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A Gaussian wavepacket can be considered a superposition of infinite sinusoidal waves, each with different frequency and amplitude.
The propagation velocity of each constituent wave can be frequency independent, which gives a non-dispersive profile for the packet. If the velocity is frequency dependent, the profile shows dispersion. In quantum mechanics, this would correspond to a wavefunction that is composed of plane waves with different momenta.

The non-dispersive is time symmetric. We can create a packet on a rope on one side and do the same on the other side. There is no difference. There is time symmetry. We have to choose the right material, for frequency independent propagation velocities of the constituent waves though.

So let's focus on dispersive Gaussian packets. Let's say we have found a medium with the right properties. The right frequency dependent propagation speeds for the sinusoidal waves. Now if we induce these oscillations somehow in the material, a dispersive GWP will propagate through it. We could also try to "swing" the right shape into the medium (can we?). The GWP is asymmetric in time (or symmetric, depending on how you look).

Now in theory is is not difficult to reverse the process in time. But can this be done in practice? I mean, can we reverse the dispersion? Or is this akin to reversing the motion of gas particles so they collect in a container from which they were released forward in time? And what would be the situation for a quantum Gaussian wavepacket?

MatterGauge
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  • Related: https://physics.stackexchange.com/q/54534/2451 – Qmechanic Feb 15 '22 at 20:04
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    Does this answer your question? Is it possible for $\Delta x$ ($\sigma_x$) of any free particle wave packet to be decreasing at any time? –  Feb 15 '22 at 20:05
  • @DvijD.C. I'm not doubting if it is possible because it is. In theory. But in practice? And not for a particle but a classical packet. – MatterGauge Feb 15 '22 at 20:08
  • @Felicia That's an engineering question? Unless you think there's a fundamental reason as to why it shouldn't be possible empirically even if it's possible in theory. If so, can you emphasize it in your question? If you're just asking how to do it in a lab, that's an engineering question IMO. –  Feb 15 '22 at 20:09
  • @DvijD.C.Well, I wonder what has to be reversed. A nondispersive wave is easily reversed. But what about dispersive and why would this be difficult? Is it like reversing gas so it localizes? I don't want to actually do it. – MatterGauge Feb 15 '22 at 20:12
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    @DvijD.C. not of engineering but a question of "possibility". Think of EM wave propagation: Are they really reversible by any physical means? Maybe if you allow infinite size mirrors but if you allow only finite size then what? Can you revert diffraction by more diffraction? Or can we say that diffraction increases entropy irreversibly? – hyportnex Feb 15 '22 at 20:23
  • Again, there is no sense in which $\psi(x,t)$ is a preferred solution of the Schrodinger equation over $\psi^\ast(x, -t)$ as long as the Hamiltonian is $T-$even which it is for a free particle. So, what's the root of your suspicion that it might be highly improbable to reverse the time evolution of a Gaussian wavepacket (like that of the delocalization of gas in a room)? –  Feb 15 '22 at 20:24
  • @DvijD.C. The evolution of a gas is also reversible. All particle collisions can be reversed. But in practice it's impossible. How do you reverse a Gaussian wavefunction in practice? – MatterGauge Feb 15 '22 at 20:41
  • But.. doesn't it reverse itself? – Cosmas Zachos Feb 15 '22 at 21:55
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    A quick search revealed that deep water waves exhibit instabilities [https://www.annualreviews.org/doi/abs/10.1146/annurev.fl.25.010193.002105]. In contrast, I don't think you could heat up a wire in such a way that the heat "focuses" over time, but thats just my intuition. Both of these lead to intersting questions: how can water waves focus, and why is heat always dispersing? But I think your question is a little too general at the moment for me to give an answer. Basically, the answer seems to be "depends on the system"! – myorbs Feb 15 '22 at 22:42
  • @CosmasZachos Time reversing the Gaussian wavepacket in QM seems obvious. It doesn't grow smaller backwards in time, as you don't reverse the "hidden variables," so the spreading is time symmetric. Just reverse the momenta of the constituent plane waves and the packet still spreads. But what about a classical packet? – MatterGauge Feb 15 '22 at 23:03
  • @maor That seems what Im talking about! – MatterGauge Feb 15 '22 at 23:06
  • I agree that the QM wave packet disperses when you evolve it backwards in time. If you time translate it $t\to t-t_0$, it still a solution to the SE. Doing so, you could start with a slightly dispersed packet, it will evolve to $t=t_0$ when it is maximally focused, then disperse again. – myorbs Feb 17 '22 at 10:48

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