Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform

TL;DR

This paper introduces a method to simplify certain quantum Hamiltonians into first-order differential operators using coherent state transforms, enabling explicit solutions for quantum dynamics.

Contribution

It generalizes the geometric dynamics approach to quantum systems, identifying Hamiltonians reducible via Gaussian and Airy beam transforms and providing explicit solutions.

Findings

Reduction of specific quantum Hamiltonians to first-order PDEs

Explicit solutions for systems involving Gaussian and Airy beams

Extension of harmonic oscillator dynamics to broader quantum systems

Abstract

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.

Equations

$$\tilde{f}(t,z)=e^{-\pi \mathrm{i}\omega t}f(0,e^{-2\pi \mathrm{i}\omega t}z).$$

$$\mathrm{i}\hslash \dot{\varphi }(t)=H\varphi (t).$$

$$|x,t⟩=|x_t⟩,$$

$$f\mapsto \tilde{f}(x)=⟨x⟩f.$$

$$\tilde{f}(t,x)$$

$$\mathrm{i}\hslash \frac{\mathrm{d}\tilde{f}}{\mathrm{d}t}=\tilde{H}\tilde{f},$$

$$|x⟩=\rho (\mathsf{s}(x))|0⟩,$$

$$\mathrm{i}\hslash \frac{\mathrm{d}}{\mathrm{d}t}\tilde{f}(t,z)=\pi \hslash \omega (\tilde{f}(t,z)+2z{\partial }_z\tilde{f}(t,z)),z\in \mathbb{C}.$$

$$[X_1,X_2]=X_3,[X_1,X_3]=X_4.$$

$$g=(x_1,x_2,x_3,x_4):={\mathrm{e}}^{x_4{X}_4}{\mathrm{e}}^{x_3{X}_3}{\mathrm{e}}^{x_2{X}_2}{\mathrm{e}}^{x_1{X}_1}.$$

$$[{\rho }_{h_2{\hslash }_4}(g)f](x_1^{\prime })={\mathrm{e}}^{2\pi \mathrm{i}(h_2{x}_2+{\hslash }_4(x_4-x_3{x}_1^{\prime }+\frac{1}{2}{x}_2{x}_1^{\prime 2}))}f(x_1^{\prime }-x_1).$$

$$\rho (h)|0⟩=\chi (h)|0⟩\text{for all}h\in H,$$

$$\tilde{f}(x)=⟨x⟩f,\text{ where }|x⟩=\rho (\mathsf{s}(x))|0⟩,x\in G/H.$$

$$\tilde{f}(x_1,x_2,x_3)=⟨x_1,x_2,x_3⟩f$$

$$={\mathrm{e}}^{-2\pi \mathrm{i}{h}_2{x}_2}{\int }_{\mathbb{R}}\mathrm{d}yf(y){\mathrm{e}}^{-2\pi \mathrm{i}{\hslash }_4(-x_{3}y+\frac{1}{2}{x}_2{y}^2)}\overline{\varphi }(y-x_1).$$

$$[{\tilde{\rho }}_{{\hslash }_4}(g)f](x_1^{\prime },x_2^{\prime },x_3^{\prime })={\mathrm{e}}^{2\pi \mathrm{i}{\hslash }_4(x_4-x_1{x}_3+\frac{1}{2}{x}_1^2{x}_2+x_1{x}_3^{\prime }-\frac{1}{2}{x}_1^2{x}_2^{\prime })}$$

$$\times f(x_1^{\prime }-x_1,x_2^{\prime }-x_2,x_3^{\prime }-x_3-x_1{x}_2^{\prime }+x_1{x}_2).$$

$$\varphi (y)=\exp \left(\frac{\pi \mathrm{i}D{\hslash }_4}{3}{y}^3-\pi E{\hslash }_4{y}^2+2\pi \mathrm{i}D{h}_{2}y\right),$$

$$\begin{array}{rl}\mathrm{d}{\rho }_{h_2{\hslash }_4}^{\mathrm{i}{X}_1+\mathrm{i}D{X}_2+E{X}_3}=\mathrm{id}{\rho }_{h_2{\hslash }_4}^{X_1}+\mathrm{i}D\mathrm{d}{\rho }_{h_2{\hslash }_4}^{X_2}+E\mathrm{d}{\rho }_{h_2{\hslash }_4}^{X_3}& \\ =-\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}y}-\pi {\hslash }_{4}D{y}^2-2\pi \mathrm{i}E{\hslash }_{4}y-2\pi h_{2}D& .\end{array}$$

