# Quantum Mechanics - Lowering Operator

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1. Jun 26, 2015

### izzmach

1. The problem statement, all variables and given/known data
let A be a lowering operator.

2. Relevant equations
Show that A is a derivative respects to raising operator, A†,

A=d/dA†

3. 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?

2. Jun 26, 2015

### theodoros.mihos

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

3. Jun 26, 2015

### izzmach

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

4. Jun 26, 2015

### theodoros.mihos

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

5. Jun 26, 2015

### strangerep

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 $k$.