- #1

t0pquark

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- Homework Statement
- How to derive ##(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma \vert 0_{\gamma} \rangle = 0 ## from ##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle##

- Relevant Equations
- Creation operator: ##\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma##

Annihilation operator: ##\hat{a}_2^\dagger = -\hat{a}_1sinh\gamma + \hat{a}_1^\dagger cosh\gamma##

Where ##e^\gamma \equiv \sqrt{\frac{m_1 \omega_1}{m_2 \omega_2}}##

I'm working through https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_06.pdf, and I'm stumped how they got from Equation 5.26 (##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle##) to Equation 5.27 (##(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma) \vert 0_{\gamma} \rangle = 0 ##).

I've tried substituting in the creation (##\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma##) and annihilation ((##\hat{a}_2^\dagger = -\hat{a}_1 sinh\gamma + \hat{a}_1^\dagger cosh\gamma##) operators and then rearranging, but I can't get what they have.

I've tried substituting in the creation (##\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma##) and annihilation ((##\hat{a}_2^\dagger = -\hat{a}_1 sinh\gamma + \hat{a}_1^\dagger cosh\gamma##) operators and then rearranging, but I can't get what they have.