# Name of this relation, and struggle proving it.

• M. next
In summary, the relation [a, a^{+(n)}] = na^{+(n-1)} can be proven by induction using the identity [A, B C] = [A, B] C + B [A, C]. This was not a homework problem, but rather a personal attempt to convince oneself of its truth. Additional hints were given by fellow contributors to assist in the proof.
M. next
[a, a$^{+(n)}$] = na$^{+(n-1)}$

1) What's the name of this relation if it has any?

2) I tried to prove this by induction, I started by saying that for n=1, this holds since [a, a$^{+}$] = 1 (as we all know and as we can all prove)

then I assumed it true for (n-1), but I didn't go too far afterwards. Can someone give me a hint concerning its proof.

Thanks!

Is it homework?

M. next said:
[a, a$^{+(n)}$] = na$^{+(n-1)}$

1) What's the name of this relation if it has any?

2) I tried to prove this by induction, I started by saying that for n=1, this holds since [a, a$^{+}$] = 1 (as we all know and as we can all prove)

then I assumed it true for (n-1), but I didn't go too far afterwards. Can someone give me a hint concerning its proof.

Thanks!

Induction works just fine.

There's an identity for working with commutators that helps:

$[A, B C] = [A, B] C + B [A, C]$

Apply to the case $A = a$, $B = (a^\dagger)^{n-1}$, $C=a^\dagger$.
Then it should be easy to prove it by induction.

Last edited:
1 person
Oh thank you! I am now convinced, I tried proving it several ways, and didn't use what you proposed, and the procedures kept turning me down! Thank you, again!

Bill_K,this wasn't a homework, I was trying to convince myself that its true. I asked my professor and he told me to prove it by induction and didn't give further hints.

Thanks guys.

## 1. What is the name of this relation?

The name of this relation is [insert name here].

## 2. What evidence supports this relation?

The evidence supporting this relation includes [list evidence here].

## 3. Why is proving this relation a struggle?

Proving this relation can be a struggle because [explain reasons here].

## 4. How does this relation contribute to the scientific field?

This relation contributes to the scientific field by [explain contributions here].

## 5. Are there any notable exceptions or limitations to this relation?

There are some notable exceptions and limitations to this relation, such as [explain exceptions/limitations here].

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