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The Schrödinger equation is

$$i \hbar \frac{\partial}{\partial t} \Psi(x, t) =- \frac{\hbar^2}{2m}\Delta \Psi(x,t)+V(x)\Psi(x, t)$$

If $\Psi(x,t) = e^{-iEt}u(x)$, we will get a ground state equation. For this case, it is $S^1$ symmetric. But as I know, in gravitational field , the orbital can be a elliptic, so it means the Schrödinger equation should have elliptic symmetric solution, is it ?

What is elliptic symmetric: for the Schrödinger equation, it is an elliptic symmetric solution, if it has form solution $\Psi(x,t) = (a\cos Et+ i b\sin Et)u(x)$, where $a,b>0$ is constant.

Enhao Lan
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  • How is $S^1$ symmetry applied to this equation? There is no ellipse here. Is $x\in R^1$ or $R^3$? – Hans Feb 27 '19 at 07:49
  • The potential $V(x)$, if considered as a gravitational potential, has surely an influence on the structure of the ground state solutions $u(x)$ and thus on $\Psi$: however, consider what happens for the vibrating drum, where the potential is the function $V_D$ which is $0$ inside the disk $D$ and $+\infty$ outside it. Even if the drum plate $D$ is circular, the eigenfunctions possess many more symmetries than the simple $S^1$. – Daniele Tampieri Feb 27 '19 at 07:56
  • What do you call an "elliptic symmetric solution" ? In the classical world, the trajectory is planar and has a three planes of symmetry. But one of them is accidental and will certainly not remain because only one focus is involved. –  Feb 27 '19 at 08:52
  • @YvesDaoust I have add the definition of elliptic symmetric solution in above. – Enhao Lan Feb 28 '19 at 02:07
  • @DanieleTampieri Thanks your example, I just want to know there are any paper about the elliptic symmetric solution of Schrödinger equation. Since not any potential can make it has symmetric solution. – Enhao Lan Feb 28 '19 at 02:10
  • So the symmetry you are searching for are to be considered respect to the time $t$: unfortunately I am not aware of any study on such solutions. – Daniele Tampieri Feb 28 '19 at 07:01

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