Talk:Probability current

Derivation

I added another line to the continuity equation derivation to make it a bit easier to follow.

$\int _{V}\left({\frac {\partial |\Psi |^{2}}{\partial t}}\right)dV=-\int _{V}\left({\vec {\nabla }}\cdot {\vec {j}}\right)dV$

--146.232.75.208 (talk) 18:56, 26 February 2008 (UTC)

Current and stationary states

The article states

Note that the probability current is nonzero despite the fact that plane waves are stationary states and hence
${\frac {d|\Psi |^{2}}{dt}}=0\,$
everywhere. This demonstrates that a particle may be in motion even if its spatial probability density has no explicit time dependence.

But doesn't this only work because the plane wave is unnormalizable? Any normalizable wavefunction would have to be localized. If it is localized and has non-zero probability current, then the "center of gravity" (i.e. the position expectation value) of the wavefunction must be moving. So the wavefunction will have time dependence. This is like saying that an infinite stream has flowing water but no movement.

However, I can imagine a particle traveling in a circle. Then I suppose it could have a non-zero flux without time dependence of the probability distribution.128.112.50.18 01:26, 12 March 2007 (UTC)

ψ* What does this mean?

I can't understand what this symbol means. I understand that, ρ=ψ*ψ, but I can't find anywhere that actually says what ψ* is or how to do calculations with it. The page in wikibooks for probability flux uses this symbol and it changes how Schrödinger's Equation is written. Can anyone help? Thanks, Brian.

ψ* stands for the Complex conjugate of the wavefunction ψ.

Last equation

I can't see how the second equality

${\frac {1}{2m}}\left(\Psi ^{*}{\vec {P}}\Psi -\Psi {\vec {P}}\Psi ^{*}\right)={\frac {\hbar }{2mi}}\left(\Psi ^{*}{\vec {\nabla }}\Psi -\Psi {\vec {\nabla }}\Psi ^{*}\right)+(1-\beta ){\frac {q}{m}}{\vec {A}}(t)|\Psi |^{2}$

in the last equation in the article follows. I would have thought that the vector potential part of one term would cancel the other due to the minus sign, rather than adding as they seem to here. This is probably an embarrassing algebraic mistake on my part, but I thought I should bring it up in case it is actually a mistake. 163.1.176.253 (talk) 14:45, 13 June 2009 (UTC)

The idea is more general...

...than the article suggests. Probability current is not specific to quantum mechanics -- it's relevant to the Fokker-Planck equation for instance, and many related topics like the forwards and backwards equations. 128.97.41.120 (talk) 00:57, 22 April 2010 (UTC)

Citation Problem and a suggestion

Getting this cited should be easy. pick up any graduate textbook on QM. All include the statement at least, and I remember Shankar's book going through the full derivation (Though I might be wrong - it's been a while).

Also, I'm maybe just being a bit nit-picky, but I've noticed that many other Wikipedia articles use the overline notation for complex conjugate, especially those dealing with complex conjugates themselves and Hilbert spaces, as well as other related articles. considering the importance of these concepts to QM, Shouldn't there be some consistency? I was taught the overline first, through mathematics, and introduced to the star notation when I took QM, so I might be somewhat biased here, but considering the clarity of the overline notation vs. the star notation IMO, perhaps it should be changed. Just a recommendation. 98.245.6.2 (talk) 07:15, 7 April 2011 (UTC)

Continuity equation

The continuity equation derivation is very nice, but doesn't really belong in this article. For that reason I have cut and pasted it into the QM section in continuity equation with an alternative derivation I produced (with reference to a book) - I hope whoever wrote it doesn't mind in case they're reading this. Nothing wrong with two alternative derivations in one place. This article should simply mention that the probability current leads to the continuity equation via the Schrödinger equation, and link to that article - also really be about what the definition implies and what quantum mechanical concepts follows from it...

F=q(E+v^B) (talk) 21:35, 18 November 2011 (UTC)

Probability 4-current from Dirac equation

See here.-- F = q(E + v × B) 08:53, 12 January 2012 (UTC)

Also the mass m in the definition doesn't have to be the reduced mass of the particle. Furthermore I removed the hamiltonian expression for the charged-particle-in-EM-field-definition - it doesn't add anything, and since is would not be a suitable stand-alone section it has been blended into the main definition section, towards the end.-- F = q(E + v × B) 09:48, 12 January 2012 (UTC)

"is complex-valued"

The article begins by saying that the current vector is complex-valued, though the given mathematical definition makes it quite obviously real. Also, the given continuity equation, which balances it against the derivative of a squared magnitude, clearly requires it to be strictly real as well. I think some clarification is needed in that opening sentence, which seems to imply that any attempt to get a real, easily-visualized vector is just hopeless. Clearly, that sentence is referring to some particular case (without saying as much), and then not actually showing us that case! 146.6.208.9 (talk) 13:57, 5 June 2012 (UTC)

Classical probability current

I searched for probability current, looking for the definition of classical probability current that shows up in continuous stochastic processes, and Wikipedia brought me to this page, but the whole article is about quantum probability current, which is a special case (or a different case, since in quantum the amplitude diffuses instead of the probability density). I think this article needs to discuss the classical probability current before getting into quantum mechanics. --Sprlzrd (talk) 19:56, 28 April 2015 (UTC)

Probability Flux

Looking at the units (prob/time/area) the term 'Probability Current' seems to be very unfortunate. — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 15:11, 6 June 2020 (UTC)

It bears looking at carefully. Unfortunate or not, it is standard: its divergence yields the probability density rate of change, in close analogy to mass flux in hydrodynamics, which, naturally, it is attempting to evoke. Cuzkatzimhut (talk) 16:55, 6 June 2020 (UTC)

A typo in section "Spin-0 particle in an electromagnetic field"

Where:

$\mathbf {j} ={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-q\mathbf {A} |\Psi |^{2}\right]\,\!$

$\mathbf {j} ={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-{\frac {q}{c}}\mathbf {A} |\Psi |^{2}\right]\,\!$

should be:

$\mathbf {j} ={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]\,\!$

$\mathbf {j} ={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2{\frac {q}{c}}\mathbf {A} |\Psi |^{2}\right]\,\!$

Units?

The intro says

"probability current is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area"

It seems to me that the probability current j has units of probability per unit time, while the space derivative of j has units of probability per unit time per unit length. Area should not appear.

I see my mistake now: in the one-dimensional system, the units of j are indeed probability per time, but in the three-dimensional system the units of j are probability per time per unit area. AxelBoldt (talk) 12:52, 22 August 2021 (UTC)

"Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid."

I think the probability is the fluid, not the probability density. j measures the rate of flow of this fluid, and rho is the density of this fluid. Is this correct? AxelBoldt (talk) 00:06, 22 August 2021 (UTC)

Extra term fails unit test

The extra term ${\frac {\mu _{S}}{s}}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )$ in Probability_current#Spin-s particle in an electromagnetic field contains an error since it fails the unit test that can be easily seen with SI unit system. The magnetic moment has the unit A*m², so contains the unit Ampere but no other factor contains the unit Volt (or something with 1/A), so it's impossible to get the unit of the probability current density j (1/m²/s).--FbiSupLabAcc (talk) 12:29, 15 April 2023 (UTC)