$$\mathcal{C}f(x_1,x_2,x_3)=0,\text{for any}f\in {\mathsf{L}}_0(\mathbb{G}/Z)$$

$$\mathcal{C}$$

$$\mathcal{S}$$

$$={\partial }_{33}^2+4\pi \mathrm{i}{\hslash }_4{\partial }_2-8{\pi }^2{h}_2{\hslash }_{4}I.$$

$$\mathrm{d}{\rho }^G=\sum _{j,k=1}^3{a}_{jk}\mathrm{d}{\rho }^{X_j}\mathrm{d}{\rho }^{X_k}\text{where }{X}_n\in \mathfrak{g}.$$

$$H_r=\mathrm{d}{\tilde{\rho }}^G+(A{\partial }_1+B{\partial }_2+C{\partial }_3+K)\mathcal{C}+F\mathcal{S}.$$

$$A$$

$$C$$

$$F$$

$$K$$

$$\displaystyle\qquad+5\pi\hslash_{4}Dx_{1}^{2}-4\pi\hslash_{4}x_{1}x_{2}\big{)}+\frac{a_{21}}{E}\,.$$

$$\begin{array}{rl}{a}_{12}& =2D{a}_{11}-a_{21},a_{22}=D^2{a}_{11},\\ a_{23}& =D(a_{13}+a_{31})-a_{32}.\end{array}$$

$$\begin{array}{rl}& f(t;x_1,x_2,x_3)={\mathrm{e}}^{\pi {\hslash }_4(-E{x}_1^2+\mathrm{i}D{x}_1^3/3)}\varphi (t;u_1,u_2)\\ & \text{for }{u}_1=D{x}_1^2+2\mathrm{i}E{x}_1-2x_3,u_2=D{x}_1-x_2+\mathrm{i}E.\end{array}$$

$$(a_{jk})=\frac{1}{m}\left(\begin{array}{ccc}1& D& 0\\ D& D^2& 0\\ 0& 0& 0\end{array}\right),$$

$$\begin{array}{rl}{H}_{\rho }& =-\frac{1}{m}(\mathrm{d}{\rho }^{X_1}+D\mathrm{d}{\rho }^{X_2}{)}^2,\end{array}$$

Full text

Sponsored by The Taif University (Saudi Arabia).

PACS: 02.20.Qs, 02.30.Jr, 03.65.-w, 03.65.Db, 42.30.Kq, 42.25.-p

1. Introduction

The coherent states (CS) were introduced by Schrödinger in 1926 [Schrodinger26a] but were not in use until much later [Bargmann61, Segal60, Glauber63a, Sudarshan63a]. Nevertheless, ideas from [Schrodinger26a] were crucial for formation of quantum mechanics [Steiner88a]. Naturally, further developments of the concept of CS manifested a remarkable depth and width [KlaSkag85, Perelomov86, Walls08, AliAntGaz14a, Gazeau09a]. The canonical CS of a harmonic oscillator have a variety of important properties, e.g. semi-classical dynamics, minimal uncertainty, parametrisation by points of the phase space, resolution of the identity, covariance under a group action, etc. As usual, different generalisations of CS start from one particular property used as a definition, other properties may or may not follow as consequences depending on circumstances. For example, the approach in [Hall94a] employed a connection of CS with the heat kernel (Gaussian) and its analytical extension, but requires the compactness of the underlying group—in contract to the original setup of CS from the Heisenberg group.

This letter revises geometrisation of quantum evolution in CS representation. More specifically, in Defn. 3 we describe a method to determine all Hamiltonians admitting a reduction to a first order partial differential equation (PDE) by means of the coherent state transform (CST) with a given fiducial vector. Thereafter, the corresponding first order PDE can be explicitly solved and the geometrised evolution is realised by a time-dependent coordinate transformations. The method is based in the development [Kisil11c, Kisil13c, Kisil17a, AlmalkiKisil18a] of the CS from group representations [Perelomov86, AliAntGaz14a, Gazeau09a], notably an analyticity-type condition in the image space of CST as explained below.

