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## 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?

## Main Question or Discussion Point

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.

##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.

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