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.