , the vector potential has the form (4.13), and the Schrödinger equation (7.3) becomes
[TABLE]
when the quadratic terms with are neglected.
Now (6.9) implies that
[TABLE]
which coincides with (A.48) except for the range of . Thus, the spectral lines (5.2) are shifted by and are given by
[TABLE]
These spectral lines coincide with the result (A.49) in Old Quantum Mechanics .
7.2 The selection rules
Let us show that the selection rules (A.50) also hold in the Schrödinger theory in the sense that the intensity of other lines vanish. This follows from formulas for the intensity of spectral lines [11, (45.13), (45.21)], [8, (7.48), (7.52)]. Namely, the intensity of the line with frequency is proportional to the sum
[TABLE]
where is the dipole moment
[TABLE]
In the spherical coordinates with the axis , all eigenfunctions of the operator have the form with integer and with the corresponding eigenvalues . Hence, with a suitable choice of the multiplier, the eigenfunctions read
[TABLE]
and the corresponding eigenvalue equals . Hence, in the spherical coordinates the internal integral over in (7.8) equals zero for , since coordinates of the vector contain harmonics only with .
Thus, the normal Zeeman effect in the Schrödinger theory can be completely explained. However, the anomalous Zeeman effect is not explained by this theory. This problem was solved only after the introduction in the Schrödinger equation of the electron spin and of an additional term (“Russell-Saunders coupling”), describing the spin interaction with the angular momentum , see [8, Section 10.3].
Remark 7.1**.**
The Schrödinger operator (7.4) is generalized in a natural way to the case of a slowly varying magnetic field ,
[TABLE]
where is the trajectory of an atom in space. **
8 Own magnetic moment of electron
The introduction of electron spin was suggested by experiments of Einstein–de Haas, Stern–Gerlach, and Lande’s vector model, and Bohr theory of the Mendeleev periodic table of chemical elements.
8.1 Einstein-de Haas experiments
In 1915, the presence of the intrinsic magnetic moment of an electron was demonstrated by experiments of Einstein–de Haas, who measured the ratio of the magnetic moment of the electrons of an atom to their angular ( mechanical ) moment. The result, very surprisingly, did not fit the classical theory. The measurements were based on the observation of torsional vibrations of a ferromagnetic rod suspended on a thin filament inside the solenoid.
The classical angular and magnetic moments of the electrons of an atom are determined similarly to (A.43),
[TABLE]
where are the positions of electrons of the atom, and are their velocities. therefore
[TABLE]
similarly (A.44). In stationary states of the atom one has , where the bar means the average over time. Therefore, formulas like (8.1) are suitable for definition of time-averaged values and regardless of the choice of the origin and position of atom. Hence, the same is true for *time-averaged * values of the angular and magnetic moments and of all electrons of a macroscopic body, and so the summation is now carried over all electrons of the rod. Thus,
[TABLE]
When a current is turned on in the solenoid, the magnetic field orients the magnetic moments of all atoms of the rod, so that the total magnetic moment of atoms instantly becomes maximal, and accordingly, the total angular momentum of electrons also instantaneously increases, since the magnetic moment of the nuclei in these experiments is negligible. But then the conservation of the total angular momentum of the rod (electrons together with the crystal lattice) leads to the opposite rotation of the lattice, to which the suspension thread is attached. Changing the frequency of the alternating current in the solenoid one can determine the resonant frequency by the maximal amplitude of the rotational oscillations of the rod, which allows us to find the ratio , which enters the equation of oscillations [10]. The result was in contradiction with (8.3):
[TABLE]
In more accurate experiments (Beck, Arvidson, Klaasen) it turned out that the gyroscopic ratio . This had led to the assumption that the electron itself has angular and magnetic moments with this ratio.
8.2 Vector model of Lande
In 1921 Lande suggested a phenomenological classical model for the description of the anomalous Zeeman effect in the framework of “Old Quantum Mechanics”, treating the electron’s own magnetic moment as the electron rotation around an axis [3, Chapter VI, Section 2], [8, Section 14.7]. Lande’s formula produced a result that coincides remarkably well with the experimental observations of the anomalous Zeeman effect.
8.3 Stern–Gerlach experiment and Goldsmith–Uhlenbeck hypothesis
In 1922, Stern and Gerlach conducted a crucial and clarifying experiment on the splitting of a beam of silver atoms in two parts in a strongly inhomogeneous magnetic field. Formula (A.47) implies that an inhomogeneous magnetic field acts on an atom with magnetic moment with the force
[TABLE]
and the Lorentz force is due to the neutrality of the atom. Hence, the splitting of atomic beam means that electrons in atoms of these two beams were in states with different magnetic moments, since the magnetic moment of the nucleus is negligible (). Later, similar experiments were conducted with hydrogen atoms.
In 1925, Goldsmith and Uhlenbeck suggested an explanation for this fact by an “own magnetic moment” of electron which can be only in two states: the magnetic moment is parallel to the magnetic field in one state, and antiparallel in the other.
Particularly puzzling was the splitting exactly in two beams because, according to the Old Quantum Mechanics , the splitting is possible only into an odd number of components. Namely, according to (A.42) the angular momentum of an electron in an atom takes discrete values with integer , and the magnetic moment at a fixed takes different values of , where with step , as in the later-developed Schrödinger theory.
Respectively, Goldsmith and Uhlenbeck interpreted the splitting in two levels in the Stern-Gerlach experience as the presence of electron states with and with the corresponding angular momentum values . At the same time, they postulated the electron ratio (8.4) with the experimentally measured value :
- The electron has its own “spin moment” of magnitude
and intrinsic magnetic moment of magnitude .
9 Spin and Pauli equation
After the emergence of the Schrödinger’s theory in 1926, a new contradiction arose with the Stern–Gerlach experiment since silver atoms were prepared in a spherically symmetric state, which was known from spectroscopy. But the last term in the Schrödinger operator (7.10) cancels such states, since is a generator of rotations around the vector . Therefore, there should be no first-order splitting!
In the language of quantum numbers, it looks like this: in the -state, the angular momentum under the action of a magnetic field is conserved, and the multiplicity of such a state is !
In 1927, Pauli proposed to modify the Schrödinger equation to double the multiplicity of all stationary states in accordance with the two-digit quantum number introduced by Pauli in his theory of periodic system (Section A.7). For this purpose, he introduced the two-component wave functions and added to the energy operator (7.4) a new spin term with coefficients corresponding to the Goldsmith–Uhlenbeck conjecture.
Namely, formula (A.46) means that the magnetic moment appends the value to electron energy in the magnetic field. Therefore, the last term in (7.4) means that an electron has the magnetic moment , in the state due to its angular momentum . Therefore,
[TABLE]
which coincides with (8.2). Hence, the Goldsmith–Uhlenbeck conjecture dictates for the Pauli two-component wave functions the replacement of by , where , and is one of the Pauli matrices
[TABLE]
As a result, (7.4) becomes the Pauli equation
[TABLE]
The eigenfunctions of the Pauli operator are obviously expressed in terms of the eigenfunctions ,
[TABLE]
Namely, the spin term shifts the eigenvalues by in accordance with the Goldsmith–Uhlenbeck conjecture, and for the multiplicity of all stationary states of the operator doubles as compared to the operator (7.4).
Thus (7.5) implies that the eigenvalues of the Pauli operator are numbered with the quantum numbers (A.51), and coincides with the eigenvalue of the operator :
[TABLE]
Obviously, , so the observables , and are conserved along the solutions of the Pauli equation (9.3).
Remark 9.1**.**
The sets of possible energies in (9.5) and (A.48) coincide with a fixed principal quantum number because . **
The Pauli equation (9.3) allowed us to explain the experiments of Einstein–de Haas and Stern–Gerlach, as well as many details of the anomalous Zeeman effect, see [79, Section 10.3].
