- #1

Sekonda

- 207

- 0

I have a question on showing how the raising operator in QM raises a particular eigenstate by 1 unit, the question is showed below:

I think I know how to do this but not sure if what I'm doing is sufficient:

[tex]\hat{N}a^{\dagger}|n>=([\hat{N},a^{\dagger}]+a^{\dagger}\hat{N})|n>[/tex]

So I considered the N operator acting on the raising operator, we know the commutation relation stated on the RHS, so this simplifies to:

[tex]\hat{N}a^{\dagger}|n>=(a^{\dagger}+a^{\dagger}\hat{N})|n>[/tex]

Letting N act on the state N we attain:

[tex]\hat{N}a^{\dagger}|n>=(1+n)a^{\dagger}|n>[/tex]

I'm not sure if it's enough to say now that by definition of the raising operator 'a-dagger', it raises the state n to n+1 and we conclude that 'a-dagger' acting on a state n is equal to some constant multiplied by (1+n) and the state 1+n.

Cheers,

SK