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The Hamiltonian I used is the classical one with no potential energy: H=p^2/2m $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} $$ I want to gain an intuitive understanding of what's happening in this differential equation.

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  • $\begingroup$ The discussion at the end of the diffusing Gaussian wavepacket is not clear to you? It details how it is driven by the uncertainty principle and basic properties of Fourier Analysis. $\endgroup$ – Cosmas Zachos Jul 2 '19 at 16:07
  • $\begingroup$ Please don't call the quantum Hamiltonian "classical"! $\endgroup$ – electronpusher Jul 2 '19 at 16:14
  • $\begingroup$ Related. $\endgroup$ – Cosmas Zachos Jul 2 '19 at 18:37
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As @CosmasZachos said in comments, the Wikipedia section on "Gaussian wavepacket in Quantum mechanics" does a good job of explaining the mathematics (I hope so cuz its too complex for me XD). What I found really interesting was the physical interpretation of this phenomena. Apparently, the spread of the wavepacket corresponds to growing uncertainty in momentum. Due to Heisenberg's relation, the localized (pinned down at a spot) particle's uncertainty in momenta is INCREASING LINEARLY with time. This is CRAZY! Getting a better idea of position is making momentum harder and harder to measure...that too WITH TIME. It's like Nature is constantly destroying information because of the particle's interaction with the environment (measurement). So if we wait long enough, how much will the uncertainty grow and how does that work out physically. WOW!

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  • $\begingroup$ Checkout Fokker-Planck equation. This type of spreading happens in classical estimation as well. There's nothing especially quantum about the uncertainty spreading. $\endgroup$ – Brick Jul 2 '19 at 17:50
  • $\begingroup$ Fokker-Planck can be used to model the uncertainty of a single, classical particle arising from classical measurement uncertainty. The comment by @CosmasZachos, seems to imply (maybe I misunderstood it), that it is somehow limited to ensemble situations, and that is not true. It's a fairly general feature of probabilistic models, not just in physics, that if you have uncertainty in the time-rate-of-change of something that your uncertainty in that something will grow with time. But I agree the underlying reasons and the equations modeling them depend on the system and are not all the same. $\endgroup$ – Brick Jul 3 '19 at 0:17
  • $\begingroup$ The spreading of the wave packet is not fundamentally a quantum mechanical phenomenon. The same equation (with other constants) describes the paraxial approximation for the propagation of an optical beam. The beam it described is a Gaussian beam. It too spreads out. The fundamental reason is found in Fourier theory. $\endgroup$ – flippiefanus May 2 at 3:27

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