9.1 Angular momentum and representations of rotation group
Commutation relations (4.15) are equivalent to the commutation of the dynamical group of the Schrödinger equation with rotations of space around the -axis. These rotations form a representation of the group . In particular, if the potentials are “spherically symmetric,” (4.12) holds with respect to any axis of space. Then commutes with representation
[TABLE]
of the whole group , where , and are the generators of the rotations around the axis:
[TABLE]
In this case, the conserved angular momentum (4.9) equals to the quadratic form of the operator
[TABLE]
Here is the generator of the representation , and so
[TABLE]
It is easy to see that, vice versa, for each representation of the rotation group commuting with the dynamical group, the quadratic form of its generator is conserved in time.
Definition 9.2**.**
For any representation of the rotation group in the phase space of the Schrödinger equation the quadratic form is called by definition the angular momentum of the state , which corresponds to this representation. **
9.2 Covariance of the Pauli equation with respect to rotations
In covariant with respect to rotations form, the Pauli equation reads
[TABLE]
where . Equivalently,
[TABLE]
where is the spin operator, and . In particular, for the case of a uniform magnetic field , the last equation coincides with (9.3) when the quadratic terms with are neglected.
Equation (9.33) can be written in the Hamiltonian form as
[TABLE]
The covariance means that the Pauli equation retains its shape after rotations with , followed by suitable transform for the potentials, the wave function and for the basis in the space of values of the wave function. Namely, let us denote for
[TABLE]
, where , and denote generators (9.30).
Remark 9.3**.**
The correspondance is a two-valued projective spinor representation of the rotation group . **
For any solution of the equation (9.33), we define
[TABLE]
Let us show that is a solution of the same equation in which , , are replaced by , respectively.
For a proof, we express the right-hand side (9.34) in terms of and apply to both sides. Since , we have
[TABLE]
since the matrix is orthogonal. Equivalently, . In other words,
[TABLE]
Hence,
[TABLE]
since the rotation of preserves the “length of the vector” .
It remains to note that the matrices obviously satisfy the same multiplication rules as , and therefore, they have the same form (9.2) in some new basis of the space . This follows similarly to the Pauli theorem for the Dirac equation [8, Theorem 11.4].
9.3 Conservation laws for the Pauli equation
Energy and momentum for the Pauli equation (9.34) are defined by the same formulas (4.2) and (4.9) as for the Schrödinger equation, and the proof of their conservation remains unchanged.
However, the definition and proof of conservation for the angular momentum require significant modification. Namely, commutation (4.15) and the conservation law (4.16) are now generally incorrect.
Definition 9.4**.**
Angular momentum for the Pauli equation is defined as
[TABLE]
Lemma 9.5**.**
Let the external potentials satisfy the invariance conditions (4.12). Then for any solution to the Pauli equation (9.34) the angular mometum (9.41) is concerved,
[TABLE]
Proof.
This follows, as above, from commutation
[TABLE]
For proof, it suffices to verify that
[TABLE]
Let us consider for example. Then
[TABLE]
because (+ cyclic permutations). Here , since the vector field is invariant under rotations around the axis, as well as . Therefore, substituting , we get
[TABLE]
It remains to verify that coefficients at and on the right-hand side vanish. First we note that the rotational invariance of the vector field , means that radial and angular components of this vector field do not depend on the angle :
[TABLE]
The differentiation in gives
[TABLE]
This easily implies the annihilation of the coefficients in (9.46). ∎
Corollary 9.6**.**
i) For the Pauli equation with uniform magnetic field the operator (9.3) commutes with the operator of the “orbital angular momentum” and with the operator of the “spin momentum” . Therefore, the quadratic forms of these operators are conserved in time.
ii) This commutation and conservation generally does not hold for magnetic fields invariant with respect to rotations around the axis. On the the hand, the quadratic form of the sum is conserved by (9.43). For spherically symmetric potentials, the quadratic forms of the operators and are also conserved, and their eigenvalues are equal to
[TABLE]
which is consistent with (9.5) in the case of a uniform magnetic field. The expressions for and stem from the general theorem on decomposition of tensor products of representations, applied to , see [17]. **
Remarks 9.7**.**
i) Let external potentials be spherically symmetric. Then commutation (9.43) implies the commutation of dynamical group with
[TABLE]
where is defined in (9.36). In this case
[TABLE]
according to (9.32), and the conserved angular momentum (9.42) corresponds (up to a factor) to this representation in the sense of Definition 9.2.
ii) For small , is the single-valued representation of the rotations , but its analytic continuation for large is a two-valued representation, as well as the spinor representation .
iii) The conservation of the momentum and angular momentum in quantum dynamics do not play an important role in quantum theory. On the other hand, the corresponding commutation relations (4.15) and (9.43) play a crucial role in the calculation of eigenvalues of related quantum observables, – for example, in the calculation of the anomalous Zeeman effect in [8, section 10.3].
10 Coupled nonlinear Maxwell–Schrödinger equations
The Maxwell equations in the entire space of in the Heaviside–Lorentz units read ([19, p. 781])
[TABLE]
The second and third equations imply the Maxwell representations and Then in the Coulomb gauge the Maxwell equations (10.1) are equivalent to the system
[TABLE]
where and are the charge and current densities, respectively, and denotes the orthogonal projection onto free-divergent vector fields from the Hilbert space .
The Schrödinger equation (3.7) describes the evolution of a wave function in a given external Maxwell field with potentials and . On the other hand, the charge and current densities (4.3), (4.7) generate their “own Maxwell field” with potentials satisfying the Maxwell equations (5.11). Hence, for a self-consistent description, these potentials and should be added to the external potentials in the Schrödinger equation (3.7). Thus, the coupled Maxwell–Schrödinger equations read as (cf. [22])
[TABLE]
where and are some external Maxwell potentials, is the electron charge and is the speed of light in a vacuum. The coupling is defined by expressing the charge and current densities in the wave function:
[TABLE]
These densities satisfy the continuity identity
[TABLE]
The system (10.3) is formally Hamiltonian, with the Hamilton functional (which is the energy up to a factor)
[TABLE]
where stands for the norm in the real Hilbert space and the brackets stand for the inner product in . The Schrödinger magnetic operator
[TABLE]
where and . The system (10.3) can be written in the Hamilton form as
[TABLE]
taking into account that , and hence, . Therefore, the energy is conserved in the case of static external potentials
[TABLE]
For instance, in the case of an atom, is the nucleus Coulomb potential, while is the vector potential of the nucleus magnetic field. On the other hand, the total charge formally is conserved for arbitrary time-dependent external potentials.
The Hamiltonian (10.6) is invariant with respect to the action of the group ,
[TABLE]
This invariance implies the charge continuity equation (10.5) by the general Noether theorem on invariants [8, Section 13.4.3]. One can also check (10.5) by direct differentiation [8, Section 3.4]. Moreover, for any solution the functions , are also a solution.
Remark 10.1**.**
The existence of global solutions to the Cauchy problems for systems (10.3) in the entire space without external potentials was proved in [20] for all finite energy initial states (10.6). The uniqueness of the solutions has so far been proved only in narrower classes of functions, [22, 23]. **
Taking into account the electron spin, the Maxwell-Schrödinger system (10.3) should be replaced by the Maxwell–Pauli system with current density
[TABLE]
where are the Pauli matrices. In this case, the Schrödinger equation is replaced by the Pauli equation
[TABLE]
and the Schrödinger operator in the Hamiltonian (10.6) must be replaced by the Pauli magnetic operator
[TABLE]
Remark 10.2**.**
The system (10.3) was introduced essentially by Schrödinger in his first articles [12], and it underlies the entire theory of laser radiation [27].
11 Quantum postulates and Maxwell–Schrödinger equations
The Maxwell–Schrödinger equations (10.3) suggest a novel interpretation for basic quantum postulates:
I. Transitions between quantum stationary orbits.
II. Wave-particle duality.
III. Probabilistic interpretation.
A rigorous dynamical description of these postulates is still unknown. This lack of theoretical clarity hinders the progress in the theory (e.g., in superconductivity and in nuclear reactions), and in numerical simulation of many engineering processes (e.g., of laser radiation and quantum amplifiers) since a computer can solve dynamical equations but cannot take into account postulates.