Our inspiration is that the canonical CS [Schrodinger26a, Glauber63a, Sudarshan63a, Bargmann61, Segal60, Walls08, Gazeau09a] are parametrised by points $z=q+\mathrm{i}p$ of the phase space in such a way that the time evolution of a quantum state $\tilde{f}(t,z)$ in a harmonic potential is given by

[TABLE]

Its key ingredient is the rigid rotations $z\mapsto\mathrm{e}^{-2\pi\mathrm{i}\omega t}z$ of the phase space—the dynamics of a classical harmonic oscillator. Thus, the geometrisation of the evolution also manifests the correspondence principle between quantum and classical mechanics.

However, various “no-go” theorems suggest that this correspondence cannot be universal. Therefore, there are many different approaches to geometrisation of quantum dynamics [Kibble79a, Prugovecki82a, BrookePrugovecki85a, Zachos02a, CarienaClemente-GallardoMarmo07a, ShalashilinBurghardt08, KaramatskouKleinert14a, Tavernelli16a, HerranzDeLucasTobolski17a, CiagliaDiCosmoIbortMarmo17a], see also [HurleyVandyck09a, Clemente-GallardoMarmo08a] for surveys and further references. Those works are often motivated by conceptual considerations of fundamental structures of Lagrangian and Hamiltonian dynamics. Our main aim here is more pragmatic: we look for a simple and effective method to express a quantum evolution through a flow of points of some set. Few connections to the above papers will be discussed at the end of this letter.

Let the dynamic of a quantum system be defined by a Hamiltonian $H$ and the respective Schrödinger equation

[TABLE]

Geometrisation of (2) suggested in [CiagliaDiCosmoIbortMarmo17a] uses a collection $\ket{x}$ of CS parametrised by points of a set $X$. Then the solution $\ket{x,t}$ of (2) for an initial value $\ket{x,0}=\ket{x}$ shall have the form

[TABLE]

where $x_{t}$ is an orbit of a one-parameter group of transformations $X\rightarrow X$. Recall that the coherent state transform (CST) $\tilde{f}(x)$ of a state $f$ is defined by

[TABLE]

It is common that CST is a unitary map onto a subspace $\mathsf{F}_{2}{}$ of $\mathsf{L}_{2}{}(X,d\mu)$ for a suitable measure $d\mu$ on $X$. If CS $\ket{x}$, ${x\in X}$ geometrise a Hamiltonian $H$ in the above sense, then for CST $\tilde{f}(t,x)=\braket{x}{f(t)}$ of an arbitrary solution $f(t)=\mathrm{e}^{-\mathrm{i}tH/\hslash}f(0)$ of (2) we have:

[TABLE]

Thus, if CS geometrise a Hamiltonian $H$ then the dynamic of any image $\tilde{f}$ of the respective CST is given by a transformation of variables.

It was already noted in [CiagliaDiCosmoIbortMarmo17a] that even the archetypal canonical CS do not geometrise the harmonic oscillator dynamics in the above strict sense due to the presence of the overall phase factor in the solution (1). The factor is not a minor nuisance but rather a fundamental element: it is responsible for a positive energy of the ground state.

To accommodate such observation with geometrising, we propose the adjusted meaning of geometrisation:

Definition 1**.**

A complete collection $\ket{x}$ of CS parametrised by points of a manifold $X$ geometrises a quantum dynamic, if the time evolution of the CST $\tilde{f}=\braket{x}{f}$ is defined by a Schrödinger equation

[TABLE]

where $\tilde{H}$ is a first-order differential operator on $X$.

To find geometrising CS one can use symmetries of the Schrödinger equation and a Hamiltonian $H$, cf. [Niederer72a, KalninsMiller74a, ATorre09a, Niederer73a, Wolf76a, AldayaGuerrero01a, AldayaCossioGuerreroLopez-Ruiz11b, AldayaGuerreroMarmo98a]. In particular [Perelomov86, AliAntGaz14a, Gazeau09a], group representations are a rich source of CS. More precisely, let $X$ be the homogeneous space $G/H$ for a group $G$ and its closed subgroup $H$. Then, for a representation ${\rho}$ of $G$ in a space $V$ and a fiducial vector $\ket{0}\in V$ the collection of coherent states is defined by

[TABLE]

where $x\in G/H$ and $\mathsf{s}:G/H\rightarrow G$ is a Borel section.

