Proof of Commutator Operator Identity

[Peter Yu]

TL;DR

Proof of Commutator Operator Identity used in Harmonic Oscillator of Quantum Mechanics

Hi All,
I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof.
Many Thanks.

 

[Gaussian97]

Well, If $k$ is a natural number (which seems to be an asumtion in your proof) I would recommend you to use mathematical induction.

 

[Peter Yu]

Thank you for your response.
Can you enlighten me on the approach.

 

[Gaussian97]

Do you know what mathematical induction is?

 

[Peter Yu]

Sorry I do not know. Can you give me some hints.

 

[PeterDonis]

@Peter Yu PF rules do not permit posting images containing your work. Please post your work directly in this thread. Use the PF LaTeX feature for equations; help on that can be found here:

https://www.physicsforums.com/help/latexhelp/

 

[Gaussian97]

Sorry I do not know. Can you give me some hints.

https://en.wikipedia.org/wiki/Mathematical_induction

Essentially, is to prove only two easier statements:
$$\left[\hat{a}^\dagger, \hat{a}\right]=-\hat{a}^0$$ and $$\left[\hat{a}^\dagger, \hat{a}^k\right]=-k\hat{a}^{k-1}\Longrightarrow \left[\hat{a}^\dagger, \hat{a}^{k+1}\right]=-(k+1)\hat{a}^{k}$$

 

[samalkhaiat]

$$\big[ \hat{a}^{\dagger} , \left( \hat{a}\right)^{k}\big] = - k \left( \hat{a}\right)^{k - 1}$$

You can easily show that the commutator $\big[ \hat{a}^{\dagger} , \hat{a} \big] = -1$ has the following representation $$\hat{a} \to a \ \mbox{id}_{C^{\infty}} , \ \ \hat{a}^{\dagger} \to - \frac{d}{da} ,$$
acting on the space $C^{\infty}$ of smooth functions of the variable $a$. So, for any function $f(a) \in C^{\infty}$ and any $p \in \mathbb{R}$, you have $$\big[ \hat{a}^{\dagger} , (f(a))^{p} \ \mbox{id}_{C^{\infty}} \big] = - p \ (f(a))^{p-1} \ \frac{df}{da} \ \mbox{id}_{C^{\infty}} .$$