Transitions between stationary orbits of atoms and accompanying radiation (Postulate I) were postulated by Bohr in 1913. The wave-particle duality (Postulate II) was conjectured by de Broglie in 1923, and the probabilistic interpretation (Postulate III) was introduced by Born in 1927. On the other hand, after the discovery of Schrödinger’s quantum mechanics, the question arose on the validity of these postulates I–III in new theory – this still remains an open problem. These and other questions have been frequently addressed in the 1920s and 1930s in discussions of Bohr, Schrödinger, Einstein and others [2]. However, a satisfactory solutions were not achieved. We propose a novel approach to these problems relying on the recent progress in the theory of attractors for Hamilton nonlinear PDEs.
The main goals of remaining part of these lectures are as follows:
i) To suggest a relation of these postulates to the theory of attractors.
ii) To survey the related results on global attractors for nonlinear Hamiltonian PDEs.
iii) To formulate novel general conjectures on the attractors.
iv) To apply these conjectures to a mathematical interpretation of Postulates I–III in the context of coupled nonlinear Maxwell–Schrödinger equations.
12 Bohr’s postulates and attractors
As a result, we expect the long-time asymptotics
[TABLE]
which holds in the -norms and -norms on every bounded region of for all finite-energy solutions to the Maxwell–Schrödinger equations (10.3).
Remark 12.1**.**
Experiments show that the time of transitions (5.1) is of order , though the asymptotics (12.1) requires an infinite time. We suppose that this discrepancy can be explained by the fact that is the time when the atom emits an overwhelming part of the radiated energy. **
Such asymptotics are still open problems for the Maxwell–Schrödinger system (10.3). On the other hand, similar asymptotics are now proved for a number of model Hamilton nonlinear PDEs with symmetry groups . In next section we state a general conjecture which reduces to the asymptotics (12.1) in the case of the Maxwell–Schrödinger system.
Definition 12.2**.**
Stationary orbits of the Maxwell-Schrödinger nonlinear system (10.3) are finite energy solutions of the form . **
Existence of stationary orbits for the system (10.3) in the whole space was proved in [21] under conditions
[TABLE]
The asymptotics (12.1) mean global attraction to the set of stationary orbits. We suggest similar attraction for the Maxwell–Dirac, Maxwell–Yang–Mills and other coupled equations. In other words, we suggest to interpret quantum stationary states as points and trajectories lying on the global attractor of the corresponding quantum dynamical equations.
12.1 The Einstein–Ehrenfest paradox
An instant orientation of the atomic magnetic moment during when turning on the magnetic field in the Stern–Gerlach experiments caused the discussion in the “Old Quantum Mechanics,” because the classical model gave relaxation time taking into account the moment of inertia of the atom [5]. In the linear Schrödinger’s theory, this phenomenon also did not find a satisfactory explanation.
However, this instantaneous orientation is exactly in line with asymptotics (12.1) for solutions to the coupled Maxwell–Schrödinger system. Namely, in the absence of a magnetic field, the ground states (with a fixed charge) form a two-dimensional manifold. When the magnetic field is turned on, the structure of the attractor (i.e., the set of corresponding stationary orbits ) instantly changes: the two-dimensional manifold bifurcates in two one-dimensional manifolds with a certain spin value. This bifurcation is not related to any moment of inertia and corresponds to the “alternative A” in the terminology of Einstein–Ehrenfest [5]: “… atoms can never fall into the state in which they are quantized not fully”.
12.2 Attractors of dissipative and Hamiltonian PDEs
Such interpretation of the Bohr transitions as a global attraction is rather natural. On the other hand, the existing theory of attractors of dissipative systems [56]–[59] does not help in this case, since all fundamental equations of quantum theory are Hamiltonian. The global attraction for dissipative systems is caused by energy dissipation. However, such a dissipation in the Hamilton systems is absent.
This is why we have developed in the 1990–2019s together with our collaborators a novel theory of global attractors for Hamilton PDEs, especially for application to the problems of Quantum Theory. Our results [60]–[78] for the Hamilton equations rely on energy radiation, which irrevocably carries the energy to infinity and plays the role of energy dissipation. A brief survey of these results can be found in Section 14, and a detailed survey in [79].
The results obtained so far indicate an explicit correspondence between the type of long-time asymptotics of finite energy solutions and the symmetry group of the equation. We formalize this correspondence in our general conjecture (13.2).
13 Conjecture on attractors of -invariant Hamilton nonlinear PDEs
Let us consider -invariant Hamilton nonlinear PDEs of type
[TABLE]
with a Lie symmetry group . Here belongs to the Hilbert space , and the Hamilton structure means that , where . The -invariance means that for all states and all transformations (more precisely, belongs to a representation of the Lie group ). In this case, for any solution to equations (13.1) the trajectory is also a solution.
Let us note that the theory of elementary particles deals systematically with the symmetry groups , , , and other, and is the symmetry group of “Grand Unification”, see [53].
Conjecture A. For “generic” -invariant equations (13.1), any finite energy solution admits a long-time asymptotics
[TABLE]
where the generators belong to the corresponding Lie algebra (more precisely, belong to a representation of ), and the above asymptotics holds in some local seminorms.
In other words, all -orbits form a global attractor for “generic” -invariant Hamilton nonlinear PDEs of type (13.1). This conjecture is a generalization of rigorous results [60]–[79] obtained since 1990 for a list of model equations of type (13.1) with three basic symmetry groups: the trivial group, the group of translations, and the unitary group . We give a brief survey of these results in Section 14.
For the case of Maxwell–Schrödinger system (10.3) with the symmetry group , the conjecture (13.2) reduces to the asymptotics (12.1).
Empirical evidence. Conjecture (13.2) agrees with the Gell-Mann–Ne’eman theory of baryons [51, 52]. Namely, in 1961 Gell-Mann and Ne’eman suggested the symmetry group and other ones for the strong interaction of baryons relying on the discovered parallelism between empirical data for the baryons, and the “Dynkin scheme” of the Lie algebra with generators (the famous “eightfold way”). This theory resulted in the scheme of quarks and in the development of the quantum chromodynamics [53], and in the prediction of a new baryon with prescribed values of its mass and decay products. This particle (the -hyperon) was promptly discovered experimentally [55]. The elementary particles seem to describe long-time asymptotics of quantum fields. Hence, the empirical correspondence between elementary particles and generators of the Lie algebras presumably gives an evidence in favour of our general conjecture (13.2) for equations with Lie symmetry groups.
Conjecture (13.2) suggests to define stationary “-orbits” for equations (13.1) as solutions of the type
[TABLE]
where belongs to the corresponding Lie algebra (more precisely, belong to a representation of the Lie algebra ). This definition leads to the corresponding “-eigenvalue problem”
[TABLE]
In particular, for the linear Schrödinger equation with the symmetry group , stationary orbits are solutions of the form , where is an eigenvalue of the Schrödinger operator, and is the corresponding eigenfunction. However, Conjecture (13.2) fails for linear equations, i.e., linear equations are exceptional, not “generic”! In the case of the symmetry group , the generator (“eigenvalue”) is -matrix, and solutions (13.3) are quasiperiodic in time.
14 Results on global attractors for nonlinear Hamilton PDEs
Here we describe rigorous results [60]–[78] obtained since 1990 on the corresponding asymptotics for a number of Hamiltonian nonlinear partial differential equations of type (13.1). We give only a brief listing of the results, see the details in [79]. The results obtained confirm the existence of finite-dimensional attractors in the Hilbert phase space, and demonstrate an explicit correspondence between the long-time asymptotics and the symmetry group of equations.
The results were obtained so far for model equations with three basic groups of symmetry: the trivial symmetry group , the translation group for translation-invariant equations, and the unitary group for phase-invariant equations.
14.1 Global attraction to stationary states
For “generic” equations with trivial symmetry group the long-time asymptotics of all finite energy solutions is the convergence to stationary states (see Fig. 1)
[TABLE]
which was proved for a number of model equations in [60]–[64]: i) for a string coupled to nonlinear oscillators, ii) for a three-dimensional wave equation coupled to a charged particle and for the Maxwell–Lorentz equations, and also iii) for the wave equation, and the Dirac and Klein–Gordon equations with concentrated nonlinearities.
