(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider two non-identical, non-interacting particles of mass M that are constrained to move on a circle of radius R. Write down the Schrodinger equation for this problem and ﬁnd the eigenfunctions and energy levels of this system.

2. Relevant equations

(see below)

3. The attempt at a solution

I started by defining a coordinate system where the circle is centered at the origin of the x-y plane (z = 0). The particles don't interact and there is no external potential so V = 0. Also, using spherical coordinates, [tex] r = R, \ \theta = \frac{\pi}{2} [/tex] . I then wrote down the Schrodinger equation:

[tex] [-\frac{\hbar^2}{2m_{1}}\nabla^{2}_{1} - \frac{\hbar^2}{2m_{2}}\nabla^{2}_{2} + V]\psi(r,\theta,\phi) = E\psi(r,\theta,\phi) [/tex]

The Laplacian in spherical coordinates is:

[tex] \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}[\frac{1}{sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial}{\partial\theta}) + \frac{1}{sin^2\theta}\frac{\partial^2}{\partial \phi^2}] [/tex]

Since r and theta are fixed, the Schrodinger equation reduces to:

[tex] \frac{1}{2}(\frac{\hbar}{R})^2(m_1 + m_2)\frac{\partial^2\psi}{\partial\phi^2} + E\psi = 0 [/tex]

letting [tex] a = \sqrt{\frac{2R^2E}{\hbar^2(m_1 + m_2)}} [/tex] the solution of the Schrodinger equation is:

[tex] \psi = Ae^{a\phi} + Be^{-a\phi} [/tex]

This is where I get stuck. I assume the next step is to apply boundary conditions which will then give the energy levels. How do you do this in this situation? I tried:

[tex] \psi(0) = \psi(2\pi) [/tex]

[tex] \frac{\partial\psi(0)}{\partial\phi} = \frac{\partial\psi(2\pi)}{\partial\phi} [/tex]

but the results didn't make any sense. Any thoughts? Thank you in advance!

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# Two Particle Schrodinger Equation in Polar Coordinates

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