I was searching for the eigensolutions of the two-dimensional Schrödinger equation

$$\mathrm{i}\hbar \partial_t \mid \psi \rangle = \frac{\mathbf{p}^2}{2m_e}\mid \psi \rangle + V \mid \psi \rangle$$

where the potential is given by $$V(\rho, \varphi)=\begin{cases} V_1 & \rho < R \\ -V_2 & \rho \geq R \end{cases}$$

using a space representation and cylindrical coordinates, $V_i \geq 0$.

I would be happy if someone could point me to a reference or even give the solution here.

Thank you in advance


Request to close the question

As I can see in the comments, questions of this kind seem to be inappropriate.
The eigensolutions are given by something like $$\psi_m(\mathbf{r},t)=e^{\mathrm{i}(m\varphi-\omega_m t)}\begin{cases} a_m J_m (k_{m,1} \rho) & \rho < R \\ b_m K_m (k_{m,2} \rho) & \rho \geq R \end{cases}$$ where the $a_m$ and $b_m$ can be calculated from the steadiness of $\psi$ and its spatial derivative in $R$. Furthermore, $k_{m,1/2} = \frac1\hbar \sqrt{\pm\,2m_e(\hbar\omega_m - V_{1/2})}$.

I am sorry for any inconvenience.


closed as off topic by David Z Jan 19 '11 at 0:18

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  • $\begingroup$ Write $p^2 = (d/dx)^2$ in cylindrical co-ordinates. Assume $|\Psi>$ can be written as a product $ \Phi(\rho)\chi(\phi)Z(z)$ and the equation splits into three. Standard separation of variables. $\endgroup$ – user346 Jan 11 '11 at 15:06
  • $\begingroup$ What do you mean by "solution". Do you actually need stationary states? $\endgroup$ – Kostya Jan 11 '11 at 15:07
  • $\begingroup$ @space_cadet: thank you for the insight. I think I am capable of solving the system myself and it is clear that solutions will have the form $J_n(k\rho)e^{\mathrm{i} (n \varphi - \omega_n t)}$ e.g. for $\rho < 0$. But I am safe to assume the solution is already known, so a nice reference is what I am looking for :) $\endgroup$ – Robert Filter Jan 11 '11 at 15:11
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    $\begingroup$ At this point it seems like just a math question, really - you're basically just looking for the solution of a known differential equation. All the physics is done. $\endgroup$ – David Z Jan 11 '11 at 15:46
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    $\begingroup$ This is a standard homework problem for an undergrad QM course. Do we really want questions like this and answers to them on MO? $\endgroup$ – pho Jan 11 '11 at 16:37