Matrix representation of certain Operator

[abcs22]

Homework Statement

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. [/B]

Homework Equations

Av=λv

The Attempt at a Solution

My problem is concerning the first part of this exercise. 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.

 

[blue_leaf77]

Hint: The matrix element $O_{ij}$ of an operator $O$ is given by $\langle i|O|j\rangle$.

I'm not really familiar with this notation of the operator

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.

 

[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.

 

[blue_leaf77]

have problem working it out with this notation

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

 

[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

 

[blue_leaf77]

I don't understand how you got i<1l1><2l2>

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.

what to do with that scalar a in front of the bracket

Sorry I forgot to add $a$. Corrected.