- #1
fluidistic
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Homework Statement
I must get the first eigenvalues of the time independent Schrödinger's equation for a particle of mass m inside a cylinder of height h and radius a where ##h \sim a##.
The boundary conditions are that psi is worth 0 everywhere on the surface of the cylinder.
Homework Equations
##-\frac{\hbar ^2}{2m} \triangle \psi =E \psi##.
Laplacian in cylindrical coordinates.
The Attempt at a Solution
I've used separation of variables on the PDE, seeking for the solutions of the form ##\psi (\rho, \theta , z)=R(\rho) \Theta (\theta ) Z(z)##.
I reached that ##\frac{Z''}{Z}=\text{constant}=-\lambda ^2##. Assuming that the Z function is periodic and worth 0 at the top and bottom of the cylinder, I reached that ##Z(z)=B \sin \left ( \frac{n\pi n}{h} \right )## where n=1,2, 3, etc.
Then I reached that ##\frac{\Theta''}{\Theta} = -m^2## where m=0, 1, 2, etc (because it must be periodic with period 2 pi). So that ##\Theta (\theta )=C \cos (m \theta ) +D \sin (m \theta)##.
Then the last ODE remaining to solve is ##\rho ^2 R''+\rho R'+R \{ \rho ^2 \left [ \frac{2mE}{\hbar ^2} - \left ( \frac{n\pi}{h} \right )^2 \right ] -m^2 \}=0##. This is where I'm stuck.
It's very similar to a Bessel equation and Cauchy-Euler equation but I don't think it is either. So I don't really know how to tackle that ODE. Any idea? Wolfram alpha does not seem to solve it either: http://www.wolframalpha.com/input/?i=x^2y%27%27%2Bxy%27%2By%28x^2*k-n^2%29%3D0.