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noospace

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I'm trying to evaluate the expectation of position and momentum of

[itex]\exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle}[/itex]

where [itex]\hat{a},\hat{a}^\dag[/itex] are the anihilation/creation operators respectively.

Recall [itex]\hat{x} = \sqrt{\hbar/(2m\omega)}(\hat{a}^\dag + \hat{a})[/itex].

The calculation of [itex]\langle \hat{x} \rangle[/itex] and [itex]\langle \hat{p} \rangle[/itex] are easy if one uses the Hadamard lemma

[itex]e^X Y e^{-X} = Y+ [X,Y] + (1/2)[X,[X,Y]] + \cdots [/itex].

I'm running into touble in the evaluation of [itex]\langle\hat{x}^2\rangle,\langle\hat{p}^2\rangle[/itex]. In particular in the calculation of

[itex]\exp\left(\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)(\hat{a}^\dag + \hat{a})^2\exp\left(-\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)[/itex].

I was able to show that

[itex]\exp\left(\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)(\hat{a}^\dag + \hat{a})^2\exp\left(-\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right) = (\hat{a}^\dag + \hat{a})^2 + \xi([\hat{a}^2,\hat{a}^\dag^2]+[\hat{a}^\dag\hat{a},\hat{a}^\dag^2-\hat{a}^2]) + \frac{\xi}{2!}[(\hat{a}^\dag+\hat{a})^2,[\hat{a}^2,\hat{a}^\dag^2]+[\hat{a}^\dag\hat{a},\hat{a}^\dag^2-\hat{a}^2]]+\cdots[/itex]

but I can't figure out how to simplify this.

[itex]\exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle}[/itex]

where [itex]\hat{a},\hat{a}^\dag[/itex] are the anihilation/creation operators respectively.

Recall [itex]\hat{x} = \sqrt{\hbar/(2m\omega)}(\hat{a}^\dag + \hat{a})[/itex].

The calculation of [itex]\langle \hat{x} \rangle[/itex] and [itex]\langle \hat{p} \rangle[/itex] are easy if one uses the Hadamard lemma

[itex]e^X Y e^{-X} = Y+ [X,Y] + (1/2)[X,[X,Y]] + \cdots [/itex].

I'm running into touble in the evaluation of [itex]\langle\hat{x}^2\rangle,\langle\hat{p}^2\rangle[/itex]. In particular in the calculation of

[itex]\exp\left(\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)(\hat{a}^\dag + \hat{a})^2\exp\left(-\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)[/itex].

I was able to show that

[itex]\exp\left(\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right)(\hat{a}^\dag + \hat{a})^2\exp\left(-\xi (\hat{a}^\dag^2-\hat{a}^2)/2\right) = (\hat{a}^\dag + \hat{a})^2 + \xi([\hat{a}^2,\hat{a}^\dag^2]+[\hat{a}^\dag\hat{a},\hat{a}^\dag^2-\hat{a}^2]) + \frac{\xi}{2!}[(\hat{a}^\dag+\hat{a})^2,[\hat{a}^2,\hat{a}^\dag^2]+[\hat{a}^\dag\hat{a},\hat{a}^\dag^2-\hat{a}^2]]+\cdots[/itex]

but I can't figure out how to simplify this.

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