How Do You Find the Time Evolution of a Wave Function in Quantum Mechanics?

[mathlete]

I'm given the value of a normalized wave function at t=0 (see attachment) and I'm asked to find the wave function at some time t. I have no idea where to even begin, the book has zero examples of anything and I'm just stuck

 

[quantumdude]

What does your book have to say about the time evolution of a wavefunction? More specifically, have you been told what the time evolution operator is?

 

[Physics Monkey]

mathlete,

I can't see yout attachment so I don't know what level you're at, therefore I'm going to give a fairly basic approach. If I have an exact energy state $$\psi_E(x)$$ that satisfies the time independent Schrödinger equation $$H \psi_E = -\frac{\hbar^2}{2m} \frac{d^2\psi_E}{dx^2} + V \psi_E= E \psi_E,$$ then how does $$\psi_E$$ change in time? Hint: use the time dependent Schrödinger equation.