(Quantum mec.) Probability of measuring a given value of an observable

[Jalo]

Homework Statement

Given the following hamiltonian and the observable $\widehat{B}$

find the possible energy levels ($a$ is a real constant). If the state is in it's fundamental state what's the probability of measuring $b_{1}$, $b_{2}$ and $b_{3}$?

Homework Equations

The Attempt at a Solution

To find the energy levels I simply calculated the eigenvalues of the matrix. I got:

$E_{1}=E_{0}-\sqrt{2}a$
$E_{3}=E_{0}$
$E_{2}=E_{0}+\sqrt{2}a$

Next I found the eigenvector associated with the eigenvalue $E_{1}$ to find the fundamental state. I got:
$v_{1}=\frac{1}{2}(1,\sqrt{2},1)$

I don't know how to solve it from here tho.. Am I doing something wrong?
The solutions are 1/4, 1/2 and 1/4, respectively.

Thanks.

 

[TSny]

Hi Jalo. All looks good. You are almost there. (If the answer were a snake it would bite you. )

Note that you can write $v_1 = \frac{1}{2}(1, \sqrt{2}, 1)$ as $v_1 = \frac{1}{2}(1, 0, 0) + \frac{1}{\sqrt{2}}(0, 1, 0) + \frac{1}{2}(0, 0, 1)$.

What do the individual states $(1, 0, 0), (0, 1, 0),$ and $(0, 0, 1)$ represent? Note that you are using a basis where B is diagonal.

 

[Jalo]

Thank you very much! I was so close, should have figured it out! :P