Here are some stationary states depending on the considered trajectory , and the convergence holds in local seminorms of type for any . The convergence (14.1) in global norms (i.e., corresponding to ) cannot hold due to energy conservation.
Example 14.1**.**
Nonlinear Huygens Principle.* *Consider solutions to 3D wave equation with a unit propagation velocity and initial data with support in a ball . The corresponding solution is concentrated in the spherical layers . Therefore, the solution converges everywhere to zero as , although its energy remains constant. This convergence to zero is known as the strong Huygens principle. Thus, the attraction to stationary states (14.1) is a generalization of this Huygens principle to nonlinear equations. The difference is that for a linear wave equation the limit behind the wave front is always zero, while for nonlinear equations the limit can be any stationary solution. **
The proofs in [62] and [63] rely on the relaxation of the acceleration
[TABLE]
Such relaxation has been known for a long time in classical electrodynamics as the radiation damping , but it was first proved in [62] and [63] for charged relativistic particle in a scalar field and in the Maxwell field under the Wiener Condition on the particle charge density. This condition is an analogue of the “Fermi Golden Rule”, first introduced by Sigal in the context of nonlinear wave- and Schrödinger equations [65]. The proof of the relaxation (14.2) relies on a novel application of the Wiener Tauberian theorem.
14.2 Global attraction to solitons
For “generic” translation-invariant equations, the long-time asymptotics of all finite energy solutions is the convergence to solitons
[TABLE]
where the convergence holds in local seminorms in the comoving frame of reference, that is, in for any . Such soliton asymptotics were proved in [67]–[70] for three-dimensional wave equation coupled to a charged particle and for the Maxwell–Lorentz equations. These results gave the first rigorous proof of the “radiation damping” in classical electrodynamics, which has been an open problem for about 100 years. The proofs in [67] and [68] rely on variational properties of solitons and their orbital stability, as well as on the relaxation of the acceleration (14.2) under the Wiener condition on the particle charge density.
More accurate soliton asymptotics in global norms with several solitons were first discovered in 1965 by Zabuzhsky and Kruskal in numerical simulation of the Korteweg–de Vries equation (KdV): it is the decay to solitons
[TABLE]
where are some dispersion waves. In [71] the results of numerical simulation were presented to confirm the soliton asymptotics (14.4) with many solitons for 1D relativist-invariant nonlinear wave equations. Later on, such asymptotics were proved by the method of inverse scattering problem for nonlinear integrable Hamiltonian translation-invariant equations (KdV, etc.) in the works of Ablowitz, Segur, Eckhaus, van Harten and others [66]. A trivial example is provided by the d’Alembert equation , for which any solution reads .
14.3 Global attraction to stationary orbits
For “generic” equations with unitary symmetry group , the long-time asymptotics are global attraction to “stationary orbits” (see Fig. 2)
[TABLE]
they were proved in [72]–[77] for the Klein–Gordon and Dirac equations coupled to -invariant nonlinear oscillators, and in [78], for discrete in space and time difference approximations of such coupled systems, i.e., for the corresponding difference schemes.
The global attraction was proved under the assumption that the equations are “strictly nonlinear”. For linear equations, the attraction can fail if the discrete spectrum consists at least of two points.
Remark 14.2**.**
Let us comment on the term generic in the results of the previous section and in Conjecture (13.2). Namely, this conjecture means that the asymptotics (13.2) hold for all solutions for an open dense set of -invariant equations.
i) For example, asymptotics (14.1), (14.3), (14.5) hold under appropriate conditions, which define some “open dense set” of -invariant equations with three types of the symmetry group : either under the Wiener condition or under the strict nonlinearity condition, etc. The asymptotics may break down if these conditions fail — this corresponds to some “exceptional” equations: for example, asymptotics (14.5) break down for the linear Schrödinger equations with at least two different eigenvalues.
ii) The general situation is the following. Let a Lie group be a (proper) subgroup of some larger Lie group . So, the -invariant equations form an “exceptional subset” among all -invariant equations, and the corresponding asymptotics (13.2) may be completely different. For example, the trivial group is a subgroup in and in , and asymptotics (14.3) and (14.5) may differ significantly from (14.1).
15 De Broglie’ wave-particle duality
In 1923, de Broglie suggested in his PhD to identify the beam of particles with a harmonic wave:
[TABLE]
This identification was suggested as a counterpart to the Einstein corpuscular treatment of light as a beam of photons. The duality (15.1) was the key source for the Schrödinger quantum mechanics.
We suggest a mathematical description of the wave-particle duality relying on a generalization of the conjecture (13.2) for the case of translation-invariant Maxwell–Schrödinger system (10.3) without external potentials, i.e., , In this case, the Schrödinger equation of (10.3) becomes
[TABLE]
Now the symmetry group of system (10.3) becomes , and our general conjecture (13.2) should be strengthened similarly to (14.4)
[TABLE]
for each finite energy solution, where and stand for the corresponding dispersion waves. The solitons (traveling wave solutions) for (10.3) were constructed in [21]. These asymptotics suggest to treat the solitons as electrons and provisionally correspond to the reduction (or collapse) of wave packets.
The asymptotics (15.3) suggest a mathematical description of the wave-particle duality under several assumptions. Namely, let us consider the wave function as initial data. Then initially the corresponding charge and current densities
[TABLE]
are uniform. Hence, initially the Maxwell field and vanish, as well as the potentials and . Therefore, the Schrödinger equation (15.2) implies that the wave function satisfies initially the free Schrödinger equation which implies
Further we expect that the space-averaged charge, the momentum and the energy densities do not depend on time due to the corresponding conservation laws. Then the density of the electrons (solitons) should be . Similarly, the density of momentum and energy of the solitons should be, respectively, and Finally, it is natural to assume that the velocities of the solitons should be identical by translation homogeneity. Then the momentum and energy of one electron are equal, respectively, to and which agrees with (15.1).
16 Born’s
probabilistic interpretation
In 1927, Born suggested the probabilistic interpretation of the wave function:
The probability of detecting an electron at a point at the time is proportional to .
We suggest below an interpretation of this postulate relying on asymptotics (15.3) for the coupled Maxwell–Schrödinger equations (10.3). However, the corresponding rigorous justification for the nonlinear equations (10.3) is still an open problem.
16.1 Diffraction of electron beams
Born proposed the probabilistic interpretation to describe the diffraction experiments of Davisson and Germer of the 1924–1927s. In these experiments, the electron beam was scattered by a nickel crystal, and the reflected beam was fixed on a photo-film. The resulting images are similar to “Lauegrams”, which were first obtained in 1912 by the method of Laue. Later on, such experiments were also carried out with transmitted electron beams scattered by a thin gold and platinum crystalline films (G. P. Thomson, Nobel Prize 1937). Recently Bach & al. for the first time observed double-slit diffraction of electrons [37], which is the first realization for electronic waves of the classical Young’s experiment of 1803.
The electron diffraction was considered for a long time as a paradoxical phenomenon incompatible with the concept of a point elementary particle. On the other hand, the diffraction phenomena are inherent properties of the linear Schrödinger equation as was commonly recognized for a long time [40]–[48]. We show in the next section (see also [50]) that there is a fine quantitative agreement of solutions to the linear Schrödinger equation for two-slit screen with the results of recent diffraction experiments [37]. Namely, according to the principle of limiting amplitude, the diffracted wave admits the asymptotics
[TABLE]
where is the frequency of the incident wave. The calculation of the diffraction amplitudes in the next section using the Kirchhoff approximation demonstrates that the maxima of on the screen agree very well with those of the diffraction pattern in experiments [37]. Thus, the diffraction of electron beams finds a natural basis in the linear Schrödinger theory.
