Probability Current for Free Particle Wave Function

In summary, the probability current for a free particle wave function is given by J(x,t) = (ih/4πm)(ΨΨ* - Ψ*Ψ). The direction of the current can be determined by the sign of J(x,t), with a negative sign indicating flow to the left and a positive sign indicating flow to the right.
  • #1
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[SOLVED] Probability Current for Free Particle Wave Function

Homework Statement


Find the probability current, J for the free particle wave function. Which direction does the probability current flow?

Homework Equations


[tex]J(x,t) = \frac{ih}{4\pi m}\left(\Psi \frac{\partial \Psi^{*}}{\partial x} - \Psi^{*} \frac{\partial \Psi}{\partial x}\right)[/tex]

[tex]\Psi_{k}\left(x, t\right) = Ae^{i\left(kx - \frac{hk^{2}}{4\pi m}t}\right)[/tex]

The Attempt at a Solution


I won't take the time to put my math into Latex, but I come up with

[tex]J(x,t) = \frac{A^{2}hk}{2\pi m}[/tex]

Is this correct or did I do the complex conjugate wrong?
How would I find the probability current flow direction?
 
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  • #2
I just read that the direction is simply the sign of J(x,t) ( - corresponds to left and + corresponds to right). If this is so, that would be great. Can anyone confirm? (it wasnt exactly a textbook source)
 
  • #3
singular said:
I just read that the direction is simply the sign of J(x,t) ( - corresponds to left and + corresponds to right). If this is so, that would be great. Can anyone confirm? (it wasnt exactly a textbook source)

Yes, that's correct (and you can tell that the wavefunction you have is a plane wave traveling to the right since the sign of the x and t terms in the exponential have opposite signs). Your current looks good if A is assumed real (you should really have |A|^2 there, not A^2 since a gets complex conjugated).
 
  • #4
kdv said:
Yes, that's correct (and you can tell that the wavefunction you have is a plane wave traveling to the right since the sign of the x and t terms in the exponential have opposite signs). Your current looks good if A is assumed real (you should really have |A|^2 there, not A^2 since a gets complex conjugated).

Great, thank you very much.
 

1. What is probability current for a free particle wave function?

The probability current for a free particle wave function is a measure of the rate at which the probability of a particle being at a certain location changes over time. It is a fundamental concept in quantum mechanics that helps describe the behavior of particles in motion.

2. How is probability current calculated for a free particle wave function?

Probability current can be calculated by taking the product of the wave function and its complex conjugate, and then multiplying by the gradient of the wave function. This represents the flow of probability in a specific direction at a given point in space.

3. What is the significance of probability current in quantum mechanics?

Probability current helps us understand the behavior of particles in quantum mechanics, as it allows us to calculate the likelihood of a particle being in a certain location at a given time. It also helps us understand the concept of wave-particle duality, as the wave function represents the probability of a particle's location in space.

4. Can probability current be negative?

Yes, probability current can be negative. This means that the probability of a particle moving in one direction is decreasing while the probability of it moving in the opposite direction is increasing. This is a common occurrence in quantum mechanics and is a result of the wave-like nature of particles.

5. How does probability current relate to the conservation of probability in quantum mechanics?

According to the continuity equation, the rate of change of probability density in a certain area is equal to the negative of the divergence of the probability current. This means that the total probability within a given area remains constant, which is a fundamental principle in quantum mechanics known as the conservation of probability.

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