Superposition representation of particle state in 1-d infitne well (SUPERPOSITION?)

[mak015]

Homework Statement

Here it is: a particle in 1-d infinite potential well starts in state $$\Psi$$(x,0) = A Sin$$^{3}$$($$\pi$$*x/a): 0$$\leq$$x$$\leq$$a.

Express $$\Psi$$(x,0) as a superposition in the basis of the solutions of the time independent Schrödinger eq for this system, $$\phi_{n}$$(x) = (2/a)$$^{1/2}$$ Sin(n*$$\pi$$* x /a).

Homework Equations

What are the steps to take to bring me to the correct answer. I'm not sure what exactly the question is asking for, or rather how to show it.

The Attempt at a Solution

I assume to know that superposition states that $$\Psi$$(x,0) = $$\sum$$$$C_{n}$$*$$\phi_{n}$$(x).

Then since they are bound (therefore orthogonal) it can be said that
$$\int\Psi(x,0)\Psi^{*}_{m}(x,0)dx$$ = 1 from 0 to a.

Can it then be said that $$\sum$$C$$_{n}$$$$\int\phi_{n}(x)\Psi^{*}_{m}(x,0)dx$$also equals 1, therefor equaling the above eqn?

From here, I don't know how to approach the goal of this problem.


More parts to the question ask for solving for $$C_{n}$$ and normalizing the first given fctn.


Any help is greatly appreciated!

Mark

 

[Redbelly98]

Welcome to PF.

Not 100% sure, but I'm thinking the idea is to use trig identities to express sin³ in terms of sin(π x/a), sin(2π x/a), etc.

Haven't worked this through to know for sure that will work though.

 

[Redbelly98]

Okay, I've looked at this one some more.

A helpful identity is

sinθ = (e^iθ - e^-iθ) / (2i)​

So sin³θ = ?