Does a wave-function always expand over time in free space (no obstruction)?

I have watched several videos and have read quite a bit on this, and I have never seen an example of a wave-function getting smaller over time. If nothing interacts with it after it "forms", will it always expand? By expand I mean we have less and less certainty on where it would be if measured over time. Are there any counterexamples where it would not (more certainty on space)?

No. As an example, look at a translating Gaussian wave packet: $$A \mathrm{e}^{- (x - vt)^2 / (2\sigma^2)}.$$ For a field that obeys the classical wave equation where waves propagate with speed $v$, this packet does not spread. You can construct similar wave packets for Schrodinger, Dirac, and Klein-Gordon equations.

For wave equations, in fact, any twice differentiable function of the form $$f(x \pm vt),$$ will satisfy it. That's a pattern translating at speed $v$ without spreading.

There is also the more advanced study of solitons that don't spread even in the presence of non-linear dispersion relations.

• wouldn't the non-linear dispersion count as "obstruction"? – ZeroTheHero Apr 28 '17 at 1:05
• I interpreted obstruction as pertaining to spatial or temporal non-uniformity in the medium. Non-linear dispersion relations, including ones that have dissipation effects, can be uniform in time and space. – Sean E. Lake Apr 28 '17 at 1:07
• yes it's a little ambiguous as to what that means. I interpreted this as just fee-wave dispersion... Good answer nevertheless. I thought Schrodinger had found some such solution but I don't remember the details... – ZeroTheHero Apr 28 '17 at 1:12
• This is going a bit above my head but you gave me alot to research so thanks! By "obstruction" I simply meant anything that could cause the wave function to collapse. I wanted to simply imagine an "evolving" wave. On another note, do any of you know a good place to start with the math behind all of this? I have read a ton but still cannot answer questions like these myself because I don't know where to start learning the math. Thanks! – Tangent Apr 28 '17 at 1:42

I'm not aware of any such examples, and I wouldn't expect to find one because of the uncertainty principle. A particle being confined in an ever-smaller space would have an ever-larger uncertainty in momentum, making it less likely to remain confined in that space; it would need to be being "squeezed" by some external potential with ever-increasing force to trap it. Contrast this to the likelihood of something spreading out over a larger volume as time goes on.

The best you can hope for is the wavefunction of an eigenstate of the Hamiltonian, which will remain constant over time (up to its oscillating complex phase). This is the situation in e.g. a hydrogen atom.

• I'm hesitant to add that I suppose falling into a black hole might do the trick, but in that case you've got bigger problems figuring out what's going on... – rwold Apr 28 '17 at 0:14
• This isn't right. Consider any wavefunction spreading out over time; its time reverse will contract. – knzhou Apr 28 '17 at 0:25
• That is actually what I was getting at Knzhou. I didn't understand why there was no direction to time at the quantum level if there was always expansion as a wave-function evolved, but if it can contract then this isn't right. I didn't expect to be right, just to be told why I was wrong! Rwold: I don't care how odd the measurement is (in terms of what it produces), I am just looking for a pattern that can only be reversed by measurement. Is this what you are saying? – Tangent Apr 28 '17 at 1:50