Quick QM Question Homework: Find Probability & Average Energy

[iamalexalright]

Homework Statement

Given an initial ave function $$\Psi(x)$$ at t=0, and a complete set of energy eigenfunctions $$\varphi_{n}(x)$$ with corresponding eigenenergies E_n for a particle, and no other information, in therms of the given find:


a. Find the state of the particle at a later time

Solution: $$\Psi(x,t) = T(t)*\varphi_{n}(x) = e^{-iE_{n}t/\hbar}\varphi_{n}(x)$$

b. what is the probability that a measurement of energy will yield a particular value E_n?
$$<E> = \sum_{n}C_{n}P(C_n)$$
Where P is the probability but I don't know where to go from here.

c. Find an expression for the average energy of the particle in terms of the energy eigenvalues:
$$<E>=1/n \sum_{i=1}^{n}E_{n}$$

All seems right besides b? Any hints on b?

 

[creillyucla]

One important point you're missing is that $$\psi$$ can be any linear combination of the energy eigenfunctions $$\phi_n$$. Try the problem armed with the knowledge that:

$$\Psi(x,t) = \Sigma_n \phi_n(x,t)$$

I would be glad to answer any further questions, but this seemed to be the major stumbling block.

 

[iamalexalright]

oh, duh...

a.
$$\Psi(x) = \sum b_{n}\varphi_{n}(x)$$

which implies

$$\Psi(x,t) = \sum b_{n}e^{-iE_{n} t/ \hbar} \varphi_{n}$$

Which means in part b the probability of yielding E_n is simply b_n, correct?

$$b_{n} = <\varphi_{n}|\Psi(x)>$$

 

[iamalexalright]

hrm, LaTeX is either messing up for me or I missed something in my formatting but hopefully you can see what I meant for Psi in my second post

 

[creillyucla]

Note that
$$b_{n} = <\varphi_{n}|\Psi(x)>$$
Can be a complex number. Does a complex probably mean anything? Hint hint...