Example 2**.**

The canonical CS [Schrodinger26a, Glauber63a, Sudarshan63a, Bargmann61, Segal60, Walls08, Gazeau09a] are produced by $G$ being the Heisenberg group [Folland89, Kirillov04a, Kisil02e, Kisil17a], $H$—the centre of $G$, ${\rho}$—the Schrödinger representation and $\ket{0}$ be Gaussian $\mathrm{e}^{-\pi\hslash m\omega x^{2}}$. These CS geometrise the harmonic oscillator dynamic with the specific potential in the Hamiltonian: $H=\frac{1}{2}\left(\frac{1}{m}p^{2}+m\omega^{2}q^{2}\right)$. Indeed, the corresponding CST maps states to the Fock–Segal–Bargmann space and the harmonic oscillator dynamics (1) satisfies the first order differential equation:

[TABLE]

The construction of CS from the group representations is fully determined by a choice of $G$, $H$, ${\rho}$ and $\ket{0}$. Thus, varying some of these components we obtain different geometrisable Hamiltonians in sense of Defn. 1. For example, the minimal nilpotent extension of the Heisenberg group $\mathbb{G}{}$ (see (8)) allows to use Gaussian $\mathrm{e}^{-\pi\hslash Ex^{2}}$ with arbitrary squeeze parameter $E$ [Walls08, Gazeau09a] as a fiducial vector $\ket{0}$ for simultaneous geometrisation of all harmonic oscillators with different values of $m\omega$ [AlmalkiKisil18a] [Child14a]*§ 8.2.

Definition 3**.**

The method of order reduction employed in this letter consists of the following steps. For a group $G$ and its unitary irreducible representation ${\rho}$:

1. (1)

Chose a fiducial vector $\ket{0}$ which is annihilated by certain form of ${\rho}$, cf. (16); 2. (2)

Calculate [Kisil11c, Kisil13c, Kisil17a, AlmalkiKisil18a] the respective operator $\mathcal{C}$, cf. (18), which annihilates the image space of the CST (4). 3. (3)

Find all Hamiltonians of the form $H=F+A\mathcal{C}$, where $F$ is a first order PDE and $A$ is any operator.

The above method is quite general and is not limited to a particular group or representation. As an illustration, we use it in the following extension of the classic setup from Ex. 2:

• •

The Heisenberg group is extended to the minimal nilpotent step 3 group $\mathbb{G}{}$ defined in (8). Consequently our CS are parametrised by an extension of the classical phase space.

• •

The fiducial vector $\ket{0}$ is changed from the Gaussian to a cubic exponent (15), cf. Fig. 1. The later is the Fourier transform of an Airy wave packet [BerryBalazs79a] which are useful in paraxial optics [ATorre09a, ATorre14a].

For the found geometrisable Hamiltonians we can write explicit generic solutions through well-known integral transforms. The letter is concluded by a discussion a wider framework for our method and some its further usage is outlined.

2. Background

Here we briefly introduce the necessary background. It was already used in [AlmalkiKisil18a], which contains a detailed presentation and further references.

Let $\mathfrak{g}$ be the minimal nilpotent step $3$ Lie algebra spanned by $\{X_{1},X_{2},X_{3},X_{4}\}$ with the only non-vanishing commutators [CorwinGreenleaf90a, Kirillov04a, AlmalkiKisil18a, AllenAnastassiouKlink97, JorgensenKlink85, Klink94, BacryLevy-Leblond68a, GhaaniFarashahi17a]:

[TABLE]

The corresponding Lie group $\mathbb{G}{}\sim\mathbb{R}^{4}{}$ is three-step nilpotent and its elements will be denoted by

[TABLE]

The group can be physically interpreted as a central extension by $X_{4}$ of the Galilean group spanned by $X_{1}$, $X_{2}$ and $X_{3}$, see [AlmalkiKisil18a] for the related discussion. The rôle of central extension in quantisation is described, for example, in [AldayaGuerreroMarmo98a].

There are two important subgroups of $\mathbb{G}{}$: the centre $Z$ and the subgroups of elements $(x_{1},0,x_{3},x_{4})$, which is isomorphic to the Heisenberg group. On the other hand $\mathbb{G}{}$ is a subgroup of the Schrödinger group [Niederer72a, KalninsMiller74a, ATorre09a, Niederer73a, Wolf76a, AldayaGuerrero01a, AldayaCossioGuerreroLopez-Ruiz11b].

