Quantum Particle in an Accelerating Box — Relativistic Treatments

Quantum Particle in an Accelerating Box — How to Treat the Problem

There isn’t a single “one-size” answer because what you do depends on how the box moves. Below are practical regimes from easiest to most realistic, with LaTeX-compatible equations.

1) Box at constant relativistic velocity (no acceleration)

If the box moves inertially at speed $v$ relative to the lab, go to the box’s rest frame, solve the usual problem, then transform back.

In the box rest frame (proper length $L_0$ ):

Non-relativistic particle (valid when $\langle p^2\rangle \ll m^2 c^2$ ):

$k_n=\frac{n\pi}{L_0},\qquad E_n^{\mathrm{NR}}=\frac{\hbar^2 k_n^2}{2m}=\frac{n^2\pi^2\hbar^2}{2mL_0^2}.$

Relativistic particle (use Klein–Gordon/Dirac dispersion with the same $k_n$ ):

$E_n^{\mathrm{rel}}=\sqrt{(mc^2)^2+(\hbar c\, k_n)^2}.$

Transform back to the lab: the box length is $L=L_0/\gamma$ with $\gamma=1/\sqrt{1-v^2/c^2}$ ; momenta and frequencies Lorentz-transform. Operationally, compute in the comoving frame and transform observables at the end.

2) Slow/weak acceleration: instantaneous comoving frame (adiabatic NR QM)

If the acceleration $a(t)$ is small enough that levels don’t mix appreciably over a level-spacing time, work in the instantaneous rest frame of the box. Keep the walls at fixed $[0,L_0]$ (assuming Born rigidity; see §4). The Schrödinger equation acquires an inertial “gravity” term:

$i\hbar\,\partial_t\psi(x,t)=\left[-\frac{\hbar^2}{2m}\partial_x^2+V_{\text{box}}(x)-m\,a(t)\,x\right]\psi(x,t),$

with infinite walls at $x=0,L_0$ .

A uniform acceleration adds a linear potential $-m a x$ , shifting energies and skewing eigenfunctions.
Level shifts remain small if $\displaystyle \frac{m a L_0}{\Delta E_n}\ll 1$ (with $\Delta E_n$ a local level spacing).
Across the box, clocks redshift by $\Delta \omega/\omega \approx a\,\Delta x/c^2$ (equivalence principle).
If the wall separation changes in time $L(t)$ , map to a fixed domain via $y=x/L(t)$ and use scaling/Lewis–Riesenfeld invariants. Generic $L(t)$ causes mode mixing and non-adiabatic transitions.

3) Relativistic acceleration: Rindler frame and field-theoretic treatment

For appreciable accelerations (or long proper times “approaching $c$ ” during the run-up), use a relativistic description in accelerating coordinates (Rindler).

Rindler coordinates for constant proper acceleration $a$

$\begin{aligned} ct &= \left(\xi+\frac{c^2}{a}\right)\sinh\!\left(\frac{a\tau}{c}\right),\\ x &= \left(\xi+\frac{c^2}{a}\right)\cosh\!\left(\frac{a\tau}{c}\right)-\frac{c^2}{a}, \end{aligned}$

Metric:

$ds^2= -\bigl(1+\tfrac{a\xi}{c^2}\bigr)^2 c^2\, d\tau^2 + d\xi^2.$

A Born-rigid cavity of proper length $L_0$ spans $\xi\in[\xi_1,\xi_2]$ with $\xi_2-\xi_1=L_0$ . Different points have different proper accelerations

$a(\xi)=\frac{a}{\,1+a\xi/c^2\,}.$

Quantize the appropriate relativistic field (Klein–Gordon or Dirac) inside the cavity with boundary conditions at $\xi_{1,2}$ . Stationary modes are defined with respect to Rindler time $\tau$ ; their frequencies vary across the cavity due to gravitational redshift.

Phenomena beyond NR QM

Unruh effect: an accelerated observer sees a thermal bath at

$T_U=\frac{\hbar\,a}{2\pi k_B c}.$

Coupling to this bath induces excitations and decoherence in the accelerated frame.
Dynamical Casimir / moving-mirror particle creation: time-dependent boundaries mix positive and negative frequency modes (Bogoliubov coefficients), creating quanta in the cavity.
No global stationary state during non-uniform acceleration; evolve the state in time and track mode mixing.

Computation steps (relativistic regime)

Choose KG/Dirac as appropriate.
Specify the cavity worldlines (Born-rigid hyperbolae are standard).
Expand the field in instantaneous cavity modes; compute Bogoliubov coefficients as the walls accelerate.
Evolve the state; extract observables in the Rindler frame or transform back to Minkowski.

4) Note on Born rigidity

Special relativity forbids perfectly rigid bodies with uniform acceleration. To keep the proper length $L_0$ fixed in the box’s momentary rest frame while speeding up, the rear must have slightly larger proper acceleration than the front. To leading order, the required difference scales as

$a_{\text{back}}-a_{\text{front}}\;\sim\; \frac{a^2 L_0}{c^2}.$

Ignoring this turns the problem into a moving-wall problem with changing $L(t)$ , which is different physics.

5) Quick decision tree

Constant $v$ : solve in rest frame; Lorentz-transform observables.
Weak/slow acceleration, NR particle: instantaneous comoving box with linear inertial potential $-m a(t) x$ ; adiabatic if $\displaystyle \frac{m a L_0}{\Delta E_n}\ll 1$ .
Strong/relativistic acceleration or long boxes: Rindler cavity + relativistic field theory; expect redshifts, Unruh temperature, and possible particle creation from moving boundaries.

If you share your intended regime ( $L_0$ , particle mass, the scale of $a$ , and whether you want NR vs. relativistic accuracy), I can write the explicit mode basis and, if needed, the Bogoliubov evolution for the accelerating cavity.