# Solution for the Finite 2D Potential Well - Rotational Symmetry [closed]

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.

Sincerely

Robert

### 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

Questions on Physics Stack Exchange are expected to relate to physics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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. – user346 Jan 11 '11 at 15:06
• What do you mean by "solution". Do you actually need stationary states? – Kostya Jan 11 '11 at 15:07
• @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 :) – Robert Filter Jan 11 '11 at 15:11
• 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. – David Z Jan 11 '11 at 15:46
• This is a standard homework problem for an undergrad QM course. Do we really want questions like this and answers to them on MO? – pho Jan 11 '11 at 16:37