- #1

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1. Homework Statement

1. Homework Statement

I've been given the operators

[itex]a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}}[/itex] and [itex]a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}}[/itex] without the constants and definition of the momentum operator:

[itex]a=x+\partial_x[/itex] and [itex]a^\dagger=x-\partial_x[/itex] with [itex]\partial_x=\frac{\partial}{\partial x}[/itex]

I have to calculate [itex][a,f(a^\dagger)][/itex], with f some arbitrary function.

## Homework Equations

Product and chain rule for derivatives.

Commutator definition.

## The Attempt at a Solution

[itex][a,f(a^\dagger)]=(x+\partial_x)f(x-\partial_x)-f(x-\partial_x)(x+\partial_x)\\

=xf(x-\partial_x)+\partial_x f(x-\partial_x) - f(x-\partial_x)x- f(x-\partial_x)\partial_x\\

=xf(x-\partial_x)+(\partial_x f)(1-{\partial_x}^2)+f(x-\partial_x)\partial_x- f(x-\partial_x)x-f(x-\partial_x)\partial_x\\ =[x,f(a^\dagger)]+(\partial_x f)(1-{\partial_x}^2)[/itex] <-This is where I'm stuck :(

Can this be simplified further? Is this even correct so far?

Kind regards

Alex