Yet another mind-blowing quantum mechanics question

[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.

 

[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?

 

[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...

 

[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.

 

[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!