Wave Functions in a Potential Well with infinite high walls

[Lunar_Lander]

Homework Statement

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

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

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

For this, insert [itex]\psi(x,t)[/itex] into the time dependent Schroedinger Equation and consider integrals of the form [itex]\int_{-\infty}^{infty}dx \varphi_m(x)...[/itex] 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:

[itex]-\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))[/itex]

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 [itex]\varphi[/itex]).

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