# Reference request: Time evolution of non-Gaussian free particle wave packet

Many quantum mechanics textbooks discuss in some detail the time evolution of the free particle wave packet with Gaussian initial state. But treatments of more general non-Gaussian cases (from what I've seen) are rare, and (if they exist) seem kind of vague/incomplete.

Wondering if anyone could point me to some resource (textbook, online lecture notes, etc.) that actually does discuss in some kind of detail the more general case of time evolution of the free particle wave packet with not-necessarily-Gaussian initial state. By "some kind of detail", I'm thinking of, at a minimum, some concrete quantitative discussion of group velocity and spreading of the wave packet over time.

I realize the Gaussian case is popular because it's possible to obtain an exact closed-form solution. But presumably there are non-Gaussian cases for which at least a decent approximation is mathematically reasonably tractable? (hopefully?)

Update To add a bit more detail: The thing I'm most interested in (that isn't explicitly covered in resources I've found) is the time evolution of $$\Delta x$$ for non-Gaussian initial states (for example with momentum-space state initially square wave, as described in this doc). In the Gaussian case, the closed-form expression for the wave function can be used to show that for large $$t$$, $$\Delta x(t) = (\Delta p / m) \cdot t$$ (where $$\Delta p$$ is constant). It seems reasonable-ish that this same linear growth of $$\Delta x(t)$$ would hold for non-Gaussian cases, but an actual mathematical proof isn't obvious to me. A book or paper that actually works that out would be ideal for my purposes (or even a proof just typed directly into this thread as an answer)

• Dec 10, 2022 at 5:32
• Are you thinking that a non-Gaussian wave packet is going to exhibit interesting effects that a Gaussian one doesn’t? I doubt that that is the case. Dec 10, 2022 at 6:10
• What do you mean by wave packet if you don't mean gaussian? Those words are used interchangeably. Every specific initial condition has a different solution so unless you get more specific there isn't a clear answer. But, in general the time-dependent Schrödinger equation has only been solved for a limited number of initial states so you are likely to be out of luck in terms of a specific solution. Dec 10, 2022 at 9:24
• Starting page 11 of this doc are some wave packet examples (not all Gaussian). One case of interest would be the square wave in momentum space (evolution, not just $t=0$ description)
– NikS
Dec 10, 2022 at 10:40
• Which features of Gaussian packet evolution apply equally to non-Gaussian cases? Don’t know, can’t think of a reason to assume one way or the other (hence the question)
– NikS
Dec 10, 2022 at 10:47

To (sort of) answer my own question: "Quantum Theory for Mathematicians" by Brian Hall (section 4.5) does state that for the general (non-Gaussian) case of an initial ($$t=0$$) wave function relatively localized in momentum space around some value $$p_{0}$$, $$\Delta x^{2}(t) = (\Delta p^2 / m^2) \cdot t^2$$ plus terms linear in $$t$$ (though parts of the proof are postponed to a later chapter and the math seems above the level of undergraduate quantum mechanics). This book is part of the Springer "Graduate Texts in Mathematics" series, so it's a pretty rigorous "mathy" treatment framed in terms of Hilbert spaces, etc.