Time dependent wave function of a particle in a gravitational field

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

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

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.

• I thought of that but it doesn't look like the kind of solution I want. I want a solution where the expectation value of the position is $x_0+v_0t-1/2gt^2$ and where there is finite uncertainty in all observables Feb 12 '21 at 22:01
• $(2)$ inevitably leads to an Airy function. I don't see how you can 'force' a classical solution like the one in your comment onto this DE and its Airy solution.
– Gert
Feb 12 '21 at 22:05
• What if I use time dependent perturbation theory? Sep 23 '21 at 6:56
• Worth a try I suppose but I'm not well versed in perturbation theory. Sorry.
– Gert
Sep 23 '21 at 16:14
• @RyanParikh if you want a solution that resembles the classical form, you could try working in the Heisenberg picture, in which you calculate time dependent operators that act on a fixed initial wavefunction. I have not looked at what happens in this case, but it would not surprise me if was solvable and gave you something like what you are after Sep 23 '21 at 16:23