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 $H$:

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

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.