Quantum Mechanics - Lowering Operator

[izzmach]

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?

 

[theodoros.mihos]

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

 

[izzmach]

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

sorry, i don't quite understand. Mind to explain it?

 

[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.

 

[strangerep]

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 $k$.