For two real parameters $h_{2}$ and $\hslash_{4}\neq 0$, we will use the unitary irreducible (UIR) representation of the group $\mathbb{G}{}$ in $\mathsf{L}_{2}{}(\mathbb{R}{})$ given by, cf. [Kirillov04a]*§ 3.3, (19) [AlmalkiKisil18a]:

[TABLE]

Note that ${\rho_{h_{2}\hslash_{4}}}(x_{1},0,x_{3},x_{4})$ coincides with the Schrödinger representation of the Heisenberg group [Kisil02e, Kisil10a, Kisil17a]. In particular, vectors $X_{1}$ and $X_{3}\in\mathfrak{g}$ correspond to momentum and position observables in the coordinate representation.

Let ${\rho}$ be a representation of $G$ and $\ket{0}$ be a joint eigenvector of operators ${\rho}(h)$ for all $h$ in subgroup $H$ of $G$:

[TABLE]

where $\chi$ is a character of $H$. Then, CST is completely determined by its values on $G/H$ [Gilmore1972a, Perelomov86]. Thus, for a section $\mathsf{s}:G/H\rightarrow G$ we define the induced coherent state transform (ICST) from a Hilbert space $\mathsf{H}{}$ to a space of functions $\mathsf{L}_{0}{}(G/H)$ by:

[TABLE]

For the subgroup $H$ being the centre $Z$ of $\mathbb{G}{}$, the representation ${\rho_{h_{2}\hslash_{4}}}$ (10) and the character $\chi(0,0,0,x_{4})=\mathrm{e}^{2\pi\mathrm{i}\hslash_{4}x_{4}}$ any function in $\mathsf{L}_{2}{}(\mathbb{R}{})$ satisfies the eigenvector property (11). Thus, for the respective homogeneous space $\mathbb{G}{}/Z\sim\mathbb{R}^{3}{}$ and the section $\mathsf{s}:\mathbb{G}{}/Z\to\mathbb{G}{};\ \mathsf{s}(x_{1},x_{2},x_{3})=(x_{1},x_{2},x_{3},0)$, ICST is:

[TABLE]

It intertwines ${\rho_{h_{2}\hslash_{4}}}$ with the representation on $\mathsf{L}_{2}{}(\mathbb{R}^{3}{})$:

[TABLE]

For a fixed $x_{2}\in\mathbb{R}{}$, the map $f\otimes\ket{0}\mapsto\tilde{f}(\cdot,x_{2},\cdot)$ is a unitary operator: $\mathsf{L}_{2}{}(\mathbb{R}{})\otimes\mathsf{L}_{2}{}(\mathbb{R}{})\rightarrow\mathsf{L}_{2}{}(\mathbb{R}^{2}{})$.

Define the cubic extension $\ket{0}$ of a Gaussian, see Fig. 1 by:

[TABLE]

where square–integrability of $\ket{0}$ over $\mathbb{R}{}$ requires that $E\hslash_{4}$ is strictly positive and $D$ is real. Physically, the parameter $E$ encodes a squeeze of CS [AlmalkiKisil18a]. State $\ket{0}$ (15) is interesting because it is a null-solution of the generic derived representation $\mathrm{d}{\rho_{h_{2}\hslash_{4}}}$ of $\mathfrak{g}$:

[TABLE]

Thus, this operator is a counterpart of the annihilation operator of canonical CS. It follows [Kisil11c, Kisil13c, Kisil17a, AlmalkiKisil18a], the image space $\mathsf{L}_{0}{}(\mathbb{G}/Z)$ of ICST (13) with the fiducial vector (15) satisfy to

[TABLE]

where

[TABLE]

which we call the analytic condition.

Another condition on $f\in\mathsf{L}_{0}{}(\mathbb{G}/Z)$ is generated by the Casimir element $C=X_{3}^{2}-2X_{2}X_{4}$ [CorwinGreenleaf90a, Kirillov04a]. The corresponding operator acts as a multiplication operator by $8\pi^{2}h_{2}\hslash_{4}$ on $\mathsf{L}_{0}{}(\mathbb{G}/Z)$, thus, for any $f\in\mathsf{L}_{0}{}(\mathbb{G}/Z)$ we have $\mathcal{S}f(x_{1},x_{2},x_{3})=0,$ where

[TABLE]

The relation (19) will be called the structural condition determined by the Casimir operator and is independent from a fiducial vector being used. Summing up, we conclude that $\mathsf{L}_{0}{}(\mathbb{G}/Z)$ is annihilated by elements of the left operator ideal $K$ generated by $\mathcal{C}$ (18) and $\mathcal{S}$ (19).

