Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was wondering what is known about the solution of the Schrödinger equation $$i h \frac{\partial}{\partial t} Ψ(x, t) =- \frac{h^2}{2m}\Delta Ψ(x,t)+V(x)Ψ(x, t)$$ for $t ∈ \mathbb{R}$. What sort of conditions are put on the potential $V$ to guarantee a solution and what space does a solution lie in? I could find information about the equation $$-i h \frac{\partial}{\partial t} Ψ(x, t) = \frac{h^2}{2m}\Delta Ψ(x,t)$$ for $x\in \mathbb{R}^n$ and $t>0$, but most everything I see about the previous equation I find hard to understand. Is there some reference where such issues are dealt with in a clear manner

share|cite|improve this question
Try to find the solution in the form $Ψ(x,t)=u(x)T(t)$. – vesszabo Oct 17 '12 at 18:59
You might perhaps get more answers to this on the Physics Stack Exchange. If you want, you can flag your question for ♦ moderator attention and ask for it to be migrated there. – Ilmari Karonen Oct 17 '12 at 19:01
@IlmariKaronen As I am a maths student and am completely unaware of Physics, I was hoping for more math solutions – user38404 Oct 17 '12 at 19:20
@vesszabo I was interested in the question of existence of solution, weak or otherwise, and I was hoping for some reference which puts the problem in a proper framework and suggests the most general conditions on $V$ which would ensure solution. This paper… mentions in the first paragraph that the Schrödinger equation has a global solution etc, but does not mention any source. – user38404 Oct 17 '12 at 19:51
Thanks for the link. I will think about it and try to ask one of my colleague. (He is an expert :-) ) – vesszabo Oct 17 '12 at 20:06
up vote 0 down vote accepted

Partial answer. Substituting $\Psi(x,t)=u(x)T(t)$ we obtain $$ ih\frac{T'(t)}{T(t)}=-\frac{h^2}{2m}\cdot\frac{\Delta u(x)}{u(x)}+V(x)=K, $$ $K$ is a constant. From this $$ T(t)=c_1 \exp\left(-\frac{iKt}{h}\right), $$ where $c_1$ is arbitrary constant. For $u(x)$ we get $$ \frac{h^2}{2m} \Delta u(x)-(V(x)-K)u(x)=0. $$ Without loss of generality we may assume that $\frac{h^2}{2m}=1$. This equation is time independent and has an enormous literature.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.