Quantum Mechanical Measurements
Measurement happens from projecting the state on an eigenstate of a non-degenerate eigenvalue.
Example: Stern-Gerlach
Measure: \(|\psi\rangle\), \(S_z|\pm\rangle = \pm\frac{\hbar}{2}|\pm\rangle\).
\(\langle S_z \rangle = \frac{\hbar}{2}\mathcal{P}(+\hbar/2) + -\frac{\hbar}{2}\mathcal{P}(-\hbar/2)\), \(P(\pm\hbar/2)=|\langle\pm|\psi\rangle|^2\).
Example
\(|\psi\rangle = \frac{1}{\sqrt{19}}|\varphi_1\rangle + \frac{2}{\sqrt{19}}|\varphi_2\rangle + \sqrt{\frac{2}{19}}|\varphi_3\rangle + \sqrt{\frac{3}{19}}|\varphi_4\rangle + \sqrt{\frac{5}{19}}|\varphi_5\rangle\).
\(H|\varphi_n\rangle = n\mathcal{E}_0|\varphi_n\rangle\).
Outcomes: \(n\mathcal{E}_0\).
\(\mathcal{P}(\mathcal{E}_0) = \frac{1}{19}/\frac{15}{19}=\frac{1}{15}\).
\(\langle H\rangle = \sum_n (n\mathcal{E}_0)\mathcal{P}(n\mathcal{E}_0) = \frac{52}{15}\mathcal{E}_0 = \left(3+\frac{7}{15}\right)\mathcal{E}_0\).
Example
Given Matrix Operator - Get Eigenvalues and then Eigenvectors.
Degenerate Eigenvalues
Guess: Projection on the degenerate states: \(P_n=\sum_k^{g_n} |\varphi_n^k\rangle\langle\varphi_n^k|\).
\(|\psi\rangle = \sum_n\sum_i^{g_n}c_n^i|\varphi_n^i\rangle\), \(c_n^i=\langle\varphi_n^i|\psi\rangle,\mathcal{P}(a_n)=\sum_i^{g_n}|c_n^i|^2\).
Example
\(L_z^2\), \(\lambda=0,1\). Probability: \(|0\rangle\Rightarrow \frac{1}{4}\), \(|1\rangle \Rightarrow \frac{3}{4}\). States: \(|0\rangle \Rightarrow \begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}, |1\rangle \Rightarrow \begin{pmatrix}a \\ 0 \\ b\end{pmatrix}\).
For \(|\psi\rangle = \begin{pmatrix}\frac{1}{2}\\\frac{1}{2}\\\frac{1}{\sqrt{2}}\end{pmatrix} = \frac{1}{2}|0\rangle + \frac{1}{2}|1_1\rangle + \frac{1}{\sqrt{2}}|1_2\rangle\).
If we measured \(1\), we would have \(|\psi'\rangle = \begin{pmatrix}1/\sqrt{3}\\0\\\sqrt{2/3}\end{pmatrix}\).