# Homework Help: Eigenvalues of an Operator

1. Nov 27, 2007

### cscott

If I have an operator of the form $$1+3\vec{e}\cdot\vec{\sigma}$$ where $$\vec{e}\cdot\vec{e}=1$$.

How can I find the eigenvalues quickly?

2. Nov 27, 2007

### malawi_glenn

you write out the matrix representation of the operator, and then you find the eigenvalues and eigenvectors to that matrix, same as you do in linear algebra.

3. Nov 27, 2007

### cscott

How can I write out the matrix representation without knowing e?

4. Nov 27, 2007

### malawi_glenn

You know this:

$$\vec{e}\cdot\vec{e}=1$$

Why not just make the ansatz:
$$\vec{e} = (a,b,c)$$
with:
$$a^2 + b^2 + c^2 = 1$$

When you dont have any numbers or explicit expressions, but you have a condition to be fulfilled, you can atleast do an asatz.

5. Nov 27, 2007

### cscott

Thanks

6. Nov 27, 2007

### Avodyne

Or, you could just choose to use a coord system in which e is in the z direction.

7. Nov 28, 2007

listen man if you download the book Schaum's Outline of Quantum Mechanics off emule, go to page 54 there's a whole section on how to represent an operator in matrix form. There are also plenty of problems on the subject in 5,6,7.
I guess you can also check these stuff in the Cohen-Tannoudji book, also availabe in emule. Good luck.
And by the way, I find those one line advices to be very unhelpful. that's why I usually turn to the books.

8. Nov 29, 2007