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I'm learning the QM propagator and the first example is of course the free particle: $\hat{H}=\frac{p^2}{2m}$, then the new wavefunction is found by: $$\psi(x,t)=\int dx_0\;K(x,t;x_0,t_0)\;\psi(x_0,t_0)$$ and $K=\langle x|U(t,t_0)|x_0\rangle;$ Thus evaluating the time-evolution operator and etc we find: $$\psi(x,t)=\int dx_0\;\frac{A}{\sqrt{t}}\exp\bigg({\frac{im(x-x_0)^2}{2t}}\bigg)\psi(x_0,t_0)$$ with $A$ being a constant.

So the problem for me is this: giving a gaussian like $\psi(x_0,t_0)$ centered at $x_0$. I supposed that the latter-time wavefunction $\psi(x,t)$ would have a maximum at $\bar x=(p/m)(t-t_0)$, but it seems that the wavefunction keeps centered at $x_0$ and just goes spreading.

Am I in the proper frame of the particle? How can I see the particle moving at all?

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  • $\begingroup$ what is the $p$ in your formula for $\bar{x}$ and how did you derive that formula? Why do you think a particle with initially Gaußian wavefunction should move? $\endgroup$
    – ACuriousMind
    Commented Apr 22, 2023 at 13:42

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If you start with a particle having a definite position,e.g., $\psi(x_0,t_0)=\delta(x_0-y)$, then its momentum ic completely uncertain - it can go either right or left with equal probability. Thus we do not expect directed motion.

If we start with a particle having a definite momentum, $\psi(x_0,t_0)\propto e^{ikx_0}$, it will remain in this state, since it is the eigenstate of the Hamiltonian - nothing changes.

Now, you can experiment with Gaussian wave packets - having finite position and momentum uncertainty - these will indeed behave as moving with speed, which is the mean speed of the initial packet. But this information is encoded in $\psi(x_0,t_0)$, not in the propagator.

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    $\begingroup$ To really tie this home: you can multiply a Gaussian wavepacket with $e^{ikx_0}$ to get a wavepacket that is moving with wavenumber $k$. $\endgroup$ Commented Apr 22, 2023 at 14:58
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You must have chosen a pure Gaußian. That has mean momentum of zero, and thus of course it would not be moving, only spreading. If you want it to be moving with some initial momentum, there are Gaußians with momentum that you can start with. Then they would move.

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