3. Geometrisation with Fourier–Fresnel and Fourier–Airy transforms

Recall, any two operators different by an element of the left operator ideal $K$ generated by $\mathcal{C}$ (18) and $\mathcal{S}$ (19) have equal restrictions to the space $\mathsf{L}_{0}{}(\mathbb{G}/Z)$. Among many equivalent operators we can look for a representative with desired properties, e.g. a first order differential operator, which geometrises dynamics in the sense discussed above.

In this section we present the complete characterisation of quadratic forms on the Lie algebra $\mathfrak{g}$ which admit geometrised dynamics by means of covariant transform with a fiducial vector (15).

Let ${\rho}$ be UIR of $\mathbb{G}{}$, for the respective derived representation $\mathrm{d}{\rho}$ of $\mathfrak{g}$ consider a general quadratic form:

[TABLE]

If $\mathrm{d}{\rho^{G}}$ admits a geometric dynamic in Airy-type CS $\ket{x_{1},x_{2},x_{3}}$ then there is a first-order differential operator, $H_{r}$ on $\mathsf{L}_{0}{}(\mathbb{G}{}/Z)$ such that $H_{r}-\mathrm{d}{\tilde{\rho}^{G}}$ is in the ideal generated by $\mathcal{C}$ (18) and $\mathcal{S}$ (19):

[TABLE]

Here $A,B,C,K$ and $F$ are certain coefficients, which are chosen to eliminate all possible second-order derivatives in $H=\mathrm{d}{\tilde{\rho}^{G}}$. In its turn, this depends on values $D$ and $E$ in the fiducial vector $\ket{0}$ (15) and the respective analyticity operator $\mathcal{C}$ (18).

Substituting the explicit expressions of the derived representation and $\mathcal{C}$ (18), $\mathcal{S}$ (19) into the relation (21) we can eliminate second derivatives in $H_{r}$ if:

[TABLE]

The last parameter $K$ is used to get imaginary coefficients in front of $\partial_{1}$ and $\partial_{3}$ to obtain a geometric action in the phase space parametrised by momentum-position variables $(x_{1},x_{3})$. Additionally to the above values (22), we have to apply the following restriction of the coefficients $a_{jk}$ on the quadratic form (20):

[TABLE]

The remaining coefficients $a_{11},a_{21},a_{13},a_{31},a_{32}$ and $a_{33}$ as well as parameters $D$ and $E$ are free variables. Thus, we have obtained the desired classification:

Proposition 4**.**

The Hamiltonian (20) can be geometrised over $\mathbb{G}{}/Z$ by Airy-type CS from the fiducial vector (15) if and only if coefficients $a_{jk}$ satisfy (23).

Note that even the case $D=0$ (that is a Gaussian as a fiducial vector) is more general than the classical result of Schrödinger from Example 2: the latter requires the exact match of the squeezing parameter $E=m\omega$ in the Hamiltonian and the CS. The larger group $\mathbb{G}{}$ can treat a harmonic oscillator through Gaussians with arbitrary squeeze, see [AlmalkiKisil18a] or presentation without groups in [Child14a]*§ 8.2.

4. Solving the geometrised equation

Geometrisable Hamiltonians described in Prop. 4 can be explicitly solved using the following steps:

1. (1)

Find the generic solutions of the analytic condition (18), which is a first-order PDE. Since it is Hamiltonian-independent, the obtained solution can be re-cycled. 2. (2)

Substitute the above analytic solution into the reduced (first-order) form of the Schrödinger equation for $H_{r}$ (21). A resulting dynamical equation is significantly simplified in analytic variables and admits an explicit generic solution. 3. (3)

Substitute the generic solution into the structural condition (19), this produces a second-order PDE solved by well-known integral representations.

Details of calculations can be found in [Almalki19a], we only indicate milestones here. The Hamiltonian-independent solution to the analyticity condition (18) is:

[TABLE]

The remaining steps will be considered on two specific examples satisfying requirements from Prop. 4.

We consider the matrix satisfying conditions (23):

[TABLE]

where $m$ is the mass. The respective Hamiltonian

[TABLE]

is the Weyl (symmetric) quantisation [Kisil02e, Kisil10a, Kisil17a] of the classical Hamiltonian

[TABLE]

The classical orbits in the phase space are presented on Fig. 2 (top).

