Particle in infinite potential well with variable walls

TL;DR

This paper explores a quantum phenomenon where a particle in an infinite potential well, after the barrier is removed, can be found with certainty at the opposite end at specific times, revealing counter-intuitive properties of quantum mechanics.

Contribution

It introduces a thought experiment analyzing the particle's behavior in a variable-well scenario, calculating specific times of certainty based on symmetry and system parameters.

Findings

Periodic certainty of particle location at specific times

Dependence of these times on particle mass and chamber size

Parameter ranges highlighting non-relativistic quantum mechanics limitations

Abstract

A Gedanken experiment is described to explore a counter-intuitive property of quantum mechanics. A particle is placed in a one-dimensional infinite well. The barrier on one side of the well is suddenly removed and the chamber dramatically enlarged. At specific, periodically recurring, times the particle can be found with probability one at the opposite end of the enlarged chamber in an interval of the same size as the initial well. With the help of symmetry considerations these times are calculated and shown to be dependent on the mass of the particle and the size of the enlarged chamber. Parameter ranges are given, where the non-relativistic nature of standard quantum mechanics becomes particularly apparent.

Equations

$$H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}.$$

$$\psi (x)=\sqrt{\frac{2}{\delta }}\sin \left(\frac{\pi x}{\delta }\right)\mathrm{with}{E}^{\prime }=\frac{{\pi }^2{\hslash }^2}{2m{\delta }^2}.$$

$${\varphi }_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)\mathrm{and}{E}_n=\frac{{\pi }^2{\hslash }^2{n}^2}{2m{L}^2},$$

$$\Psi (x,t)=\sum _{n=1}^{\infty }{a}_n{\varphi }_n(x)e^{ıE_{n}t/\hslash },$$

$$\displaystyle a_{n}=\frac{2}{\pi}\sqrt{\frac{\delta}{L}}\sin\Big{(}\frac{\pi n\delta}{L}\Big{)}\bigg{(}1-n^{2}\frac{\delta^{2}}{L^{2}}\bigg{)}^{-1}=\frac{2}{\pi}\,\,\frac{\sin(\pi n\eta)}{1-n^{2}\eta^{2}}\,\,\,\sqrt{\eta}\,\,$$

$${\int }_{L-\delta }^{L}dx|\Psi (x,t){|}^2=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\int }_{L-\delta }^{L}dx{a}_n{a}_m^{\ast }{e}^{ı(E_n-E_m)t}\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)\sqrt{\frac{2}{L}}\sin \left(\frac{m\pi x}{L}\right).$$

$$\displaystyle{\rm for\,\,\,\,n=2k}:\exp\Big{(}\frac{\pi^{2}\hbar(2k)^{2}}{2mL^{2}}\hat{t}\Big{)}=\exp(2\pi\imath)=1$$

$$\displaystyle{\rm for\,\,\,\,n=2k+1}:\,\,\,\,\exp\Big{(}\frac{\pi^{2}\hbar(2k+1)^{2}}{2mL^{2}}\hat{t}\Big{)}=\exp(\pi\imath)=-1,$$

$$\hat{t}:=\frac{2m{L}^2}{\pi \hslash },$$

$$\hat{\delta }\sim$$

$$\hat{L}\sim$$

$$\hat{m}\sim$$

Full text

Quantum mechanics contains a tangle of unresolved foundational issues, involving the role of probability, the measurement process, the reduction of the state vector, and the relation to relativity, and provides ample topics for investigation. In this note a simple Gedanken experiment is considered to study an implication of the lack of upper limit for the speed of information propagation in the Schrödinger equation - a diffusion equation.

A particle is put in a one-dimensional infinite well of size $\delta$. One of the barriers is removed at time $t=0$ and the particle is able to spread out over a larger well of size $L$. After a well-defined time, dependent on the size of the post-expansion well and the particle mass, the wave function will again be concentrated in an interval of the initial size of the box, but this time at the other edge of the enlarged well. The instantaneous well expansion has been extensively studied by Aslangulasa and in papers cited therein.

A particle of mass $m$, trapped in a one-dimensional infinite square well of width $\delta$, is described by the Hamiltonian

[TABLE]

The boundary conditions are such that the wave function $\psi(x)$ vanishes at $x=0$, $x=\delta$ and outside the well. The starting wave function $\psi(x)$ is chosen to be

[TABLE]

For more details about quantum mechanics in an infinite well see the paper by Bender et al.bbm .

After removing the barrier at $t=0$ the infinite potential is resized and reaches from the origin to $L$ with $L>\delta$. The Hamiltonian stays unchanged, but the boundary conditions are replaced by the wave function vanishing at $x=0$ and $x=L$. The new stationary states and the corresponding energy eigenvalues are

[TABLE]

where $n$ is any whole number. The new wave function in the enlarged well depends on the choice of the starting eigenfunction in the smaller well $\psi(x)$ and has the form

[TABLE]

with the transition probability between the original test function and the new basis at the time the barrier is raised of the form

[TABLE]

with $\eta$ defined as $\delta/L$. The probability of finding the particle in the interval $[L-\delta,L]$ at any time $t>0$ is

[TABLE]

