Probability Current for Free Particle Wave Function

[singular]

[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

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

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

The Attempt at a Solution

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

$$J(x,t) = \frac{A^{2}hk}{2\pi m}$$

Is this correct or did I do the complex conjugate wrong?
How would I find the probability current flow direction?

 

[singular]

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)

 

[kdv]

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).

 

[singular]

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.