QM: Problem with an assignment using bra and ket notation

[Vilashjorthen]

Homework Statement

This problem is about the 1-dimensional harmonic oscillator.
The normalized energy levels are labeled |n>, n=0,1,2...
Two operators are given by

\widehat{A} = \alpha (\widehat{a}₊+\widehat{a}_-)
\widehat{B} = i\beta (\widehat{a}₊²+\widehat{a}_-²)

where \alpha and \beta are real numbers and \widehat{a}₊ and \widehat{a}₊ are the step up and step down operators

Calculate \widehat{A}|n> and \widehat{B}|n>

Homework Equations

\widehat{a}₊|n> = √(n+1)|n+1>
\widehat{a}_-|n> = √(n)|n-1>

The Attempt at a Solution

I have inserted the above equations and got an answer for \widehat{A}|n> (although I am not sure if it is sufficient):
\widehat{A}|n>=\alpha (√(n+1)|n+1> + √(n)|n-1>)
But with \widehat{B}|n> I am stuck at
\widehat{B}|n>= i\beta (\widehat{a}₊√(n+1)|n+1> - \widehat{a}_-√(n)|n-1>)

Any help will be much appreciated. Thank you in advance.

P.S.: This is my first post, so please let me know if I have posted something "the wrong way".

 

[BruceW]

hey, welcome to physicsforums :)
Your work looks good so far. What is it you are stuck with? In the last line, you have operators acting on something that is a number times a vector. What is the general rule for this?