(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to prove:

[tex]a'(z)=S(z)aS^\dagger(z)[/tex]

Where [tex]S(z)[/tex] is the squeeze operator and [tex]a'(z)[/tex] is the pseudo-lowering operator.

2. Relevant equations

[tex]S(z)=e^{\frac{1}{2}({z^\ast}a^2-za^{\dagger 2})}[/tex]

[tex]e^x=\sum_n{\frac{x^n}{n!}}[/tex] ; I don't think 2 simultaneous Taylor series expansions will get you very far here...

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[tex]a'={\mu}a+{\nu}a^\dagger[/tex]

[tex]\mu=cosh(r)[/tex]

[tex]\nu=sinh(r)e^{i\theta}[/tex]

[tex]cosh(r)=F(\frac{1}{2}},\frac{x^2}{4}})[/tex]

[tex]sinh(r)=xF(\frac{3}{2}},\frac{x^2}{4}})[/tex]

where [tex]F(a,x)[/tex] is a power series expansion

3. The attempt at a solution

From the second set of equations (which operate on the left side of the original problem), I can make a'(z) into something that looks like a Taylor series expansion. I think you should use the operator expansion theorem now... except I don't know what it is, concretely, or how to use it. Please explain briefly or even better link to a site that provides an explanation (I looked hard: apparently QM isn't that popular).

This is due on Wed and this site is a last resort. Please respond soon.

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# Homework Help: Difficult quantum mechanics question

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