# Quantum Mechanics - Lowering Operator

## Homework Statement

let A be a lowering operator.

## Homework Equations

Show that A is a derivative respects to raising operator, A†,

A=d/dA†

## The Attempt at a Solution

I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can someone please explain briefly what i should do in the next step?

## Answers and Replies

##A_+|n> = |n-1> \, , \, A_-|n+1> = |n> ## so
$$A_+\,A_-\, |n> = |n> \,\Rightarrow\,\left(A_+A_- + [A_+,A_-]\right)|n> = |n>$$

##A_+|n> = |n-1> \, , \, A_-|n+1> = |n> ## so
$$A_+\,A_-\, |n> = |n> \,\Rightarrow\,\left(A_+A_- + [A_+,A_-]\right)|n> = |n>$$

sorry, i dont quite understand. Mind to explain it?

## |n> = \Psi_n ## my English is very poor for this theme. ## \frac{d}{dA_+}\left(A_+A_-+1\right) ## is that you need.

strangerep
Science Advisor
I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can someone please explain briefly what i should do in the next step?
You didn't write out the commutation relations in your initial post. I.e., ##~[A, A^\dagger] = \dots ~?##

Start by working out ##[A, (A^\dagger)^2]##, and ##[A, (A^\dagger)^3]## using the Leibniz product rule for commutators. This should help you see the pattern. Then try to prove ##[A, (A^\dagger)^k] = k (A^\dagger)^{k-1}## by induction on ##k##.