Proving commutator relation between H and raising operator

guyvsdcsniper
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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,
Screen Shot 2022-08-25 at 10.06.11 PM.png


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]?

IMG_1106.JPG
 
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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
 
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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.
 
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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##.
 
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Thanks to all, I have seen the trivial mistake I made. I was able to get the correct answer now.
 
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