Time Evolution Of A 1-D Gaussian Wave Packet Under The Gravitational Potential

[TheFinalThy]

Hello Colleagues,

I am curious about a problem in Quantum Mechanics that incorporates the evolution of a Gaussian Wave Packet under the Gravitational Potential.

What I am interested in is equation (3) in the following paper:
"On the quantum analogue of Galileo's leaning tower experiment"
(Unfortunately, I can not post a direct link due to forum restrictions...)

I have worked with Gaussian Packets representing a Free Particle before. In that case, I was able to time evolve the function trivially since I had time translation invariance on my side s.t. t ---> (t - t-initial) or (t - t_final).

However, with equation (3), I do not believe that I have time translation invariance. How, then, can I time evolve this function to some later or some earlier time? What Propagator would I use?

Thank you for any effort or time put into my question. If I can elaborate about what I am asking, please let me know.

 

[fzero]

If there were no time translation invariance, we could not write down a conserved Hamiltonian. In this case, the gravitational potential is just modeled as a linear potential and the solutions to the time-independent Schrödinger equation, $\phi_n(z)$ are related to the Airy function, see http://www.physics.csbsju.edu/QM/fall.03.html for some concrete results.

The propagator can presumably be computed from the usual formula

$$ K(x',t';x,t) = \sum_n \phi_n^*(x')\phi_n(x) e^{-iE_n(t'-t)/\hbar} .$$

You will need to look up identities for the Airy function to perform the sum and later integrals. Things like the integral representation or representation in terms of Bessel functions would be useful.

 

[TheFinalThy]

So, you are saying that it is not as simple as plugging in (t - t') for my original "t" as in equation (3), as I suspected, correct? I think that this is the case, but I am not positive.

Then, assuming that my assumption about not having time translation invariance is correct, if I want to solve for a time before or after time "t" in the formula, I must first solve for the propagator and use that in the normal fashion to find my new function?

And, if this is true, then what will be my \phi_n(x)? Will it be equation (1) in "On the quantum analogue of Galileo's leaning tower experiment"?

Or, am I just making a mess of things and is it true that I could insert (t - t') in for "t"?
I am sorry for the confusion, but I fear that it is the case that I am quite puzzled about how this works.

 

[fzero]

No, you can't just substitute t-t' for t. You have to solve the time-independent Schrödinger equation to find the $\phi_n(x)$. That link I gave explains how to do this. You leave the energy eigenvalue $E$ arbitrary and find that the solutions look like $\mathrm{Ai}(x-E)$. (I am leaving various details out.) Applying the boundary conditions fixes the allowed values of $E$ in terms of the zeros of this function. Once you have the energy eigenfunctions and eigenvalues, you can use the standard formula for the propagator.

Now this is the most straightforward method. There may be a simpler way to find the answer by using the symmetry of the Schrödinger equation: $x\rightarrow x + a, E\rightarrow E - mga$.

 

[TheFinalThy]

Ah, indeed. Thank you for your help! I shall try this.