Creation Operator of Harmonic Oscillator

[KFC]

For creation operator of hamonic oscillator, we have

[tex]a^\dagger |n> = \sqrt{n+1}|n+1>[/tex]

if I consider the creation operator operate on the bar vector, should I also get the same thing? namely

[tex]<n|a^\dagger = \sqrt{n+1}<n+1|[/tex]

 

[Avodyne]

No. Take the hermitian conjugate of your first equation; what do you get?

 

[malawi_glenn]

You have to consider the dual correspondence:

[tex] X|\alpha > [/tex] corresponds to [tex] < \alpha | X^{\dagger} [/tex]

Also you can consider the number operator: [tex] N = a^{\dagger}a [/tex], with [tex] N|n> = n |n> [/tex]

Sandwhich it between: [tex] <n|N|n> [/tex], try it!