Probability current direction

In summary, the probability current for a free particle is given by J = (hbar/2m) * (Ψ*∇Ψ - Ψ∇Ψ*), and it always flows in the positive x direction due to the constant and uniform nature of the wave function. This can be explained by the fact that the gradient of the wave function is also constant and points in the same direction as the wave vector.
  • #1
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Homework Statement


Find the probability current of a free particle. What direction does the probability current flow?

Homework Equations



The Attempt at a Solution


The book also gives the wave equation and probability current flow in terms of Psi so it is just plug and solve. I have the correct answer for this [tex] J = A^2 \frac{\hbar k}{m} [/tex]

I also know that the answer for probability current flow is in the positive x direction but I don't understand why, given that it seems to be uniform and independent of x.
 
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  • #2
Can someone explain why the probability current flows in the positive x direction?

The probability current for a free particle can be derived from the Schrodinger equation and is given by J = (hbar/2m) * (Ψ*∇Ψ - Ψ∇Ψ*), where Ψ is the wave function. This equation shows that the probability current is dependent on the gradient of the wave function.

In the case of a free particle, the wave function is a plane wave, which has a constant magnitude and phase throughout space. This means that the gradient of the wave function is also constant and points in the same direction as the wave vector, which is in the positive x direction.

Since the probability current is proportional to the gradient of the wave function, it also flows in the positive x direction. This can also be visualized by imagining the wave function as a probability density cloud, and the gradient as the direction of flow of this cloud. In the case of a plane wave, the cloud is uniform and flows in the same direction as the wave vector.

Therefore, the probability current for a free particle always flows in the same direction as the wave vector, which is in the positive x direction. This is true for all points in space, as the wave function and its gradient are constant throughout.
 

1. What is probability current direction?

Probability current direction refers to the direction of flow of probability in a system. It is a concept used in quantum mechanics to describe the movement and behavior of particles.

2. How is probability current direction calculated?

The calculation of probability current direction involves using the mathematical equation J = (ih/2m)(Ψ*∇Ψ - Ψ∇Ψ*), where J is the probability current, i is the imaginary unit, h is Planck's constant, m is the mass of the particle, Ψ is the wavefunction, and ∇ is the gradient operator. This equation is derived from the Schrödinger equation.

3. What does a positive probability current direction indicate?

A positive probability current direction indicates that the probability of finding a particle in a particular location is increasing with time. This can happen when the particle is moving towards that location, or when it is spreading out or diffusing in that direction.

4. Can probability current direction be measured?

No, probability current direction cannot be directly measured. It is a theoretical concept used to describe the behavior of particles in quantum mechanics. However, the effects of probability current can be observed through experiments and calculations.

5. How does probability current direction relate to other quantum mechanical concepts?

Probability current direction is closely related to other concepts in quantum mechanics, such as wavefunctions, position, and momentum. It helps to explain the behavior of particles and their movement in a system. It is also used in the study of quantum tunneling and quantum transport phenomena.

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