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Schrödinger's equation for hydrogen atom in free space can be easily solved by switching to center of mass frame, introducing reduced mass and separating variables in the resulting 3D problem. But what if we confine electron and proton to some sort of a box with homogeneous Dirichlet boundary conditions?

Is there any shape (cubic, spherical, any shape with finite diameter) for such a box, which would allow the problem to be solved analytically — or at least separate the variables so as to get a finite set of ODEs?

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  • $\begingroup$ hi! Assume a spherical, perfectly reflecting mirror centered on the H nucleus - see figure in the picture. Is this problem solvable. The boundary conditions should be that the wave-function vanishes on the mirror. Since the atom is neutral, I assume that no charges appear on the mirror surface - am I right? I don't see a reason why the Schrodinger equation shouldn't be separable by variables. Do you see such a reason? $\endgroup$ – Sofia Jan 23 '15 at 15:37
  • $\begingroup$ Do you mean a box around the proton, or a box that both particles bounce around in? For the former, a spherical box will give the spectrum in terms of the zeros of Laguerre polynomials. The latter will entangle the relative motion with the COM motion, which will not be analytically treatable. $\endgroup$ – Emilio Pisanty Jan 23 '15 at 15:41
  • $\begingroup$ @EmilioPisanty : good question, box around the atom, not around the nucleus. I rely on that the atom is neutral (I hope that I can rely on this) otherwise on the internal side of the mirror will appear a negative charge and the things will complicate. $\endgroup$ – Sofia Jan 23 '15 at 15:48
  • $\begingroup$ @Sofia Note that the 'wall' in the OP is a surface at which homogeneous Dirichlet boundary conditions (i.e. $\psi=0$) apply. A real mirror will be much more complicated and you will have interactions, even for a neutral atom; these are known as Casimir-Polder interactions. $\endgroup$ – Emilio Pisanty Jan 23 '15 at 15:52
  • $\begingroup$ @EmilioPisanty : very well, then what you suggest? Please fill free to replace my suggestion, and I will adapt the picture. $\endgroup$ – Sofia Jan 23 '15 at 16:07

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