Time dependent wave equation trouble

[Physser]

Time dependent wave equation trouble!

Homework Statement

I'm having heaps of trouble getting my head around the time dependent wave function and the use of operators to find measurement/probabilities etc...

I'm having trouble with something like the following,

If have a 1D inf potential well with region -a <=x <= a , and at a certain time have the following wave function,

ψ= √(5a) cos (∏x/2a) + 2.√(5a) sin (∏x/a)

and you were asked to find the possible energy levels and their associated probabilities?

Homework Equations

The Attempt at a Solution

As far as I know, can find energy levels by Hu_n=Eⁿ.u_n
using u₁= √(5a) cos (∏x/2a)
and u₂ = 2.√(5a) sin (∏x/a)
to find the two distinct energy levels,

basically my question is, is this the correct way to do this? and perhaps I'm asking why? and then what has me really confused is finding their associated probabilities... I feel it's supposed to be obvious/ easy but i really lack some understanding here.

 

[Fightfish]

Your wavefunction there is in a linear superposition of two energy eigenstates.
Your task is to ascertain "how much" of each energy eigenstate it contains, and then use Born's probability interpretation to obtain the corresponding probabilities.

 

[vela]

Homework Statement

I'm having heaps of trouble getting my head around the time dependent wave function and the use of operators to find measurement/probabilities etc...

I'm having trouble with something like the following,

If have a 1D inf potential well with region -a <=x <= a , and at a certain time have the following wave function,

ψ= √(5a) cos (∏x/2a) + 2.√(5a) sin (∏x/a)

and you were asked to find the possible energy levels and their associated probabilities?

Homework Equations

The Attempt at a Solution

As far as I know, can find energy levels by Hu_n=Eⁿ.u_n
using u₁= √(5a) cos (∏x/2a)
and u₂ = 2.√(5a) sin (∏x/a)
to find the two distinct energy levels,

basically my question is, is this the correct way to do this? and perhaps I'm asking why?

Sort of, but not really.

A measurement of some physical quantity corresponds to an operator, and the possible outcomes of the measurement are the eigenvalues of the operator. In the case of energy, the operator is the Hamiltonian, so you want to solve the eigenvalue equation $\hat{H}\phi = E\phi$ to find the eigenfunctions and the eigenvalues.

When you do this for the infinite square well, you find a set of normalized eigenfunctions $\phi_n(x)$ and their corresponding energy $E_n$. This problem is probably done somewhere in your textbook or was covered in your lecture, so you should already have expressions for the normalized eigenfunctions and their energies.

If you're given a system in state $\psi(x)$, you can expand this state in terms of the normalized eigenfunctions:
$$\psi(x) = \sum_{n=1}^\infty c_n\phi_n(x).$$ The probability of measuring $E_k$ turns out to be given by $|c_k|^2$. So what you want to do is first look up or calculate what the normalized eigenfunctions are, and then write the state you were given in terms of these eigenfunctions.

What you did was verify that $u_1$ and $u_2$ are proportional to the normalized eigenstates $\phi_1$ and $\phi_2$ and thus found the corresponding energies. The constants in front of the sine and cosine, however, are a combination of the normalization constant and $c_n$. You need to figure out how to separate the two factors.