- #1
Wavefunction
- 99
- 4
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.