# Yet another mind-blowing quantum mechanics question

1. Nov 4, 2007

### guruoleg

1. So, here is the problem: Suppose the state of the field is described by a state vector->

$$|\psi> = \sqrt(3)/2|n>_{m} + 1/2|n+2>_{m}$$

( $$|n>_{m}$$ means "n" photons in mode "m" and zero photons in every other mode)

If a measurement is made to determine th number of photons in mode "m"
a) What is going to be the expected result?
b) What is going to be the variance?

So, I was taking a look at it and I realized (or at least I think) that this might be a pure state because all the photons reside in a single mode. So, we could find the density operator, $$\rho$$, and it will be either one or zero since $$\rho=\rho^2$$. So, the mean will equal Tr($$\rho$$n) however that will equal infinity which is does not seem to make sense. Then, alternatively I tried deriving another expression for the density operator- using some crazy assumptions such as $$(a^+)^2 |n>= \sqrt((n+1)(n+2))|n+2>$$ so that is what I have now and I don't know what to do with this...

So, what am I missing here? Is this really a pure state or is this a mixed state? Or do you have to assume something else? My main problem is dealing with the |n+2> part which is greatly bothering me. Once I get the mean it should be straightforward to find the variance. Any hints and comments would be greatly appreciated.

Last edited: Nov 4, 2007
2. Nov 5, 2007

### Galileo

Since you are given a state vector, it's a pure state. A mixed state means a classical ensemble of state vectors. Here you know exactly what state you are in.

You want to measure the number of photons, what observable (operator) corresponds to that measurement?

3. Nov 5, 2007

### guruoleg

Hey Galileo, thanks for the input. N is the observable that I believe corresponds to that measurement. I just don't know how to handle the |n+2> situation...

4. Nov 5, 2007

### Galileo

There's nothing strange with |n+2>.

If you were given that the state was $$|\psi> = \sqrt(3)/2|2>_{m} + 1/2|4>_{m}$$
would you be able to calculate the expectation value of N?
What if the state was $$|\psi> = \sqrt(3)/2|34>_{m} + 1/2|36>_{m}$$?

You're just looking at the same problem for a general value of n.

5. Nov 6, 2007

### guruoleg

Actually I figured it out the answer to the mean: you can use the following expressions:

$$n\hat|n> = n|n> n\hat|n+2> = n+2|n>$$

So, when it is all said and done the mean should be $$n + 1/2$$
Thanks a lot for the help!