Proving commutator relation between H and raising operator

[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^†]?

 

[topsquark]

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

 

[malawi_glenn]

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.

 

[Orodruin]

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]

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