Understanding Probability Density Equation & Example

[ehrenfest]

I am confused about the the probability density equation:

\vec j = \frac{\hbar}{2mi}\left(\Psi^* \vec \nabla \Psi - \Psi \vec \nabla \Psi^*\right)

Psi is not a ket on the right side, correct?
If not how can perform a del on it and get a vector on the right side?

Can someone give me a concret example of what you would plug into this expression:
\Psi^* \vec \nabla \Psi

 

[Gokul43201]

Psi is not a ket, it is a position-space wavefunction, given by the inner product: \psi _n(x) = \langle x|n \rangle

E.g., in the 1D infinite square well of width L, \psi_n(x)=\sqrt{2/L}~sin(n \pi x/L)

The gradient of a scalar field is a vector field.