Ground state energy of a particle-in-a-box in coordinate scaling

[hilbert2]

TL;DR Summary

How does the ground state energy of one particle in n-dimensional box change in uniform coordinate scaling? What happens if the box has a finite depth in potential energy units?

The energy spectrum of a particle in 1D box is known to be

$E_n = \frac{h^2 n^2}{8mL^2}$,

with $L$ the width of the potential well. In 3D, the ground state energy of both cubic and spherical boxes is also proportional to the reciprocal square of the side length or diameter.

Does this apply for an n-dimensional box of any shape, including those that are not simply connected? And if the depth of the potential well is not infinite, does it approach some other scaling law in the limit of small well depth? The first of these claims should probably be true just because of the scaling property of the n-dimensional kinetic energy operator and Laplacian in a uniform coordinate scaling.

Edit: And, if there's a finite potential well of side length $L$, depth $V_0$, and $d$ is the approximate distance the particle in the ground state is able to sink inside the potential walls. will the $E_0$ be proportional to $L^{-2}$ if $L$ is made much larger than $d$? I can solve this numerically myself but it would be interesting to see a source where it has been discussed before.

 

[anuttarasammyak]

Summary:: How does the ground state energy of one particle in n-dimensional box change in uniform coordinate scaling? What happens if the box has a finite depth in potential energy units?

Does this apply for an n-dimensional box of any shape, including those that are not simply connected?

In 3D, for rectangle box
$$E_{n_x,n_y,n_z}=\frac{h^2}{8m}(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2})$$

 

[hilbert2]

Yes. that's the spectrum of a 3D parallelepiped potential well, but I was thinking about one of any shape, like elliptical or half-sphere.

Edit: Solving numerically the ground states of several potential wells of same finite but large depth $V_0$, it seems that the $E_0$ decreases faster than $L^{-2}$ (possibly exponentially) when $L\rightarrow\infty$.

 

[anuttarasammyak]

Summary:: How does the ground state energy of one particle in n-dimensional box change in uniform coordinate scaling? What happens if the box has a finite depth in potential energy units?

And if the depth of the potential well is not infinite, does it approach some other scaling law in the limit of small well depth?

For 1D well potential Shrodinger equation is written as
$$\frac{d^2\psi}{dx^2}+(\epsilon-v)\psi=0$$
where x,$\epsilon$, v are dimensionless parameters, x=X/L, L is width of the well, and $\epsilon$=E/e, v=V/e
$$e=\frac{\hbar^2 }{2mL^2}$$
So limit to infinite well is in exact mathematical sense
$$\frac{\hbar^2 }{2mL^2V}\rightarrow +0$$
Wide well or heavy mass does similar effect.

 

[hilbert2]

Thanks. I'll have to calculate the $E_0$ of several potential wells of same finite depth $V_0$ but different $L$. Then I'll plot the $E_0 (L)$ function in log-log coordinates to see how small $V_0$ can be without the result deviating from $L^{-2}$ behavior. Any power law graph looks like a straight line when drawn in double logarithmic coordinates, so that's a way to find out if $E_0 (L)$ decreases exponentially and not like $L^{-a}$ with $a>0$ for some value of $V_0$.