QM:Expectation values and calculating probabilities

[Wavefunction]

Homework Statement

An operator [itex]\mathbf{A}[/itex], corresponding to a physical quantity [itex]\alpha [/itex], has two normalized eigenfunctions [itex]\psi_1(x)\quad \text{and}\quad \psi_2(x)[/itex], with eigenvalues [itex]a_1 \quad\text{and}\quad a_2[/itex]. An operator [itex]\mathbf{B}[/itex], corresponding to another physical quantity [itex]\beta[/itex], has normalized eigenfunctions [itex]\phi_1(x)\quad \text{and}\quad \phi_2(x)[/itex], with eigenvalues [itex]b_1 \quad\text{and}\quad b_2[/itex]. [itex]\alpha [/itex] is measured and the value [itex]a_1[/itex] is obtained. If [itex]\beta[/itex] is then measured and then [itex]\alpha [/itex] again, show that the probability of obtaining [itex]a_1[/itex] a second time is [itex]\frac{97}{169}[/itex].

Homework Equations

The eigenfunctions are related via:

[itex] \psi_1 = \frac{(2 \phi_1+3 \phi_2)}{\sqrt{13}}[/itex]
[itex]\psi_2 = \frac{(3 \phi_1-2 \phi_2)}{\sqrt{13}}[/itex]

The Attempt at a Solution

Okay now I know I can represent [itex]|\psi\rangle[/itex] by:
[itex]|\psi\rangle = \frac{1}{\sqrt{13}}\begin{pmatrix}2&3\\3&-2\end{pmatrix}|\phi\rangle[/itex]

I also know that initially:

[itex]\mathbf{A}|\psi\rangle = a_1|\psi\rangle = \frac{a_1}{\sqrt{13}}\begin{pmatrix}2&3\\3&-2\end{pmatrix}|\phi\rangle[/itex] which I can then bra through by [itex]\langle \psi|[/itex] in order to get [itex] a_1 \langle \psi |\psi \rangle [/itex]

Here's where I'm stuck, but I think maybe I should repeat the above process with the respective operators to get something like [itex]\langle\mathbf{A}_{\alpha}\rangle \langle\mathbf{B}_{\beta}\rangle \langle\mathbf{A}_{\alpha}\rangle[/itex]

However, I'm unsure because I'm not very familiar with QM and I'm trying to prepare for the class before it begins this fall. Thanks for your help everyone.

 

[strangerep]

It's a lot simpler than that -- I suspect you're over-thinking the problem.

I'm not sure how much I should give away in a first response, so I'll just offer an initial hint:

What is the probability that a state prepared in state ##\psi_i## will be detected as state ##\phi_j## ?
In other words, what is ##P(\phi_j|\psi_i)## (which is read as "probability of ##\phi_j##, given ##\psi_i##) ?

BTW, which textbook(s) are you working from? If you have Ballentine, then you might be able to deduce the answer to my question from his eq(2.28).

 

[Wavefunction]

It's a lot simpler than that -- I suspect you're over-thinking the problem.

I'm not sure how much I should give away in a first response, so I'll just offer an initial hint:

What is the probability that a state prepared in state ##\psi_i## will be detected as state ##\phi_j## ?
In other words, what is ##P(\phi_j|\psi_i)## (which is read as "probability of ##\phi_j##, given ##\psi_i##) ?

BTW, which textbook(s) are you working from? If you have Ballentine, then you might be able to deduce the answer to my question from his eq(2.28).

Well the text I have is Modern Quantum Mechanics 2nd Edition by J.J. Sakurai; however, I'm just picking random problems from different sources to get a feel for QM. This particular problem is from here: http://farside.ph.utexas.edu/teaching/qmech/Quantum/node44.html

In regards to the problem:[itex] P(\phi_j|\psi_i) = |\langle \phi_j|\psi_i\rangle|^2 [/itex] correct? So then the probability of measuring [itex] b_1[/itex] when the system is in the state [itex] \psi_1 [/itex] is [itex] |\langle \phi_1|\psi_1\rangle|^2=\frac{4}{13} [/itex]

 

[CAF123]

In regards to the problem:[itex] P(\phi_j|\psi_i) = |\langle \phi_j|\psi_i\rangle|^2 [/itex] correct? So then the probability of measuring [itex] b_1[/itex] when the system is in the state [itex] \psi_1 [/itex] is [itex] |\langle \phi_1|\psi_1\rangle|^2=\frac{4}{13} [/itex]

Correct, so having measured ##b_1##, the system is now in the state ##\phi_1##. Then a third measurement is made, this time for A. What is the probability that, for your system in the state ##\phi_1##, that you measure ##a_1##?

 

[Wavefunction]

Correct, so having measured ##b_1##, the system is now in the state ##\phi_1##. Then a third measurement is made, this time for A. What is the probability that, for your system in the state ##\phi_1##, that you measure ##a_1##?

Ah okay that would be [itex] |\langle \psi_1|\phi_1\rangle|^2 =\frac{4}{13}[/itex]

 

[CAF123]

Ah okay that would be [itex] |\langle \psi_1|\phi_1\rangle|^2 =\frac{4}{13}[/itex]

Correct, can you finish the problem?

 

[Wavefunction]

Correct, can you finish the problem?

I think so because I think now I need to multiply the two probabilities together to get [itex] \frac{16}{169} [/itex] Then I need to repeat the process for the state [itex] \phi_2 [/itex]:

[itex] |\langle \phi_2|\psi_1\rangle|^2 = \frac{9}{13} [/itex]

Measuring [itex] \mathbf{A} [/itex] again:

[itex] |\langle \psi_1|\phi_2\rangle|^2 = \frac{9}{13} [/itex]

Multiplying together: [itex] \frac{81}{169}[/itex]

Now adding the two together yields [itex] \frac{16+81}{169} = \frac{97}{169} [/itex]

 

[CAF123]

Yes, looks fine.