Finding the probability of energy measurements

[Denver Dang]

Homework Statement

Hi... My problem says:

In a given experiment the system, at the time t = 0, in the normalized state is given by:

\phi(t =0) = \frac{1}{\sqrt{5}}(i\psi_{210} + 2\psi_{211})

What possible outcomes is possible if you do an energy measurement on the system in this state, and with what probability does this occur ?

Homework Equations

I know that the states \psi_{21-1}, \psi_{210} and \psi_{211} is given by the matrix H:

\[ \left( \begin{array}{ccc}<br /> E_{2} + 2\gamma\hbar^{2} &amp; 0 &amp; 0 \\<br /> 0 &amp; E_{2} + 2\gamma\hbar^{2} &amp; 0 \\<br /> 0 &amp; 0 &amp; E_{2}\end{array} \right)\]

Don't know if I need to be telling more ?

The Attempt at a Solution

Well, I know the outcome has to be the two last states (In the order I wrote my states), which means: \psi_{210} and \psi_{211}.
This means that I have to calculate the probability for those two, which means: P(E_{2} + 2\gamma\hbar^{2}) and P(E_{2}).

My problem is, that I'm not quite sure how to calculate that :/

My book says:

P(j) = \frac{N(j)}{N},
but I have no idea how to make use of that in this case.

I know the answer should be: P(E_{2} + 2\gamma\hbar^{2}) = \frac{1}{5} and P(E_{2}) = \frac{4}{5},
but again, not sure how to do it.

So I was hoping someone could give me some pointers towards this, probably, easy question :)


Regards.

 

[nickjer]

I will use the bracket notation...

You are given a normalized wavefunction |\Psi\rangle. So the probability of measuring a specific eigenstate |\psi_j\rangle is just:

P(j) = \left|\langle\psi_j|\Psi\rangle\right|^2

Each eigenstate corresponds to a specific energy. So you know the probability of measuring a specific energy.

 

[Denver Dang]

Ahhh, ofc...
I think I have it now :)

Thank you very much.