How Does a Particle's Energy State Change in a Non-Uniform 2D Corridor?

[hilbert2]

Suppose we have a particle of mass $m$ moving freely in the xy-plane, except for being constrained by hard walls to have $-L/2 < y < L/2$. Now, the energy eigenstates would be something like

$\psi (x,y) = C \psi_x (x) \psi_y (y) = C e^{-ikx}\cos\left(\frac{n\pi y}{L}\right)$,

where $n$ would have to be an odd integer, it seems. So, now the total energy $E$ would be a sum from the free-particle motion in the x-direction and particle-in-box motion in y-direction.

Now, suppose that the 2d "corridor" the particle is moving in is not of constant width $L$, but has some width $L_1$ when $x < x_1$ and other width $L_2$ when $x > x_2$. Also, between $x_1$ and $x_2$ the width increases linearly from $L_1$ to $L_2$.

In the region $x < x_1$ the energy eigenfunctions would be like $\psi (x,y) = C_1 e^{ik_1 x}\cos\left(\frac{n\pi y}{L_1}\right)$ and in region $x > x_2$ they would be like $\psi (x,y) = C_2 e^{ik_2 x}\cos\left(\frac{n\pi y}{L_2}\right)$. If $L_2 > L_1$, it would seem to be required that $k_2 > k_1$ to make the total energy eigenvalue the same in both regions despite the ground-state energy of the particle-in-box like y-motion being smaller when the width of the corridor is larger.

Questions:

1. If the distance between $x_1$ and $x_2$ is very large compared compared to the difference $| L_1 - L_2 |$, can the wavefunction between $x_1$ and $x_2$ be approximated by something like

$\psi (x,y) = Ce^{ik(x)x}\cos\left(\frac{n\pi y}{L(x)}\right)$,

where the functions $L(x)$ and $k(x)$ change linearly between $x = x_1$ and $x = x_2$ ?

2. Is this kind of a problem described in any textbook or journal article ?

 

[mfb]

1. I would expect so, but I'm not sure if that approximation tells you what you want, because it doesn't reflect the dynamics of the particle. The slope transfers momentum between the two directions.

What about the odd states, by the way (sine instead of cosine)?

 

[hilbert2]

Oh, yes, there are also the odd states. I was thinking of this as some kind of an exercise I invented myself. I think it's possible to show that the approximation approaches an exact eigenstate when $\left|\frac{L_2 - L_1}{x_2 - x_1}\right| \rightarrow 0$, by showing that the energy standard deviation approaches zero at that limit.