Wave Functions in a Potential Well with infinite high walls

[Lunar_Lander]

Homework Statement

Consider a potential well with infinite high walls, i.e. V(x)=0 for -L/2\leq x \leq +L/2 and V(x)=\infty at any other x.

Consider this problem (the first task was to solve the stationary Schroedinger equation, to get the Energies and Wave Functions, especially for n=1 and n=1, which gave \varphi_1=\sqrt{2/L}\cos(\pi/L\cdot x) and \varphi_2=\sqrt{2/L}\sin(2\pi/L\cdot x)) for the time-dependent Schroedinger Equation. At the time t=0, the wave function is given as \psi(x,0)=\frac{1}{\sqrt{2}}\varphi_1(x)+\frac{1}{\sqrt{2}}\varphi_2(x).

The further development of \psi(x,t) in time can be given as \psi(x,t)=a_1(t)\varphi_1(x)+a_2(t)\varphi_2(x). Determine the coefficients a_1(t), a_2(t).

For this, insert \psi(x,t) into the time dependent Schroedinger Equation and consider integrals of the form \int_{-\infty}^{infty}dx \varphi_m(x)... with m=1, 2. From that you will obtain a simple differential equation for each a₁and a₂, which you can solve easily with the boundary conditions given for t=0.

Homework Equations

Schroedinger Equations, other equations given above.

The Attempt at a Solution

I have first tried to follow the instructions and tried to insert the ψ into the Schroedinger Equation, which looked like this:

-\frac{\hbar^2}{2m}(a_1(t) \frac{d^2}{dx^2}\varphi_1(x)+a_2(t) \frac{d^2}{dx^2} \varphi_2(x))=i\hbar(a_1'(t)\varphi_1(x)+a_2'(t) \varphi_2(x))

But I have to admit that I do not know how to go on from there. How can I get those integrals that are mentioned in the problem? And how do I get the differential equations?

 

[BruceW]

First, you should get the left-hand side in terms of things you know (i.e. the energy values and \varphi).

This will give you an equation involving \varphi, a, a'. From here, you want to get rid of the \varphi, because you want an equation with just a and a'. Think of a way you could do this.