Solve Equation of Continuity Using Schrodinger Equation

[opous]

Homework Statement

Use the Schrödinger Equation to show that

$$\frac{\partial}{\partial t}(\Psi^{*} \Psi) = - \underline{\nabla}. \underline{j}$$

Homework Equations

$$\underline{j} = \frac{-i}{2m} \left[\Psi^{*}(\nabla \Psi) - (\nabla \Psi^{*})\Psi]\right$$

$$\frac{\partial}{\partial t}n(x,t) = -\underline{\nabla}. \underline{j}(x,t)$$

$$n(x,t) = \Psi^{*}(x,t)\Psi (x,t)$$

I'm not sure how the Schrödinger equation comes into play here... can anyone offer any suggestions?

 

[CompuChip]

On the left hand side I see a time derivative, while on the right hand side there is a space derivative (gradient). The Schrödinger equation contains both, so you could use it to go from one to the other (and you might also want to use the complex conjugate of it)