The Shrödinger equation for adjusted $H_{r}$ (21) with substitution of (22) is a rather complicated first order PDE on $\mathbb{R}^{3}{}$. However, it significantly simplifies in analytic coordinates (24):

[TABLE]

The method of characteristics solves (28) to

[TABLE]

Finally, the structural condition (19) turns into the following Schrödinger equation of a free particle:

[TABLE]

for $\xi=\frac{1}{u_{2}}$, $\eta=\frac{u_{1}}{u_{2}}$. A generic solution of (30) is:

[TABLE]

for $g(s)$ determined by the initial conditions. Thus, the function $f$ (24) with substitution of $\phi$ (29) and $\psi$ (31), represents the dynamics of (26).

We can similarly treat the Hamiltonian

[TABLE]

which is different from (27) by a quadratic potential with $a=m\omega>0$. The classical orbits of this Hamiltonian are presented on Fig. 2 (bottom).

The analytic coordinates (24) simplify the respective first order PDE for (21); the generic solution is:

[TABLE]

The structural condition (19) reduces to a heat-like equation:

[TABLE]

A generic solution to (34) is given by the integral:

[TABLE]

where $k$ is determined by the initial value. Thus, the solution $f$ in form (24) is obtained by substitution of $\phi$ (33) and $\psi$ (35).

5. Discussion and conclusion

Hamilton equations describe classical dynamics through a flow on the phase space. This geometrical picture inspires numerous works searching for a similar description of quantum evolution starting from the symplectic structure [Kibble79a], curved space-time [Prugovecki82a, BrookePrugovecki85a, Zachos02a, KaramatskouKleinert14a, Tavernelli16a, HerranzDeLucasTobolski17a], differential geometry [HurleyVandyck09a, CarienaClemente-GallardoMarmo07a, Clemente-GallardoMarmo08a] and quantildezer–deltaquantildezer formalism [Zachos02a, CiagliaDiCosmoIbortMarmo17a], coherent states dynamics [ShalashilinBurghardt08]. A common objective of those researches is a conceptual similarity between fundamental geometric objects and their analytical counterparts, e.g. the symplectic structure on the phase space and derivations of operator algebras.

This letter is focused on more practical aims. We use an appropriate coherent state transform (CST) to reduce the order of the Schrödinger equation as described in Defn. 1. As specified in Defn. 3 the application of our method is completely determined by a choice of a group $G$, its subgroup $H$, a representation ${\rho}$ of $G$ in a vector space $V$ and a fiducial vector $\ket{0}\in V$. Once these elements are fixed one can give a characterisation of Hamiltonians admitting geometrisation. For those systems explicit solutions can be obtained through standard procedures. Mathematically our work is close to the framework of transmutations of PDE [Hersh75a, Carroll85a, KravchenkoTorba16a, KatrakhovSitnik18a, SitnikShishkina19a].

In Prop. 4, we described all Hamiltonians which can be geometrized by the minimal nilpotent extension $\mathbb{G}{}$ (8) of the Heisenberg group and the most general fiducial vector $\ket{0}$ (15) annihilated by the derived representation of the Lie algebra (16). This illustrates, that geometrisation of quantum mechanics may require a set which is significantly bigger than the classical phase space, despite of common anticipations [Zachos02a].

Our solutions manifest an advantage of the bigger group $\mathbb{G}{}$ over the Heisenberg group even in the simplified case $D=0$. The Hamiltonian (26) with $D=0$ describes a free quantum particle. Its solution is geometric in the well-known plane wave decomposition, which emerges from (31) if we set $D=E=0$ in analytic coordinates $(u_{1},u_{2})$ (24). However, canonical CS do not provide a transparent solution of a free particle through squeezed states with $E\neq 0$.

Similarly, for $D=0$ the Hamiltonian (32) reduces to the harmonic oscillator. Its geometric dynamic in terms of canonical CS is only possible for the particular value $E=m\omega$ of the squeezing parameter. Yet the larger group $\mathbb{G}{}$ allows to obtain a geometric dynamic for a range of $E$ as shown in [AlmalkiKisil18a], which also can be recovered from $\ket{0}$ (33) for $D=0$ in (24).

