Undergrad Operator acting on ket state n

[Jean-Mathys du bois]

TL;DR

how does the operator (N+1)^-1/2 α act on a |n> state of harmonic osciliator? N is the number operator N|n>=n|n> and α anihilation operator

I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?

 

[andresB]

If $$\hat{N}\left|n\right\rangle =n\left|n\right\rangle $$
then, by the very definition of function of operators, we have that
$$f(\hat{N})\left|n\right\rangle =f(n)\left|n\right\rangle $$

 

[PeroK]

Summary:: how does the operator (N+1)^-1/2 α act on a |n> state of harmonic osciliator? N is the number operator N|n>=n|n> and α anihilation operator

I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?

To make sense of a non-polynomial function of an operator, you can interpret it as a Taylor series:
$$(1 + N)^{-1/2} = 1 - \frac 1 2 N + \frac 3 8 N^2 \dots $$

 

[Jean-Mathys du bois]

The operator (N+1)^-1/2 α , i think is called Susskind