Time dependent wave function of a particle in a gravitational field

Question

I found this great question about the solution of the Schrodinger equation for a particle in a constant gravitational field, but the solution they wanted is to the time independent Schrodinger equation.

$$E \psi=\frac{-\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}+mgx\psi$$

Wave function of a particle in a gravitational field

I want a solution for the time dependent Schrodinger equation for a particle in a constant gravitational field, one with dispersion, where the energy is not exactly known. How do I get it? Basically I am trying to get a solution to this equation

$$i\hbar \frac{\partial\psi}{\partial t}=\frac{-\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}+mgx\psi$$

Hi I am coming back to edit this question in hopes that I can direct students in the right direction. The reason I asked this question in February 2021 was due to a fundamental misunderstanding of quantum mechanics and the Schrodinger equation. I did not understand the role of the time independent Schrodinger equation and I did not see the use in decomposing the wave function into a sum of stationary states. Here is a resource that greatly helped me understand why you would want to do it and HOW to do it. I have linked to the specific page that made things click for me. https://farside.ph.utexas.edu/teaching/qmech/Quantum/node101.html

Answer 1

The time-dependent SE in PDE shorthand:

$$i\hbar \psi_t=-\frac{\hbar^2}{2m}\psi_{xx}+mgx\psi$$

To solve it, we use separation of variables, by assuming the Ansatz: $$\psi(x,t)=\Psi(x)\phi(t)$$ Inserting into the PDE: $$i\hbar \Psi(x)\phi'(t)=-\frac{\hbar^2}{2m}T(t)\Psi''(x)+mgx\Psi(x)T(t)$$ Dividing both sides by $\psi(x,t)$ we get: $$i\hbar\frac{\phi'}{\phi}=-\frac{\hbar^2}{2m}\frac{\Psi''}{\Psi}+mgx=E$$ Where $E$ is a separation constant.

So we obtain two separate DEs, one in $t$ and one in $x$: $$i\hbar\frac{\phi'}{\phi}=E\tag{1}$$ $$\Rightarrow \phi(t)=e^{-\frac{{ E i t}}{\hbar}}$$ $$-\frac{\hbar^2}{2m}\frac{\Psi''}{\Psi}+mgx=E\tag{2}$$ $(2)$ is basically the DE you'll find in your link.