Hamiltonians (27) and (32) are similar to charged particle in a magnetic field. The Fourier transform, which swaps the coordinate and momentum pictures, relates $\ket{0}$ (15) to Airy wave packets [BerryBalazs79a, ATorre09a] which geometrizes the dual Hamiltonian:

[TABLE]

Quantisations of Hamiltonians (27), (32) and (36) may be relevant for paraxial optics [ATorre09a, ATorre14a]. The cubic parameter $D$ of the fiducial vector $\ket{0}$ (15) is dictated by the Hamiltonian, while the squeezing parameter $E$ is not fixed. However, the convergence of integrals (35) requires that $E\geq a=m\omega$.

The research can be continued in many directions, e.g.:

• •

Keeping the present group $\mathbb{G}{}$ and fiducial vector $\ket{0}$ (15) one may look for Hamiltonians beyond the quadratic forms (20).

• •

Keeping the group $\mathbb{G}{}$ look for another fiducial vectors, which will be null-solutions to more complicated analytic conditions than (16).

• •

Finally, many different groups can be considered instead of $\mathbb{G}{}$ with the Schrödinger group [Niederer72a, KalninsMiller74a, ATorre09a, ATorre14a, Niederer73a, Wolf76a, AldayaGuerrero01a, AldayaCossioGuerreroLopez-Ruiz11b] to be a very attractive choice.

Despite of pragmatic nature of this letter, our method and obtained results offer a deeper view on correspondence between classical and quantum mechanics. We use geometrisation, which is very close to that proposed in [CiagliaDiCosmoIbortMarmo17a]. However, our Defn. 1 has notable distinctions, e.g. it is compatible with important requirement of positive energies of ground states.

Although we were not focused on the broader context of geometrisation of quantum mechanics [Kibble79a, Prugovecki82a, BrookePrugovecki85a, Zachos02a, CarienaClemente-GallardoMarmo07a, KaramatskouKleinert14a, Tavernelli16a, HerranzDeLucasTobolski17a, CiagliaDiCosmoIbortMarmo17a, HurleyVandyck09a, Clemente-GallardoMarmo08a], this aspect is implicitly present in this work. For example, the symplectic structure flashes through the obtained solutions (29) and (33). Indeed, initially our CST (13) is parametrised by points $(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}{}$ and this odd-dimensional manifold cannot have a non-degenerate symplectic structure. However, solutions (29) and (33) are defined in terms of function $\psi$ of two complex variables. The complex structure in the first variable is compatible with the symplectic structure on the classical phase space with coordinate $(x_{1},x_{3})$. On the other hand, the complex structure in the second variable requires an analytic extensions, see discussion of this in [AlmalkiKisil18a].

Another prominent theme of geometrisation is a correspondence between classical and quantum mechanics. Hamiltonians (27) and (32) resemble particles in magnetic field with velocity-dependent forces, which do not produce a work. Classical dynamics determined by (27) and (32) in the configuration space is independent of $D$. with $D=0$ (no field). However, classical trajectories in the phase space for $D\neq 0$ are significantly different from the rigid rotation of the phase space familiar from the harmonic oscillator, see Fig. 2 (bottom). Correspondingly, the quantum ground state $\ket{0}$ (15) in the coordinate representation has the same density $\left|\ket{0}\right|$ for every $D$ and it coincides with the Gaussian, see Fig. 1. However, quantum phase factors of the ground state $\ket{0}$ (15) depends on $D$ and it can be visualised by interference of two ground states slightly displaced in space, see Fig. 3. This again illustrates the fundamental correspondence between classical momenta and quantum phases.

At a wider scope, the link between classical and quantum mechanics is a two-way road: geometrisation of quantum mechanics (discussed in this letter) is naturally complemented by “non-commutativisation” (as a form of quantisation) of classical theory. Indeed, since Dirac’s paper [Dirac26b] non-commutativity is deemed to be the main distinguishing assumption of quantum mechanics from the classical theory. The degree of non-commutativity is measured by a non-zero Planck constant $\hslash$ and the correspondence to classical mechanic is implemented through the semi-classical limit $\hslash\rightarrow 0$. It was only recently shown [Kisil12c, Kisil17a] that there is a natural formulation of classical mechanics with non-commutative observables and a non-zero Planck constant. This quantum-like form of classical mechanics was based on the Heisenberg group representations, which is linked it to the present work. Further research in this direction seems to be promising.

Acknowledgements: We are grateful to Prof. D. Shalashilin and anonymous referees for several useful suggestions.