So I bugged the folks in General Physics about the latter form of the question a while back, and got some rather unconvincing "can't be done" replies. To state the problem specifically (and my motivation):(adsbygoogle = window.adsbygoogle || []).push({});

Let's say that the probability density of finding a particle at any place/time is given (ρ(r, t)) -- from that, shouldn't we be able to get *a* wave function? Of course, it will probably not be unique (certainly not up to a phase), but it seems like it might be something that would be useful to do.

More useful in my mind -- though basically the same problem (see below for justification) -- would be to derive the (say, electric) current given the time-dependence of the global (electric) charge density ρ(r, t). This is easy if you are dealing with point particles, where Ji = ρvi, where vi is the (four-) velocity of particle i (or if you had a "velocity field", v(r, t), it would just be J = ρv)...but in general, such a field (or breakdown into individual particles) won't exist - and even if it did, in those particular cases, you could derive the current from the charge density, provided that no partcles are adjacent:

It's not covariant, but you could take the gradient of the change in the field as time increases -- e.g. Ji = (ρ,t),i (here i is the coordinate -- e.g. x/y/z).

For example, take a single point charge in 1D moving to the right at 1 unit/s: ρ(x, t) = δ(x-t). So here's the charge (integrated around a single point to get rid of the delta) at various times/places:

t | x | ρ | ρ,t

0 | 0 | 1 | 0 (assume it started there)

0 | 1 | 0 | 0

1 | 0 | 0 | -1 (charge left)

1 | 1 | 1 | 1 (charge appeared)

So, the "gradient" of ρ,t at t=1 points from low to high, or in the direction the charge is moving, proportional to the speed it's moving and to the strength of the charge -- e.g., it's the current, up to a scalar factor.

However, if you instead had a line of charge on the x axis and the charge was moving along that axis, ρ,t would be 0, so it breaks down.

HOWEVER, it still seems like the time-dependence of the charge density -- perhaps with suitable boundary conditions -- should be sufficient to determine the current.

To get back to my claim that this was the same as finding a wave function for a given probability density, there I'm appealing to standard QM -- say, unquantified Dirac electron theory -- where the wave function is directly tied to the current via J = ψ∇ψ* - ψ*∇ψ (pulled from the wikipedia's probability current article -- it may not hold exactly for the Dirac equation, but IIRC it's similar -- but in either case, it's not important).

So, I started working on the problem of expressing a given density (probability or charge) as the norm squared of a wave function: e.g. Given ρ(r, t) find ψ(r, t) such that ψ*ψ = ρ, *AND* it gives a nontrivial current (ψ∇ψ* - ψ*∇ψ). I found that it would have the general form of

ψ = ε + i√(ρ + ε^2)

And ε ≠ 0, because then j = 0. I found some other interesting things, like if you restrict ε,x = 0 you get j = ψ,x ε...but as far as something I could use, I got nowhere.

However, I did get a fresh breath of hope from that probability current article : there it mentions that when ψ is a plane wave exp(i[kx - ωt]), ρ is constant, and j turns out to be ρ*hk/m, but momentum = hk, and v = momentum / m, sp j = ρv for a plane wave -- and *everyone* knows that any function can be expressed as a superposition of plane waves! So my problem was solved! That is, until I sat down to try to actually do it--here's where I mention I never got past the intro physics courses in college. So after an hour or so of transforming, twiddling/fiddling and inverse-transforming, I began to wonder again if I wasn't on a fool's errand, and decided to try you crazy-smart folks in the QM, in hopes that someone can look at this and say "oh yeah, everyone knows that's (impossible/trivial/useless/all of those)".

So...any thoughts?

Thanks -

Justin

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# Wavefunction from probability, OR current from charge density?

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