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?