A Particle moving on a constrained path

[hilbert2]

There seem to be many kinds of examples where the behavior of a quantum particle having been constrained to move on a curve or surface is investigated. The simplest is the case of a particle on a circular path or a spherical surface, where the energy eigenstates are equal to the angular momentum eigenstates.

Reading some article about a particle on an elliptical path gave me an idea of constraining an electron on a squircle, or a square with rounded corners, and then gradually making the corners less and less round, approaching a situation where it's very close to an actual square with sudden 90 degree turns at its corners.

Is it likely that this kind of a system would converge towards some kind of a sensible solution at the limit of "a particle on the circumference of a square" ? A sudden turn in the direction of motion sounds quite strange even in the context of classical mechanics, and in the QM case it's difficult to have any intuition about what would happen in this situation.

Edit: The simplest way to model this would probably be to construct a piecewise defined Hamiltonian where the rounded square is made of straight line segments and quarter-circles, with the radius of curvature of the corners becoming smaller as it approaches an actual square. Then the kinetic energy should be written as a function of $d\theta/dt$ (with $\theta$ the polar angle measured from the center of the rounded square), so that the quantum Hamiltonian could be written. A curious thing about this system is that knowledge about the momentum of the particle would actually also give knowledge about its position; if it has a nonzero x-component of momentum, it could not be located on the part of the square that is normal to the x-axis.

 

[hilbert2]

I calculated this for a simpler situation where a particle coming from the direction of negative x-axis takes a quarter-circular 90 degree turn and continues to the direction of positive y-axis. If the radius of curvature of the turn is $R_c$, the circular, vertical and horizontal parts of the path can be written in parametric form:

1. $\theta\in[-\pi/2,0] \rightarrow (x,y) = (1-R_c +R_c \cos(\theta), -1 +R_c +R_c \sin(\theta))$ (circular part)
2. $\theta\in[0,\pi/2] \rightarrow (x,y) = (1,-1+R_c +R_c \tan(\theta))$ (vertical part)
3. $\theta\in[-\pi,-\pi/2] \rightarrow (x,y) = (1-R_c -R_c /\tan(\theta),-1)$ (horizontal part)

(Sorry for the non-dimensionalized position variables) For $R_c = 0.5$ this can be plotted in Mathematica by command

"ParametricPlot[{HeavisideTheta[-x] HeavisideTheta[Pi/2 + x] {1 - 0.5 + 0.5 Cos[x], -1 + 0.5 + 0.5 Sin[x]}, {1, -1 + 0.5 + 0.5 Tan[x]}, {1 - 0.5 - 0.5/Tan[x], -1}}, {x, -Pi, Pi/2}, PlotRange -> {{-1, 1.5}, {-1.5, 1}}]".

As the solution for the circular part is the same as for a free particle on a ring, we have a wavefunction

$\psi_1 (\theta) = Ae^{\pm ik_c \theta}$, where $k_c$ and $A$ are constants.

On the vertical part the wave function is

$\psi_2 (\theta) = Be^{iky} + Ce^{-iky} \\ = B\exp\left[ik (R_c \tan(\theta) + R_c -1)\right] + C\exp\left[-ik (R_c \tan(\theta) + R_c -1)\right]$

and on the horizontal part it is

$\psi_3 (\theta) = De^{-ikx} + Ee^{-ikx} \\ = D\exp\left[ik(1-R_c -R_c /\tan(\theta))\right] + E\exp\left[-ik(1-R_c -R_c /\tan(\theta))\right]$.

The energy calculated at the circular part is $E = \frac{\hbar^2}{2MR_{c}^{2}}\dot k_c (k_c + 1)$,

and at the linear parts it is $E = \frac{\hbar^2 k^2}{2M}$.

$M$ is the mass of the particle. Now the two energies must be equal, and the wave function must be continuously differentiable at the border points of the three regions, which sets some conditions for the parameters in the wave functions.

Now it wouldn't seem that anything strange happens when the $R_c$ is made to approach zero, representing the turn approaching a sharp 90 degree corner, so it looks like the particle doesn't care at all if there are discontinuous changes of direction on the path it is constrained to move on.