Number Operator in Matrix Form

[martyg314]

Hi-

I have a basic QM problem I am trying to solve. We are just starting on the formalities of Dirac notation and Hermitian operators and were given a proof to do over Spring Break. I am stuck on how to set up the operators and wave equation in matrix and vector form to complete the proof as requested. We have done very basic matrix operations in class (ie \hat{H} =(h g;h g) or a wavefunction in terms of |1> and |0>, but nothing like the following, nor is it covered in the text (Griffiths).

Homework Statement

Consider the number operator \hat{N} =\hat{a+}\hat{a-}
for the HO problem.
1) Express the operator in matrix form and show that it is Hermitian.
2) Express the basis states of the HO problem |\psin> in column vector form and use the matrix form of \hat{N} to show that the matrix-mechanics version of the eigenvalue equation ( \hat{N}|\psin> = n|\psin> ) works out.

Homework Equations

The Attempt at a Solution

for 1) I determined the matrix form of the ladder operators as the square roots of n, n+1 etc. off the diagonal (ie: {0 0 0; \sqrt{1} 0 0;0 \sqrt{2} 0}). This is easy enough to show as Hermitian.

However I'm stumped as to the column vector form of the HO basis states. I'm sure it's something simple I'm overlooking, but I would appreciate any tips.

Thanks,
M

 

[Chopin]

If you've constructed the matrix form for the ladder operators, then you must already know how to express the basis states as column vectors, because you can't write down a matrix until you know the basis vectors. Can you go into more detail about how you arrived at those matrix forms?

Also, just to check...you are already familiar with the "conventional" way of constructing the basis states for the HO (i.e. using the ladder operators + a vacuum state), correct?