Qantum mechanics condition for time independence

[Mechdude]

Homework Statement

$$\psi(x,t) = \psi_1 e^{-i E_1 t/2} + \psi_2 e^{-iE_2 t/2}$$
under what conditions is the probability density time independent?

Homework Equations

$$|\Psi(x,t)|^2 = \psi(x,t)* \psi(x,t)$$

The Attempt at a Solution

i found a statement in pg 71 of prof Richard Fitzpatrick's notes on quantum mechanics (university of Texas at Austin) that says :
" If a dynamical variable is represented by some Hermitian operator A which
commutes with H (so that it has simultaneous eigenstates with H), and contains
no specific time dependence, then it is evident...that the expectation value and variance of A are time independent. In this sense,the dynamical variable in question is a constant of the motion."


is this the condition that is being sought for?
because by defenition getting the probability density will involve the relevant equation I've written down and the time variable exits out of the expression

 

[Oerg]

Actually, what you have written in your attempt at a solution is referring to the expectation of some dynamical variable represented by the operator A.

All you have to do is plug in your equation for \Psi into the expression for the probability density. What do you find then and under what conditions do the time dependent part drop off?