16.2 Discrete registration of electrons
However, in 1948 the probabilistic interpretation received new content and confirmation after the experiments of Biberman, Sushkin and Fabrikant [33]. In these experiments with very low electron beam intensity, the diffraction pattern was created as an averaging of random discrete registration of individual electrons. Later on, similar experiments were carried out by Chambers, Tonomura, Frabboni, Bach & al. [34, 35, 36, 37]. To explain this phenomenon, there are at least two possibilities, which are both related to random fluctuations:
i) Random interaction with counters. One possibility to explain the discrete registration is a random triggering either a) of registration counters located at the screen points, or b) of atoms of the photo emulsion. We suppose that the probability of triggering is proportional to the current, which is given by
[TABLE]
by (17.19) and according to (10.4) with . Let us note that we set since there is no external fields between the scatterer screen and the screen of observation. The term with in (10.4) is also neglected since it is relatively small.
Therefore, the averaged diffraction pattern should have maxima at the screen points with maximal electric current, which coincide with the screen points with maximal amplitude by (16.2). This coincidence is confirmed in [50] by calculations of and by comparison with experiments [37]. Thus, the discrete registration of electrons also admits an interpretation in the linear Schrödinger theory.
ii) Random reduction of wave packets. Another possibility to explain the discrete registration is the soliton-conjecture (15.3) for translation-invariant Maxwell–Schrödinger system (10.3). This conjecture is inspired by the asymptotics (14.4), which was proved for translation-invariant integrable nonlinear PDEs. Respectively, we suppose that the decay (15.3) should hold between the scatterer screen and the screen of observation, where the external fields vanish, and hence, the system (10.3) is translation-invariant, see Fig. 3. Such a decay into solitons should be considered as a random process, as it is subject to microscopic fluctuations.
An averaged registration rate of electrons at a point of the screen should be proportional to the current (16.2) if the contribution of the dispersion waves is negligible. This follows from the charge conservation law. Therefore, the averaged diffraction pattern again should have maxima at the screen points with maximal amplitude . Thus, this treatment of the discrete registration of electrons requires the soliton-conjecture (15.3) for nonlinear Maxwell–Schrödinger equations.
17 On diffraction of electrons and Aharonov–Bohm shift
17.1 Introduction
The diffraction of the electron beams was observed first by Davisson and Germer in the 1924–1927s [28] (see also [8, Section 5.2]). In these first experiments, the electron beam was scattered by the nickel crystal and the reflected beam was fixed on a film. The resulting images are similar to X-ray scattering patterns (lauegrams), first obtained in 1912 by the method of Laue. Later, such experiments were also carried out with transmitted electron beams passing through thin crystal films of gold and platinum (Thomson 1927). Only recently Bach & al carried out for the first time the two-slit diffraction of electrons [37].
The influence of the “excluded magnetic field” on the diffraction process, which was first predicted in 1949 by Ehrenberg and Siday [39], became widely discussed after the work of Aharonov and Bohm [40], and was confirmed experimentally in 1960 by Chambers [34]. There is a huge amount of works concerned with various aspects of the A-B effect: the shift of the diffraction pattern, the change in the scattering matrix and in the discrete spectrum etc. The survey and the references can be found in [49, 41, 42].
The diffraction of electrons was initially recognized as a paradoxical phenomenon incompatible with the concept of a point elementary particle (now the non-point nature of electrons and of other elementary particles is well established experimentally). The paradoxical appearance of the “particle diffraction” inspired the introduction of the probabilistic interpretation of the wave function by Born in 1927 (though the actual need of the probabilistic interpretation arises at a novel stage – when interpreting a discrete registration of individual electrons in the Biberman–Suchkin–Fabrikant experiments of 1949 with super-weak intencities of falling beams).
On the other hand, the electron diffraction completely agrees with the wave nature of the electron, which was formalised by the Schrödinger theory based on the wave-particle duality of de Broglie. We apply the Fresnel-Kirchhoff theory to the calculation of the diffraction amplitude and check that the formula agrees satisfactory with the results of recent experiments [37]. Our paper aroses exactly because of the possibility of such a comparison.
Similarly, the main topic of discussions on A-B shift is always the distinguished role of the Maxwell potentials in quantum theory, which is considered as the testimony of the nonlocal nature of the interaction of electrons with the Maxwell field. However, these discussions suggest implicitly again that electrons are point particles that do not pass through the region of the magnetic field.
In present paper, we calculate the diffraction amplitude of the transmitted electron beam for general aperture in the plane screen. The incident electron beam is described by a plane wave. We assume i) the limiting amplitude principle and ii) the Sommerfeld radiation condition for the limiting amplitude, though both these assumptions are not rigorously justified for the plane scatterers. These assumptions allows us to express the limiting amplitude via its Cauchy data by the Kirchhoff approach which relies on the Green integral formula. The Green function is selected via the “limiting absorption principle” providing the Sommerfeld radiation condition, which cancels the integral over the large half-sphere.
The Cauchy boundary data of the limiting amplitude are approximated by the Cauchy data of the incident wave. This approximation is commonly recognized in Optics after Kirchhoff works on the diffraction. It is still not justified rigorously for the Schrödinger equation, though it is broadly used in the physical literature, see the survey [41, pp. 390–419] and the references therein. The resulting formulas for the diffraction amplitude include a geometric factor known from the Fresnel–Kirchhoff diffraction theory. We check that this approximation agrees satisfactory with recent experiments (Bach & al [37]) in the particular case of two-slit aperture, which confirms to some extent the validity of the Kirchhoff approximation.
We extend these calculations to the case of a localized “excluded magnetic field” to justify the Aharonov-Bohm effect. Many theories of this effect rely on a gauge transform, which reduces the perturbed Schrödinger equation to the unperturbed one. However, this fact alone does not imply automatically the same correspondence (known as Aharonov-Bohm Ansatz [47]) between the diffraction amplitudes of the free and perturbed equations. Such correspondence was justified in [45, 46, 47] in the context of scattering of finite-energy wave packets moving in external electric and magnetic potentials. Different appearances of the A-B effect were studied in [48, 49, 44, 43].
We justify the A-B Ansatz for the corresponding diffraction amplitudes in the case of small magnetic fields in the original framework of scattering of plane waves [40] applying the Kirchhoff approximation. Such a scattering problem in the same approximation was solved in [40] for the case when the aperture is the whole plane and the magnetic field is supported by an infinite straight line (a “magnetic string”).
We assume the limiting amplitude principle and the Sommerfeld radiation condition for the limiting amplitude. Our main result is an expansion of the magnetic Green function satisfying the Sommerfeld radiation condition in the case of a small magnetic field. This expansion relies on a novel reduction to the case of a continuous potential with rapid decay and subsequent application of the Agmon–Jensen–Kato stationary scattering theory.
The Sommerfeld condition specifies the long-range asymptotics of the Green function and plays the role of boundary conditions at infinity. The role of boundary conditions in the A-B effect was pointed out by Peshkin [42, p. 21].
Remark 17.1**.**
Our calculation shows that the A-B shift holds for small magnetic field in the zero order approximation, while in higher order approximations the change of the diffraction pattern provisionally does not reduce to a shift. **
17.2 The electron diffraction
Schematically, the experiment on the diffraction of a transmitted electron beam is shown on Fig. 3. The incident beam of electrons with energy falls from the left on the scattering plane with the apertures . The transmitted waves interfer on the observation screen , located in the half-plane (see Fig. 3).
The diffraction problem. The incident beam of free electrons is described by the wave
[TABLE]
in accordance with the de Broglie wave-particle duality. It satisfies the free relativistic Klein–Gordon equation
[TABLE]
This equation is derived from the relativistic energy formula by the canonical quantization , . For , this relativistic energy is close to the nonrelativistic one,
[TABLE]
The canonical quantization leads to the free Schrödinger equation with the shift
[TABLE]
This shift is easily removed by the gauge transformation , where . Now satisfies the standard free Schrödinger equation
[TABLE]
and the incident wave is replaced by , where and
[TABLE]
Let us choose a coordinate system in which the scattering screen lies in the plane , and the electron beam falls from the region . The external potentials vanish everywhere in the connected domain , and hence the diffraction problem reduces to solving the free Schrödinger equation (17.5) in with the “boundary condition”
[TABLE]
We assume the limiting amplitude principle to hold
[TABLE]
It is well established for the Schrödinger equation with a short-range potential and for boundary value problems outside a smooth bounded domain [29, 30], [9, Ch. 28]. The limiting amplitude satisfies the free Schrödinger equation (17.5)
[TABLE]
due to (17.6). The solution can be expressed through its Cauchy data on using the Green formula
[TABLE]
where the Green function is the integral kernel of the free Helmholtz operator on :
[TABLE]
The integral over the large halh-sphere vanishes as due to the Sommerfeld radiation condition [15, formulas (34.5)], which holds for this Green function and for the amplitude
[TABLE]
The last conditions are obvious for the Green function (17.11). We assume these conditions also for the limiting amplitudes, since they are well established for the Schrödinger equation with a short-range potential and for boundary value problems outside a smooth bounded domain, see [29, 30] and also formulas (28.1), (27.7) and (27.8) in [9].
