Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,547 questions , 1,198 unanswered
4,552 answers , 19,366 comments
1,470 users with positive rep
411 active unimported users
More ...

Really could use some advice with this PDE

+ 2 like - 0 dislike
62 views

so I'm faced with a partial differential equation that I have been able to solve only in one limiting case. I have asked the mathematics stack exchange twice for literally any help whatsoever to no avail, so I thought I might finally appeal to the experts on PDEs ;). The equation arises out of quantum mechanics. Specifically, it is a differential form of a condition for a specific system I'm studying to minimize the Robertson-Schrödinger relation (sometimes called the "generalized uncertainty principle") for angular momentum. As a result, the solutions are wavefunctions and the usual boundary conditions apply (i.e. fading out at the boundaries (in this case infinity), normalized, etc.) I can give you a semi-coordinate-free version of the equation, and also a version in parabolic coordinates that I have found to be the most fruitful (I have tried it also in elliptic cylindrical coordinates, cylindrical, cartesian and of course spherical, among other odd coordinate systems I defined in the process, and by an enormous margin parabolic coordinates produces the most compact version of the equation). I say "semi-coordinate-free" because in order to make it coordinate-free I have to define a vector that can be written in Cartesian coordinates as: $$ \mathbf{u}_\pm = \hat{\mathbf{x}} \pm i\hat{\mathbf{y}} = (1,\pm i,0) $$ But, in doing so, I can give you guys the following form, on the loosest leash I've made an effort to obtain:

$$ \frac{1}{2}(\mathbf{u}_-\cdot\mathbf{r})(\rho^2\psi + \nabla^2\psi) - (1+\mathbf{r}\cdot\nabla)(\mathbf{u}_-\cdot\nabla\psi) = (\gamma/\hbar^2)\psi $$ where $\mathbf{r}$ is the position vector, $\rho \in \mathbb{R}^+$ and $\gamma \in \mathbb{C}$ are constants.

In parabolic coordinates, by which I mean $$ x = \sigma\tau\cos\varphi\\ y = \sigma\tau\sin\varphi\\ z = \frac{1}{2}(\sigma^2 - \tau^2) $$ it becomes $$ \frac{1}{2}\hbar^2\left[(\partial_\sigma - \frac{i}{\sigma}\partial_\varphi)(\partial_\tau - \frac{i}{\tau}\partial_\varphi) - \rho^2\tau\sigma\right]\psi = \gamma e^{i\varphi}\psi $$

I was able to solve the case $\gamma = 0$ analytically, and the solutions are of the form $$ \psi(\sigma,\tau,\varphi) = Ae^{-\frac{\rho}{2}(\sigma^2 + \tau^2)}(\sigma\tau)^{\lambda}e^{-i\lambda\varphi} \hspace{6mm} \lambda = 0,1,2,3,... $$ Beyond this, however, I'm sort of at a loss. I've been reading up on numerical solutions to PDEs, something that, as an aspiring physicist, I assume I will require a solid familiarity with. My question about this is pretty straightforward - almost any information that might lead to a general solution or a visual depiction of a solution to the 'inhomogeneous' (not sure if that word is technically applicable outside of ODEs) case $\gamma \neq 0$, would be greatly appreciated. If anyone can classify or describe the form of the equation, that would be extremely helpful. Pointers for good places to look for a method of obtaining solutions that might be particularly helpful for this specific equation would be excellent. If you can provide a solution, well then, of course that would also be wonderful, but even then I would really love to know how you got it and what you looked for in doing so. My apologies for a vague-ish request, but I it's not so much that I am not looking for an answer, I am just accepting a very wide array of them equally. I can only hope that you all will be forgiving about that and want to attempt to answer my question anyway. That said, I admit I am rather new to the whole stack exchange thing (which seems pretty sweet in general, by the way), and understand if this is a little too broad or narrow...Thanks for taking the time to read through!


This post imported from StackExchange Physics at 2014-03-06 21:22 (UCT), posted by SE-user Nicola

asked Mar 5, 2014 in Mathematics by Nicola (10 points) [ revision history ]
recategorized Apr 11, 2014 by dimension10

1 Answer

+ 1 like - 0 dislike

It is a linear PDE, hence the superposition principle holds. You can eliminate the φ dependence by making the ansatz $ψ(σ,τ,φ)=\Psi(\sigma,\tau)e^{−iλφ}$. Then you are left with a linear hyperbolic equation in σ,τ that can probably be solved by integrating one variable at a time.

answered Apr 12, 2014 by Arnold Neumaier (11,395 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverf$\varnothing$ow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...