I am an undergraduate with the background of a first course in Quantum Mechanics. I want to find out if there exist non-unique solutions to Schrodinger equation. So I have to find potentials (Hamiltonians) for which there exist multiple solutions to a Schrodinger equation. For separable solutions, the problem is analogous to the classical problem of non-uniqueness in Newton's equation addressed by Abhishek Dhar in the paper http://dx.doi.org/10.1119/1.17411. But I would like to pursue the same for non-separable analytical solution to the Schrodinger equation. Can anyone suggest how to go about it or what resources could I use to attempt the problem?


I don't want to be too much precise, but the Schrödinger equation ($i \dot{\psi}= H\psi$ to avoid confusion) has at most an unique solution under very general assumptions on the Hamiltonian operator $H$, even if you see it as a liner equation in the more general setting of Banach spaces.

In particular, it is not a priori necessary that $H$ is self-adjoint for the solution to be unique (and anyways if it is, then it is unique).

| cite | improve this answer | |
  • $\begingroup$ What do you mean by very general assumptions? $\endgroup$ – illusion Jun 23 '15 at 15:14
  • $\begingroup$ @illusion Mathematical assumptions that I am quite sure you do not want to see, for they are quite technical ;-) The main point is, however, that you have uniqueness for the solution of the Schrödinger Equation whenever the generator $H$ is self-adjoint (and also in other more general situations). And the self-adjointness of $H$ is a rather unavoidable assumption in standard QM. $\endgroup$ – yuggib Jun 23 '15 at 15:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.