Quantum Homework Question: Simplifying Xjk using A|k> and A+|k>

[bon]

Homework Statement

Please see attached question

Homework Equations

The Attempt at a Solution

So i tried to work out Xjk by pre and post multiplying by energy eigenstates <j| and |k>

Then i simplified the expression using A|k> = root k |k-1> and A+ |k> = root (k+1) |k+1>

giving a final answer of Xjk = L(root k delta j, k-1 + root (k+1) delta j, k+1)

where delta is the kronocker delta. My problem is that all of the entries in my matrix seem to be too big by 1.

So for X_21 i get root 2 instead of root 1 etc..

what's gone wrong?

Thanks

 

[bon]

any ideas? is my notation clear?