The Kirchhoff approximation. The main idea of the Kirchhoff method [4] is the approximation of the Cauchy data of solutions by the same of the incident wave (17.1)
[TABLE]
where
[TABLE]
For the Green function (17.11) we have for and
[TABLE]
where . Substituting these approximations into the Green formula (17.10), we get
[TABLE]
where is the “Fresnel angle of diffraction” between the vectors and , see [4, Fig. 8.1]. This result can also be obtained from the Kirchhoff–Fresnel formula (17) of [4, Section 8.3] for the case of an incident spherical wave with a point source , when it is removed to infinity .
Remarks 17.2**.**
i) Physically the Kirchhoff approximation (17.13) means that the electron density at the points of the aperture equals , while the density of the momentum equals . These assumptions are commonly recognised in physical literature, see the survey [41, pp. 390–419] and the references therein. In particular, this approximation is used in the paper [40]. However, this approximation is not sill justified rigorously for the scattering problem with the “boundary condition” (17.7).
ii) Formulas of type (17.16) can be found in almost all publications on the diffraction of electrons, however only with .
Fraunhofer asymptotics. For bounded apertures , the integral (17.16) admits the asymptotics in each fixed direction as ,
[TABLE]
where . This asymptotics corresponds to the Fraunhofer diffraction [4, Section 8.3.3, formula (28)]. In particular, for one circular aperture of radius centered at the integral (17.16) has been calculated in 1835 by Airy [4, Section 8.5.2, formula (13)],
[TABLE]
where .
17.3 Comparison with the experiment
The two-slit diffraction of electrons was observed experimentally first in 2013 (Bach & al [37]): the formula (17.16) is in satisfactory agreement with the results of these experiments. Namely, in the particular case, when the aperture consists of two slits centered at the points , formula (17.16) gives
[TABLE]
In these experiments the distance between the planes and is m nm, the distance between the centers of the slits is nm, and the wavelength nm. Then for
[TABLE]
Therefore, for such deviations, we can set , and then the formula (17.19) gives
[TABLE]
Let points , be symmetric with respect to the origin. Choose the -axis orthogonal to the plane of drowing, so that and , where , see Fig. 5. Then the amplitude (17.20) has the maxima at the points with coordinates determined by the Bragg rule
[TABLE]
Expanding the square root in a Taylor series, we get the equation . For we get the central maximum , and for we get the second maximum m. An experimental value, as obtained in [37], is approximately the m [38, Fig. S2 a].
Remark 17.3**.**
Formula (17.20) satisfactory reproduces positions of the maxima, but not their magnitudes. On the other hand, the picture [38, Fig. S2 a] shows complete coincidence of the measured diffraction amplitude with the graph obtained by numerical integration over paths. This integration in the limit should coincide with the exact solution of the diffraction problem (17.5), (17.7) with the incident wave (17.1). **
17.4 The Aharonov–Bohm shift
The two-slit arrangement is superimposed by a magnetic field concentrated in a closed tube lying between the plane of scattering and the pane of observation . The tube passes parallel to the -axis in the region near the aperture and encloses sufficiently far from the aperture.
In this case the diffraction pattern of the set-up of Fig. 3 is shifted downwards. This shift was first experimentally observed by Chambers [34]. The set-up was close to that of Fig. 4. A complete survey of experimental observations and various theoretical interpretations can be found in [42].
Let us calculate this AB-shift applying the Kirchchoff approximation (17.13), (17.14) of the Cauchy data. Now the free Schrödinger equation (17.5) for changes to the equation
[TABLE]
where is the vector potential of the external magnetic field in the Coulomb gauge:
[TABLE]
Respectively, the stationary Helmholtz equation (17.9) for the limiting amplitude changes to
[TABLE]
Now the Green formula (17.10) becomes
[TABLE]
where the Green function is now the integral kernel of the limit operator on . To justify this integral representation we will prove i) the existence (in a sense) of this limit operator and ii) the Sommerfeld radiation condition for similar to (17.12) for . We assume that this condition also holds for the limiting amplitude . The radiation conditions of type (17.12) for and cancel the integral over the large half-sphere in (17.25) as well as in (17.10).
Further we will substitute into (17.25) the Kirchhoff approximation (17.13), (17.14) of the Cauchy data of the amplitude and then we will check the A-B shift. We will accomplish this program for a small magnetic field.
17.5 Reduction to short-range magnetic potential.
We plan to construct the magnetic Green function using the Agmon–Jensen–Kato stationary scattering theory [31, 32]. First we reduce the problem to magnetic potentials with compact support. This is necessary for estimates (17.40) below. The potential decays slowly. For example, for an infinite cylindrical tube
[TABLE]
for any disk in if its boundary embraces the tube . Hence, the potential decays slowly like , where is the distance from the tube. For a finite tube, the decay is faster but it is also insufficient for our purposes.
Remark 17.4**.**
There were many attempts to avoid the slow decay by constructing “non-Stokesian potentials” supported by the tube [42, pp 65–70]. Then the “Stokes identity” (17.26) breaks down, which means that these potentials do not satisfy the equation in the sense of distributions in the region . **
To reduce the problem to the case of potentials with rapid decay, we choose the magnetic potential solving the equations (17.23) for all assuming and . Then we have , which implies that by the Sobolev embedding theorem, and moreover,
[TABLE]
for the potential vanishing at infinity. Further, the magnetic field vanishes outside the tube , and hence,
[TABLE]
Therefore,
[TABLE]
where is a bounded surface in (a “film”) such that the domain is simply connected, see Fig. 5, while is any path in connecting with a fixed point.
Let us denote by a bounded neighborhood of . By (17.29) we can split the vector potential as
[TABLE]
where
[TABLE]
Moreover, the estimate (17.27) implies that
[TABLE]
where is the radius of a ball which contains the tube . The splitting (17.30) allows us to factorize the operator as follows:
[TABLE]
Now the theory [31, 32] implies the existence of the limits
[TABLE]
for all values of , except a discrete set. The limit exists in the space of linear operators acting in suitable weighted Sobolev spaces, see Theorem 19.2 of [9]. Then for the corresponding integral kernels
[TABLE]
17.6 Asymptotic expansion for a small magnetic field
Let us construct the series for which converges for a small magnetic field . First, , where , and . Second, formally,
[TABLE]
where
[TABLE]
Respectively,
[TABLE]
where is the integral kernel of the operator . For example,
[TABLE]
Similar expressions hold for with arbitrary . Hence, each term is smooth for similarly to the Green function , and has the singularity near the diagonal only for .
Let us note that in the representation of the limiting amplitude (17.25) we need the Green function only for . Moreover, the diffraction is observed for sufficiently large . So in all cases . For such the integral representations of type (17.39) imply that
[TABLE]
On the other hand, the derivation of the representation (17.25) requires the Sommerfeld radiation condition
[TABLE]
The estimates (17.40) imply that these conditions hold for each term and hence for when . Moreover,
[TABLE]
which follows from (17.27) by the partial integration. Hence, the conditions (17.41) hold by (17.35), so the integral representation (17.25) is justified for small by the inequality (17.32).
