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I am trying to solve for the wavefunction for a particle inside a 2 dimensional disk of radius r_0.

The conditions are:

[tex]\int^{2\pi}_0 \int^{r_0}_0 |\psi|^2\, r \,dr\, d\theta = 1[/tex]

Then I tried separations of variables (set psi = R(r)Theta(theta)) to solve the Schrodinger equattion. didnt' really understand how it worked, but tried it anyway.

I got [tex]R(r)=A_r r^{i\sqrt{2mE_r}\over\hbar}+B_r r^\frac{-i \sqrt{2mE_r}}{\hbar}[/tex] and [tex]\Theta(\theta) = A_\theta \cos \frac{\sqrt{2mE_\theta}}{\hbar} \theta) +B_\theta \sin \frac{\sqrt{2mE_\theta}}{\hbar} \theta) [/tex]

(substituting [tex]K_i = \frac{\sqrt{2mE_i}}{\hbar}[/tex]) I plugged the wave function to the probability integral. After painstakingly doing that integral I got:

[tex]\left [ |A_\theta|^2 \left ( \pi+\frac {\sin 4\pi K_\theta \theta}{2K_\theta} \right ) +|B_\theta|^2 \left ( \pi-\frac{\sin 4\pi K_\theta \theta}{2K_\theta}\right ) \right ] |A_r^2+B_r^2| \ln r_0 \left ( \frac{(-1+2 \ln r_0)r_0^2}{2}\right ) \left ( \frac{r^{2iK_r+2}}{2iK_r+2}-\frac{r^{-2iK_r+2}}{2iK_r-2} \right ) = 1[/tex]

Now i can't solve for the energy eigenvalues because they are stuck in the deep parts of the equation (I set E = E_r + E_theta). Also where is the quantization? I know I screwed up somewhere.

The conditions are:

[tex]\int^{2\pi}_0 \int^{r_0}_0 |\psi|^2\, r \,dr\, d\theta = 1[/tex]

Then I tried separations of variables (set psi = R(r)Theta(theta)) to solve the Schrodinger equattion. didnt' really understand how it worked, but tried it anyway.

I got [tex]R(r)=A_r r^{i\sqrt{2mE_r}\over\hbar}+B_r r^\frac{-i \sqrt{2mE_r}}{\hbar}[/tex] and [tex]\Theta(\theta) = A_\theta \cos \frac{\sqrt{2mE_\theta}}{\hbar} \theta) +B_\theta \sin \frac{\sqrt{2mE_\theta}}{\hbar} \theta) [/tex]

(substituting [tex]K_i = \frac{\sqrt{2mE_i}}{\hbar}[/tex]) I plugged the wave function to the probability integral. After painstakingly doing that integral I got:

[tex]\left [ |A_\theta|^2 \left ( \pi+\frac {\sin 4\pi K_\theta \theta}{2K_\theta} \right ) +|B_\theta|^2 \left ( \pi-\frac{\sin 4\pi K_\theta \theta}{2K_\theta}\right ) \right ] |A_r^2+B_r^2| \ln r_0 \left ( \frac{(-1+2 \ln r_0)r_0^2}{2}\right ) \left ( \frac{r^{2iK_r+2}}{2iK_r+2}-\frac{r^{-2iK_r+2}}{2iK_r-2} \right ) = 1[/tex]

Now i can't solve for the energy eigenvalues because they are stuck in the deep parts of the equation (I set E = E_r + E_theta). Also where is the quantization? I know I screwed up somewhere.

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