# Proving commutator relation between H and raising operator

• guyvsdcsniper
In summary, the conversation discusses an attempt to prove the commutator relation [H,a*]=hwa* and the confusion over why [a†,a] instead of [a,a†] is appearing. The conversation also includes a hint for solving the problem and a correction made by one of the participants. Ultimately, the mistake is identified and the correct answer is obtained.

#### guyvsdcsniper

Homework Statement
Prove the commutator relation [H,a*]=hwa*
Relevant Equations
[H,a*]=hwa*
I am going through my class notes and trying to prove the middle commutator relation,

I am ending up with a negative sign in my work. It comes from [a,a] being invoked during the commutation. I obviously need [a,a] to appear instead.

Why am I getting [a,a] instead of [a,a]?

quittingthecult said:
Homework Statement:: Prove the commutator relation [H,a*]=hwa*
Relevant Equations:: [H,a*]=hwa*

I am going through my class notes and trying to prove the middle commutator relation, View attachment 313257

I am ending up with a negative sign in my work. It comes from [a,a] being invoked during the commutation. I obviously need [a,a] to appear instead.

Why am I getting [a,a] instead of [a,a]?

View attachment 313258
Hint: Calculate ##[H, a^{\dagger} ] |1>## using ##H|n> = (n + 1/2) \hbar \omega |n>## and ##a^{\dagger} |1> = c |2>##. What happens?

-Dan

guyvsdcsniper
Seems to me the step (2) is wrong, you are changing the order of operation there

In step (1) you have ## (a^\dagger a + \frac{1}{2})a^\dagger - a^\dagger(a^\dagger a + \frac{1}{2}) ##
But in step (2) you have ## a^\dagger (a^\dagger a + \frac{1}{2} - a^\dagger a - \frac{1}{2})##

Redo step (1) to (2), keep the order of operators unaltered.

Last edited:
guyvsdcsniper
It seems to me step (1) is wrong. The commutator disappeared…

Too early in the morning, you just expanded the commutator. I would not do this, I would apply commutator rules for ##[AB,C] = A[B,C]+[A,C]B##.

guyvsdcsniper, malawi_glenn and vanhees71
Thanks to all, I have seen the trivial mistake I made. I was able to get the correct answer now.

vanhees71