Remark 17.5**.**
We assume that to have which implies the representations (17.33) and (17.34). We suppose that similar representations also hold for – however this is an open question. **
17.7 The zero-order approximation
In the zero order approximation, the magnetic Green function reads
[TABLE]
Respectively, the Cauchy data (17.15) change to
[TABLE]
Finally, for small magnetic fields the terms with in (17.25) and (17.44) can be neglected with respect to (let us note that cm*-1* for pm as in the experiment of [37]). Now the integral (17.25) becomes similar to (17.16),
[TABLE]
In particular, for the two-slit diffraction we get, similarly to (17.19),
[TABLE]
The A-B shift. Now to determine the maxima of the diffraction amplitude we should add the phase contribution into the equation (17.21),
[TABLE]
Accordingly, the Taylor decomposition now gives the equation . This means that the coordinates of all the maxima are shifted by the same value ; i.e., the whole diffraction pattern shifts without changing its shape.
Remark 17.6**.**
i) The phase difference is approximately if the distance is sufficiently small.
ii) The limiting amplitude principle (17.8) and the Sommerfeld radiation condition (17.12) for the limiting amplitudes and hold in actual physical situation due to the known results from [29, 30], [9, Ch. 28], since the real scatterer is not two-dimensional.
iii) The formulas (17.45), (17.46) give the solution in the zero order approximation in the magnetic field. Respectively, the A-B shift holds in this approximation. To calculate higher order approximations, one should modify the Cauchy data (17.44) using the series (17.38). Then the diffraction pattern in presence of the magnetic field provisionally does not reduce to a shift of the pattern corresponding to .
Appendix A Old Quantum Mechanics
We recall the main achievements of the classical atomic Lorentz–Thompson theory of 1897–1913 and of “Old Quantum Mechanics” (1913–1925), as developed by Planck, Einstein, Bohr, Debye, Sommerfeld, Pauli, and many others [3, 16]. In particular, we recall the introduction of the Bohr–Sommerfeld quantization rules, selection rules and the theory of normal Zeeman effect. This is necessary for the introduction of the Schrödinger Quantum Mechanics and of the Pauli theory for the electron spin.
A.1 The Thompson electron and the Lorentz classical theory of the atom
The existence of electric currents in atoms was predicted by Amper in 1820. In 1855, Geissler and Plücker discovered “cathode rays” in a vacuum tube. In 1895, Perrin experimentally proved that these rays carry a negative charge. In 1893–1897, Thompson conducted a series of ingenious experiments on the deflection of these rays in electrical and magnetic fields, and came to the conclusion that the cathode rays consist of negatively charged particles – electrons – that obey the Lorentz equation
[TABLE]
where is the electron charge, is its mass, and and are the Maxwell external fields. In these experiments, Thompson for the first time measured with great accuracy the ratio , which enters equation (A.1). For these experiments, Thompson was awarded the Nobel Prize in 1906.
Equation (A.1) was applied to atomic electron in a constant uniform magnetic field , replacing the first term on the right by an elastic force:
[TABLE]
This linear equation led to a remarkable explanation of basic atomic phenomena (polarization, dispersion and others), and served as a basis for the Drude classical theory of metals (1900).
In 1897, Zeeman discovered a splitting of spectral lines of atoms in magnetic fields. Part of these experimental results (“normal Zeeman effect”) was explained in 1897 by Lorentz based on equation (A.2). This equation results in the Larmor precession of the electron orbit around the direction of the magnetic field. This linear equation (A.2) is easy to solve. Namely, choosing , we get
[TABLE]
where is the Larmor frequency (7.2). Hence, , and for , we get the equation
[TABLE]
The substitution gives the characteristic equation , whence
[TABLE]
Therefore, for small we get two solutions with frequencies , modified by a magnetic field (we assume that ).
So, the spectrum of the solution contains three different frequencies that make up the triplet
[TABLE]
which corresponds to the splitting of the “atomic spectral line” into three components. Such splitting was observed experimentally by Zeeman in 1895.
This classical interpretation of atomic radiation turned out to be only asymptotically exact for highly excited atoms, as it was shown later by Bohr correspondence principle (1913), see Section A.3.
Moreover, Lorenz calculated the corresponding angular distribution of atomic radiation and its polarization [8, Section 14.5]. All these calculations were in a perfect agreement with experimental observations of Zeeman, and Lorenz together with Zeeman received the Nobel Prize in 1902.
A.2 Bohr postulates and Debye’s quantization rule
In 1913, Bohr formulated postulates (5.1) and (5.2) on transitions between quantum stationary orbits. The question immediately arose of determining the energies of the stationary orbits . For the hydrogen atom, the Balmer empirical formula (1885) gives with great accuracy the values (6.4). The greatest surprise was caused by the discreteness of the set of possible energies. In 1911–1913, Ehrenfest conjectured a relation of the discreteness with adiabatic invariance of the energy under slow variation of parameters [1, section 52].
In particular, all basic integrals of dynamical equations are adiabatic invariants: energy, momentum, and angular moment. Debye in 1913 calculated the action for periodic orbits of classical electron in an hydrogen atom and discovered that the Balmer formula (6.4) is equivalent to the “quantization rule”
[TABLE]
(see [8, p. 20]), where the integral is taken over the period of the classical electron orbit with energy . Namely, the classical model of the atom is a classical particle (electron), in the Coulomb nucleous field. This nucleous can be considered fixed, since its mass is about 1840 times the mass of an electron. However, all calculations below allow a generalization to the two-body problem, which takes into account the motion of the nucleus. The dynamics of an electron is described by the Lorentz equation (A.1) with the Coulomb field (in Gaussian non-rationalized units cgs) and :
[TABLE]
Multiplying by , we obtain the energy integral . Any trajectory lies in a certain plane, for example, in the plane . In particular, for circular trajectories equation (A.8) gives
[TABLE]
and the energy integral turns into . Eliminating , we get
[TABLE]
which inplies that . Hence, for a circular orbit with period the action reads
[TABLE]
since , where is the Lagrangian of the Lorentz equation (A.8). Therefore, (A.7) is equivalent to
[TABLE]
which coincides with the Balmer formula (6.4) with the Rydberg constant (6.5), as in the later Schrödinger theory. This constant, as calculated from available values, was in an excellent agreement with experimental data, which was a genuine confirmation of the Bohr–Debye theory.
A.3 Correspondence Principle and Selection Rules
In a general form, the Correspondence Principle was formulated by Bohr in 1920, although he applied it in various aspects since 1913. For example: the frequency of atomic emission (5.2) for large quantum numbers should go into the frequency of revolution of classical electron.
This frequency of revolution was identified until 1913 with the frequency of atomic radiation as in the Lorentz theory of normal Zeeman effect in Section A.1. The revolution frequency in circular orbits is calculated from the Lorentz equation (A.8). Namely, eliminating and from (A.10) and (A.12), we get the classic frequency
[TABLE]
On the other hand, quantum radiation frequencies (5.2), with Balmer’s terms (6.4), for large and bounded differences admit the asymptotics
[TABLE]
by (6.5). The minimal frequencies correspond to , and all other frequencies (called “obertones”) are multiples to the minimal one. On the other hand, the classical frequency (A.13) has no obertones. Therefore, the correspondence principle dictates the selection rule
[TABLE]
for large quantum numbers .
A.4 Bohr–Sommerfeld Quantization
In 1915–1916, Sommerfeld and Wilson suggested to extend the Debye quantization rule (A.7) to solutions of Hamiltonian systems with several “periodic” degrees of freedom and corresponding canonically conjugate momenta . In this case, the suggested quantization rules read
[TABLE]
In particular, for three-dimensional classical atomic system with the Coulomb potential the Lorentz equation (A.1) takes the form
[TABLE]
The corresponding Lagrange and Hamilton functions are
[TABLE]
Any trajectory of lies in some plane passing through the nucleus. Therefore, we can assume that this equation reads in appropriate spherical coordinates (radius), (longitude), (latitude) with the origine in the nucleus. Each bounded trajectory is periodic, and the quantization rules (A.16) take the form of integrals over a period
[TABLE]
because .
