- #1

- 46

- 2

## Homework Statement

Prove the following relations (for ##\zeta:=r e^{i\theta}##):

[tex]

\begin{align}

D(\alpha)^\dagger a D(\alpha)&=a+\alpha\\

S(\zeta)^\dagger a S(\zeta)&= a \cosh r- a^ \dagger e^{i\theta} \sinh r

\end{align}

[/tex]

## Homework Equations

##|\alpha\rangle## is the coherent state. ##a## and ##a^\dagger## are the creation and annihilation operators. ##\alpha## is the eigenvalue of the annihilation operator.

Thus, ##a|\alpha\rangle=\alpha|\alpha\rangle##.

## The Attempt at a Solution

I really cannot attempt a solution here, because I don't seem understand expression (1). If ##a## is an operator, and ##\alpha## is a scalar, I really don't know how to interpret the expression ##a+\alpha##.

Thanks for your help!