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My question has to do with time reversal symmetry and the arrow of time in quantum mechanics.

We know that the Schrodinger equation is invariant under time reversal. However, if we take an initial Gaussian wave function for a free particle, it always spreads forward in time. So the width of the Gaussian becomes bigger as time progresses, until the wave function is flat everywhere.

Doesn't this imply a preferred direction in time? For example, if a movie of spreading of probability density were played backward, one would see accumulation of probability around a small region of space instead of spreading. This obviously does not happen in nature. Also, wouldn't information be erased in such spreading? For example, if we take a completely flat wave packet (i.e. a small constant), we cannot evolve it back into a Gaussian using the Schrodinger equation.

Does this mean we can tell the direction of time by watching the evolution of a probability density? If so, how is this consistent with time reversal symmetry in QM?

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No, this situation does not break time reversal symmetry, because the Gaussian also spreads out backwards in time. Its width is at a minimum at $t = 0$, and it increases as $t$ either increases or decreases. There's no preferred time direction at all.

One can intuitively understand this by looking at the Fourier components. The time $t = 0$ is the unique time all of the components "line up" to produce a purely real Gaussian.

Moreover, given a wavepacket that is time-asymmetric (e.g. it only spreads out going forward in time), you can always take the spread-out state and apply time reversal by complex conjugating it. This yields a wavepacket that narrows in time. This confirms that quantum mechanics really doesn't have a built-in arrow of time.

Physically, such a reversed wavepacket might not be realistic, in the exact same way that seeing an object spontaneously jump up from the floor isn't. This is the issue of the arrow of time, but it's much deeper than just looking at wavepacket spreading. You can't derive the arrow of time with just one-particle quantum mechanics.

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  • $\begingroup$ Related details in 2nd answer to physics.stackexchange.com/q/54534 $\endgroup$
    – udrv
    Commented Sep 15, 2016 at 2:54
  • $\begingroup$ Thank you for the response. I would like to extend the question on time evolution and time reversal symmetry slightly by introducing measurement of particles position at some future or past time t (or -t). For example, let's take a Gaussian wave packet at t=0, propagate it forward in time, so that the width of the packet increases. If a measurement of particle's position is made at some future time, t, the wave function changes irreversibly. The act of measurement can be made using another particle (such as a photon) that itself follows laws of QM and time reversal symmetry. $\endgroup$
    – Oti
    Commented Sep 15, 2016 at 6:20
  • $\begingroup$ Thus, time reversal is broken by the act of measurement, and we cannot run Schrodinger's equation backward to recover the Gaussian wave packet. The interaction with the photon changes the wave function of the particle in such a way that all previous information is lost. The wave function stops evolving in time (we can consider the new wave function to be a very highly localized state, like a spike). Is this still consistent with time reversal symmetry? $\endgroup$
    – Oti
    Commented Sep 15, 2016 at 6:27
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    $\begingroup$ @Oti That's not how quantum measurement works. Measurement is interaction with a macroscopic system. Interaction with a single other particle is perfectly time-reversible by the exact same procedure I said -- just complex conjugate the joint wavefunction of the two particles. $\endgroup$
    – knzhou
    Commented Sep 15, 2016 at 17:06

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