# Quantum Mechanics: Expectation values

1. Dec 22, 2008

### Nusc

1. The problem statement, all variables and given/known data

I need to find the expectation value for E but I don't know how b acts on the vacuum state.

2. Relevant equations
$$b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t)$$
$$| \psi(t)\rangle = b^\dagger| 0\rangle$$

3. The attempt at a solution
$$\langle \psi(t) | \hat{{\cal E}}^\dagger\hat{{\cal E}}| \psi(t)\rangle =$$

2. Dec 23, 2008

### Nusc

What did I not make clear?

3. Dec 23, 2008

### Avodyne

Everything! What is the system supposed to be? How are your operators and other variables defined? Are you integrating over time? Then how is it that the state is time dependent? What exactly are you supposed to compute the expectation value of???

4. Dec 23, 2008

### Nusc

This represents the single photon output level and I'm supposed to determine the
mean value and standard deviation of the single photon amplitude.
$$\hat{{\cal E}} = e^{-\kappa \tau}+ e^{-\kappa t}\int^{t}_{0}e^{\kappa \tau} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(\tau)dt$$

I'm integrating with respect to time.

$${\cal E}$$is an operator in Heisenberg picture.

b^+ creates a photon in the temporal mode $$\phi(t)$$

Does that make sense?

5. Dec 24, 2008

### Avodyne

Sorry, it does not make sense to me. Perhaps someone else will be able to help.