Instead of evaluating the sum, we shall calculate the probability at specific times, where symmetry properties come into play. The simplest case is to choose $\hat{t}$ in such a way that $\exp(\imath E_{n}\hat{t}/\hbar)$ has for even and odd $n$ always the values $+1$ and $-1$, respectively:

[TABLE]

and

[TABLE]

where $k$ is any natural number. The smallest possible solution is

[TABLE]

others can be generated by multiplying the original result with any odd whole number. The time $\hat{t}$ could be called a ‘revival time’, since the reflected wave function in its original form is reconstituted in its entirety at a new time and place. At time $\hat{t}$ the wave function is concentrated solely in the interval $[L-\delta,L]$, since it corresponds, except for a overall phase, exactly to the mirror-symmetric starting wave-function at time $t=0$ in the interval $[0,\delta]$. In the rest of the box, corresponding to the interval $[0,L-\delta]$, the wave function is zero. A more detailed explanation is given next. Odd and even elements of the $\sin(n\pi x/L)$ basis functions have different behaviour at the beginning and the end of the interval. The even basis functions change sign at the two extreme ends of the interval, i.e. $\sin(2k\pi x/L)=-\sin(2k\pi(L-x)/L)$, whereas all the odd basis functions have the same sign, i.e. $\sin((2k+1)\pi x/L)=\sin((2k+1)\pi(L-x)/L)$. At time $\hat{t}$ the energy pre-factor $\exp(\imath E_{n}\hat{t}/\hbar)$ toggles between plus and minus one for even and odd terms, which compensates the sign change of the basis elements, resulting in $|\psi(x)|^{2}=|\Psi(x,0)|^{2}=|\Psi(L-x,\hat{t})|^{2}$.

At time $\hat{t}$ the particle has reconstituted itself in the $\delta$-interval at the far end of box. If at this time light has not reached the other end of the well, i.e. $L/c\gg\hat{t}$, and $\delta$ is small compared to $L$, then this implies a contradiction between relativity with its limit on the velocity of signals and the prediction of quantum mechanics. The inequality above can be rewritten as $\hbar\pi\gg 2mLc$. One might be interested to inquire, if there are parameter ranges for which the inequality is satisfied. As an example, one might select for the particle mass and the two length scales involved

[TABLE]

The time $\hat{t}$ in this particular case is of the order of $10^{-22}s$, while the time required for light to transverse the distance $\hat{L}$ is around $10^{-13}m/(3\times 10^{8}m/s)\sim 10^{-21}s$. This suggests a natural barrier at which quantum mechanics scrapes against the relativistic limit, i.e. fails to comply with restrictions placed by relativity, since $\hat{t}$ is smaller by a factor of roughly ten than the time it takes light to transverse across $\hat{L}$. One could contemplate carrying out such an experiment to confirm the location of the particle at the time specified. Naturally, this is simpler to propose than to realise.

In this note a limiting case was explored to understand the superluminal aspect of non-relativistic quantum mechanics. This was studied with the help of a Gedanken experiment and is possibly instructive, since the notion that super-luminal communication has been banished from non-relativistic quantum mechanics is assumed by some in a cavalier fashion.

Idealisations have been liberally employed. Three in particular spring to mind: First, the removal of the barrier to allow the expansion of the wave function onto the whole length of the interval happens instantaneously. Compressing an extended process into a point in time, like the occurrence of an instantaneous measurement, is not an unusual calculation tool in quantum mechanics. The case of a moving wall has been analysed extensively, e.g. see the recent paper by Cooney cooney , which includes a review of different approaches going back to the work by Hill and Wheeler in the 1950s. A challenge is to find solutions, where the energies stay real. The removal of a wall differs from moving a wall with finite velocity, but similarities exist. Second, the energy of the particle post barrier removal does not differ from the initial energy, but the energy variance changes dramatically. How does the situation change, if the barrier removal is handled in a more realistic way? Third, the choice of parameters to meet the constraints of the inequality above are not necessarily experimentally feasible.

The approach described has some similarities to Maxwell’s fishpond kisler from classical physics. In both proposals an excitation, e.g. pebble thrown in a pond, is reconstituted after a well-defined time in a specific, but separate, place.

This elicits the question, what applications one could devise, besides checking the validity of quantum mechanics. One could use such a device to transport besides information also energy. A series of these boxes applied in sequence could be used to amplify the effect. Experimental relevance will be considered separately, since the emphasis of this note is on simplicity not applicability.

The paper by Aslangulasa gives a detailed analysis of the instantaneous well expansion, while this note tries to point out a potential implication for quantum mechanics.

While being cognisant of the various idealisations employed, the Gedanken experiment presented puts bounds, dependent on the post-expansion size of the well and the mass of the test particle, on the applicability of non-relativistic quantum mechanics, if super-luminal communication is to be avoided.

Helpful comments by L.P. Hughston and an email by Prof. Aslangul drawing my attention to his paperasa are gratefully acknowledged.

1. (1) C. Aslangul, J. Phys. A 41, 075301 (2008).

(2) C. M. Bender, D.C. Brody& B.K. Meister, Proceedings of the Royal Society London A461, 733-753 (2005),  arXiv:quant-ph/0309119.

(3) K. Cooney,  arXiv:1703.05282.

(4) P. Kinsler et al., Eur. J. Phys. 33 1737 (2012),  arXiv:1206.0003.