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 enrgy 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:

<br /> \Psi(x,t) = \Sigma_n \phi_n(x,t)<br />

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} = &lt;\varphi_{n}|\Psi(x)&gt;

 

[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
<br /> b_{n} = &lt;\varphi_{n}|\Psi(x)&gt;<br />
Can be a complex number. Does a complex probably mean anything? Hint hint...