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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 in to my question. If I can elaborate about what I am asking, please let me know.

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

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