Calculating Uncertainty in a Quantum Oscillator Mixed State

[ppyadof]

Homework Statement

Consider a quantum oscillator in a mixed state described by the density operator [itex]\rho = \frac{1}{2}( |\alpha><\alpha| + |-\alpha><-\alpha| ) [/itex]. Calculate [itex] \Delta (\hat{X}^2)_1 [/itex] and [itex] \Delta (\hat{X}^2)_2 [/itex] in this case.

Where X1 and X2 are the dimensionless position and momentum operators:
[tex] \hat{X}_1 = \frac{1}{2}(a + a^{\dagger}) [/tex]
and
[tex] \hat{X}_2 = \frac{1}{2i}(a - a^{\dagger}) [/tex]

Also, the state [itex] |\alpha> [/itex] (and -alpha) are coherent states, and so are eigenfunctions of the creation/annihilation operators.

Homework Equations

Well, I know how to calculate the uncertainty in a measurement, given the expectation values of X1, X2, X1^2 and X2^2. I also know that [itex] < \hat{A} > = Tr[ \hat{\rho} \hat{A} ] [/itex].

Since the states in the density operator are coherent, we know that:
[tex] <\alpha|\beta> = e^{ -|\alpha^2|/2 - |\beta^2|/2 + \beta^{\ast}\alpha} [/tex]

The Attempt at a Solution

Based on what I know, to get the uncertainty in X1 and X2, I need 2 calculate the expectation values for X1, X1^2 and X2, X2^2, which means I need to expand everything out into the 'a' and 'a-dagger' relations... right? Taking X1 as an example:

[tex]<X_1> = \frac{1}{4} ( \sum{m} <m|\alpha><\alpha| a + a^{\dagger} |-\alpha><-\alpha|X_1|m> [/tex].

I was wondering if this is the way to do this question, because that's a lot of work to expand that out and yet this is the easier expression! I then have 2 expand out X1^2 in terms of a and a\dagger's, and work all that out, so before I really get down to doing pages of maths, I just want to check that this is the right way to do this question.

 

[Jhero]

Thank you for your question. You are on the right track in your approach to calculating the uncertainties in this case. However, there are a few things to consider.

Firstly, when calculating the expectation value of an operator, you need to sum over all possible eigenstates of that operator. In this case, X1 and X2 are operators related to the creation and annihilation operators, so you would need to sum over all possible states of those operators. This may seem like a lot of work, but there are some tricks you can use to simplify the calculations, such as using the identity <m|n> = \delta_{mn} for the coherent states.

Secondly, when calculating the expectation value of X1^2 and X2^2, you would need to use the expression <\alpha|\beta> = e^{-|\alpha|^2/2 - |\beta|^2/2 + \beta^{\ast}\alpha} to expand out the states in the density operator. This will make the calculations easier, as you will only need to consider the terms involving the coherent states.

Finally, once you have calculated the expectation values for X1, X1^2, X2, and X2^2, you can plug them into the formula for the uncertainty \Delta (\hat{X}^2) = \sqrt{<\hat{X}^2> - <\hat{X}>^2} to get the uncertainties in X1 and X2.

I hope this helps. Good luck with your calculations!