Let us show that these quantization rules imply the Debye condition (A.7) and, respectively, the Balmer formula (6.4) holds for the energies. First, in the polar coordinates on the plane , the Lagrangian reads as , which implies and since . Therefore,
[TABLE]
The Euler–Lagrange equation for gives
[TABLE]
whence integration yields . It is easy to verify that ; i.e., is the energy of the trajectory. Hence,
[TABLE]
The minimal and maximal values of the radius on the trajectory are obtained from the condition . The energy should be negative, since, for bounded trajectories, the radical expression should be non-negative only on a finite interval . Moreover, according to (A.22). Therefore, by integrating [8, c. 262] the first quantization condition (A.19) implies
[TABLE]
Hence, due to (A.20). Therefore,
[TABLE]
which coincides with (6.4), (6.5). Obviously coincides with the magnitude of the “angular momentum vector” , and so
[TABLE]
Finally, note that the Debye quantization condition (A.7) also holds if define as the invariant canonical differential form , since in this case .
A.5 Atom in a magnetic field
The Bohr–Sommerfeld quantization (A.16) can also be applied to the classical three-dimensional model of atom with the Coulomb potential of the nucleus in a uniform magnetic field . Namely, the Lorentz equation (A.1) now takes the form
[TABLE]
where . The corresponding Lagrange and Hamilton functions read as [8, (12.81), (12.90)]:
[TABLE]
where the vector potential is , so that , and the momentum is [8, (12.82)]
[TABLE]
The homogeneous magnetic field can be eliminated by transition to the “rotating coordinates” : the trajectory of obviously satisfies the equation (A.26) with the magnetic field (“Larmor’s theorem”). Hence, the Lagrangian in the rotating frame reads
[TABLE]
which coincides with (A.18). Respectively, any trajectory lies in some plane of passing through the nucleus, and has the equation in appropriate spherical coordinates (radius), (longitude), (latitude) with origin at the nucleus. Due to the spherical symmetry of the Coulomb potential, the corresponding vector of angular momentum is conserved.
[TABLE]
according to (A.29). In particular, . Each bounded trajectory is periodic, and the quantization conditions (A.16) take the form of integrals over the period
[TABLE]
because . Here . As was shown above, these quantization conditions imply formulas (A.24) and (A.25):
[TABLE]
Respectively, the trajectory lies in the rotating plane , and generally is not periodic, since the conservation of the angular moment implies the precession of the angular momentum around the direction of the magnetic field ,
[TABLE]
On the other hand, the orbit of is periodic in the limit . Therefore, Sommerfeld generalizes the quantization conditions (A.16) to this case
[TABLE]
where .
Sommerfeld derived the following important identity [16, Chapter II, Section 8, formula (4)]
[TABLE]
which follows from the invariance of integrals of the canonical form along closed curves with respect to the canonical maps [1, Section 44]. Namely, the transition from canonical variables to conserves the canonical form , and therefore,
[TABLE]
which implies (A.35) according to the quantization rules (A.31) and (A.34). Hence,
[TABLE]
(see [16, Chapter II, Section 8, formula (3)]) since , and , because . Now the conditions (A.32) with can be rewritten as
[TABLE]
The quantization rules (A.19), (A.34) also imply the fundamental relation [16, Chapter II, Section 8, formula (5)])
[TABLE]
where is the angle between the vectors and , which is also equal to the angle between the vectors and due to the precession (A.33). This relation follows from the formula
[TABLE]
which is obtained by computing in spherical coordinates. Indeed,
[TABLE]
Similarly, because . Both momenta and are conserved, and at maximal latitude , which implies (A.40). Finally, (A.40) and (A.19), (A.34) imply (A.39) due to the conservation of and .
Corollary A.1**.**
(“Spatial Quantization”.) The angle is quantized by (A.39). Precession (A.33) implies that , and the projection of vector onto direction of the magnetic field equals . Moreover, according to (A.32). Therefore, (A.39) implies that , where . So, denoting and , we get from (A.38) that
[TABLE]
Corollary A.2**.**
Quantum stationary states of an electron in atom are numbered with three quantum numbers (main, azimuth and magnetic quantum number, respectively), and their energy, angular momentum and its projection onto the direction of the magnetic field are given by formulas (A.24) and (A.42).
Remark A.3**.**
Quantized values (A.42) were obtained for “infinitely small” magnetic field . However, the same expressions should hold also for finite small magnetic field, since the values of quantum numbers are discrete, while the action integrals (A.19) are adiabatic invariants. **
A.6 Normal Zeeman effect
In 1916, Sommerfeld obtained from the quantization rules (A.42) the same results for the normal Zeeman effect, as Lorentz from equation (A.2). Namely, the classical angular and magnetic moments of the electron are determined by the formulas
[TABLE]
Therefore
[TABLE]
The classical energy (A.27) of an electron can be expanded as
[TABLE]
Thus, in a magnetic field the classical energy acquires an additional amount
[TABLE]
where is the Larmor frequency (7.2). Hence, the total classical energy (A.45) reads
[TABLE]
The corresponding quantum version follows by substitution here expression (6.4) instead of the classical energy and (A.42) for . As a result, we get the following possible values of energy, of angular momentum and of its projection on the direction of the magnetic field
[TABLE]
which coincides with results of the Schrödinger theory (7.5) except for the range of values for . Hence, the spectral lines (5.2) a quire an additional amount . So, the spectral lines in the magnetic field are given by
[TABLE]
which coincides with the result of the Schrödinger theory (7.6). Finally, Bohr’s correspondence principle dictates selection rules, similar to (A.15),
[TABLE]
see [16, Supplements 7 and 8]). As a result, each spectral line (5.2) becomes the Zeeman triplet (7.1).
Remark A.4**.**
The selection rules (A.50) also hold in Schrödinger’s theory, see Section 7. **
A.7 Bohr–Pauli theory of periodic table
The Bohr and Pauli theory of periodic table of elements (1921–1923) was the highest achievement of the “Old Quantum Mechanics”. The theory relies on the following postulates:
-
Electrons in many-electrons atoms are weakly coupled, so that their stationary states can be described separately by the Bohr-Sommerfeld quantization rules.
Stationary orbits of each electron are numbered by four quantum numbers , where . Energy, angular momentum and its projection on the direction of the magnetic field are determined by formulas (A.24), (A.42). However, now the value is excluded:
[TABLE]
-
The sets of quantum numbers are different for different electrons in accordance with the Pauli exclusion principle.
-
Electrons with the same energy belong to the same shell: the shell corresponds to , corresponds to , – to , – to , … The atomic shells were introduced by Mosley in his interpretation of scattering of X-rays by atoms (1913).
The bound was found empirically in the “Old Quantum Theory”, and it holds automatically in the Schrödinger theory by (6.9). This bound, together with the Pauli exclusion principle 3), leads to wonderful explanation of the periods in the table of elements.
For example, the ground state of the atom corresponds to the minimal energy of the electron configuration. Hence, in the ground state the electrons should belong to shells with possible minimal values of .
The Pauli exclusion principle implies that for the shell can contain at most two electrons with the least energy corresponding to quantum numbers , and .
For , the shell can contain at most electrons with , , and .
Similarly, for a shell with any number , the maximal number of electrons is . These accupation numbers coincide with the famous “Kabbalistic sequence” of lengths of periods in the table of chemical elements. For this exact reason Pauli introduced in 1923 the fourth two-digit quantum number . Otherwise, all periods of the table would turn out to be two times shorter than necessary.
A.8 Hamilton–Jacobi equation and Optico-Mechanical Analogy
After 1915 the quantisation rules (A.7) and (A.16) attracted common attention to the study of action and to its calculation using the Hamilton–Jacobi equation (3.12). Many attempts were made to derive a new electron dynamics from this equation using Hamilton’s optico-mechanical analogy (1840), which is based on the parallelizm berween the Fermat and Maupertuis variation principles [1, p. 246]. This idea was first realized by Schrödinger in 1926, who identified the role of the Hamilton–Jacobi equation with the role of the eikonal equation in optics as the equation for the phase function of shortwave solutions. The Schrödinger equation (3.7) formally follows from optico-mechanical analogy [6], and exactly this analogy was in the center of Schrödinger’s first papers on quantum mechanics [12, II] and of his Nobel lecture [13].
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