Matrix representation of certain Operator

Tags:
1. Aug 21, 2016

abcs22

1. The problem statement, all variables and given/known data
Vectors I1> and I2> create the orthonormal basis. Operator O is:
O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number.
Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are orthonormal.

2. Relevant equations

Av=λv

3. The attempt at a solution

My problem is concerning the first part of this excercise. I'm not really familiar with this notation of the operator and not sure how I should get the matrix. I have tried improvisation and got the matrix

2a^2 -2a^2
-2a^2 2a^2

When I tried to calculate eigenvalues, I didn't get anything reasonable, so I believe that my matrix is wrong. Please help me regarding this problem, once I have the right matrix I will not have the problem finding eigenvalues nor eigenvectors.

2. Aug 21, 2016

blue_leaf77

Hint: The matrix element $O_{ij}$ of an operator $O$ is given by $\langle i|O|j\rangle$.
That is actually the vector form of the relation for a matrix $M$
$$M = \sum_i\sum_j M_{ij} c_i r_j$$
where $c_i$ is a column matrix containing 1 as the i-th element and zero otherwise and $r_j$ is a row matrix containing 1 as the j-th element and zero otherwise.

3. Aug 21, 2016

abcs22

Thank you very much for your reply. I know the formula for the matrix element but have problem working it out with this notation. I was trying to find examples which include this notation, but without any luck.

4. Aug 21, 2016

blue_leaf77

What's the problem, for example $\langle 1 |O| 2 \rangle = i a\langle 1| 1 \rangle \langle 2 |2 \rangle = ia$.

Last edited: Aug 21, 2016
5. Aug 21, 2016

abcs22

I don't understand how you got i<1l1><2l2> and also, what to do with that scalar a in front of the bracket

6. Aug 21, 2016

blue_leaf77

From $\langle 1 |O| 2 \rangle$, replace $O$ with the form you are given with in the first post and then make use of the fact that $|1\rangle$ and $|2\rangle$ are orthonormal.
Sorry I forgot to add $a$. Corrected.