Probability Current Density

In summary, there is a formula for Probability Current Density in Quantum Mechanics that equals the velocity multiplied by the wavefunction squared. This was introduced in continuum mechanics and can also be derived from Schroedinger's equation. In Bohmian theory, this expression directly represents the velocity of the particle, but in other interpretations, it can be seen as a conditional probabilistic description of velocity.
  • #1
einstein1921
76
0
hello, everyone. I have a question about Probability Current Density. I read a book which says that Probability Current Density equals velocity multiply wavefunction^2 .How to prove it? thank you!j=v*ψ^2.
best wishes!
 
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  • #3
Demystifier said:
Can you say what book it is?

Quantum Mechanics-Landau
chapter77 The required probability w is proportional to the current density along the z-axis. In
the classically accessible region, this is vzψ^2

thankyou!
 
  • #4
The definition is much different but you know that j is defined as ρv.now ρ in quantum mechanics should be probability density which is |ψ|2,and hence current density
should be (along z axis) |ψ|2vz
 
  • #5
einstein1921,

the relation

[tex]
\mathbf j = \rho \mathbf v
[/tex]

was introduced already in continuum mechanics (Eulerian description). It gives amount of mass that will flow through a small planar surface of area [itex]\Delta S[/itex] and perpendicular to unit vector [itex]\mathbf n[/itex] after time interval [itex]\Delta t[/itex]:

[tex]
amount~of~mass = \mathbf j \cdot (\Delta S \mathbf n) \Delta t
[/tex]

Have a look into beginning chapters of some textbook on hydrodynamics, they explain this in greater length.

The mass density [itex]\rho[/itex] and current density [itex]\mathbf j[/itex] satisfy the equation of continuity

[tex]
\partial_t \rho + \nabla \cdot \mathbf j = 0.
[/tex]


It turns out that Schroedinger's equation for one particle allows similar current density [itex]\mathbf f[/itex] to be defined, with the difference that now it gives the "amount of probability that flows through small area in unit time" in space, instead of giving directly amount of mass.

In theory based on Schroedinger's equation, the equation of continuity (of "probability flow") is

[tex]
\partial_t (\psi^*\psi) + \nabla \cdot \mathbf f = 0,
[/tex]

where [itex]\mathbf f[/itex] is a triple of numbers given by

[tex]
\mathbf f = Re \{\frac{1}{m} \psi^* \hat{\boldsymbol \pi} \psi \}.
[/tex]

Here [itex]\hat{\boldsymbol \pi} = \hat{\mathbf p} - \frac{q}{c}\mathbf A(\mathbf r)[/itex] is operator of kinetic momentum (mv) of the particle. All this can be derived from Schroedinger's equation.

We can even retain the same formula for current density as in continuum mechanics

[tex]
\mathbf f = \rho \mathbf v,
[/tex]

provided we define
[tex]
\rho = \psi^*\psi,
[/tex]

[tex]
\mathbf v = \frac{Re\{ \frac{1}{m} \psi^* \hat{\boldsymbol \pi} \psi \}}{\rho}.
[/tex]
 
  • #6
I have another question, for everybody here. Can this quantity in the probability current be interpreted as velocity if one does NOT adopt the Bohmian interpretation of quantum mechanics?
 
  • #7
In Bohmian theory the above expression for [itex]\mathbf v[/itex] is understood directly as the velocity of the particle. But the actual value of velocity of the particle cannot be so simple; the velocity, if it exists, surely has to fluctuate (due to background radiation)). But I think we can understand the above expression in this way:

The expression

[tex]
\mathbf v(\mathbf r) = Re\{ \frac{1}{m} \psi^*(\mathbf r) \hat{\boldsymbol{\pi}} \psi(\mathbf r) \}
[/tex]

gives expected average velocity for particle that is at [itex]\mathbf r[/itex]. So it may be a kind of conditional probabilistic description of velocity.
 
  • #8
thank you all!
 

What is probability current density?

Probability current density is a concept in quantum mechanics that describes the flow of probability through a given point in space. It is represented by a vector field that shows the direction and rate of change of probability at that point.

How is probability current density calculated?

Probability current density is calculated using the continuity equation, which states that the change in probability density over time at a given point is equal to the divergence of the probability current density at that point. This can be represented mathematically as ∂ρ/∂t = -∇⋅J, where ρ is the probability density and J is the probability current density.

What is the significance of probability current density in quantum mechanics?

Probability current density is a fundamental concept in quantum mechanics that helps us understand the behavior of particles at the quantum level. It is crucial in determining the evolution of quantum systems and plays a key role in the Schrödinger equation, which describes the time evolution of quantum states.

How does probability current density relate to the uncertainty principle?

Probability current density is closely related to the uncertainty principle, which states that it is impossible to simultaneously know the precise position and momentum of a particle. This is because the more accurately we know the position of a particle, the less certain we are about its momentum, and vice versa. Probability current density helps us understand this relationship by showing how the probability of finding a particle at a given point in space changes over time.

Can probability current density be measured?

While probability current density cannot be measured directly, it can be inferred from other measurements such as the probability density and the wave function of a quantum system. It is an abstract concept that helps us understand the behavior of particles at the quantum level, rather than a physical quantity that can be measured with a specific instrument.

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