Quantum Mechanical Measurements

Measurement happens from projecting the state on an eigenstate of a non-degenerate eigenvalue.

Example: Stern-Gerlach

Measure: $| ψ ⟩$ , $S_{z} | \pm ⟩ = \pm \frac{ℏ}{2} | \pm ⟩$ .

$⟨ S_{z} ⟩ = \frac{ℏ}{2} P (+ ℏ / 2) + - \frac{ℏ}{2} P (- ℏ / 2)$ , $P (\pm ℏ / 2) = | ⟨ \pm | ψ ⟩ |^{2}$ .

Example

$| ψ ⟩ = \frac{1}{\sqrt{19}} | φ_{1} ⟩ + \frac{2}{\sqrt{19}} | φ_{2} ⟩ + \sqrt{\frac{2}{19}} | φ_{3} ⟩ + \sqrt{\frac{3}{19}} | φ_{4} ⟩ + \sqrt{\frac{5}{19}} | φ_{5} ⟩$ .

$H | φ_{n} ⟩ = n E_{0} | φ_{n} ⟩$ .

Outcomes: $n E_{0}$ .

$P (E_{0}) = \frac{1}{19} / \frac{15}{19} = \frac{1}{15}$ .

$⟨ H ⟩ = \sum_{n} (n E_{0}) P (n E_{0}) = \frac{52}{15} E_{0} = (3 + \frac{7}{15}) E_{0}$ .

Example

Given Matrix Operator - Get Eigenvalues and then Eigenvectors.

Degenerate Eigenvalues

Guess: Projection on the degenerate states: $P_{n} = \sum_{k}^{g_{n}} | φ_{n}^{k} ⟩ ⟨ φ_{n}^{k} |$ .

$| ψ ⟩ = \sum_{n} \sum_{i}^{g_{n}} c_{n}^{i} | φ_{n}^{i} ⟩$ , $c_{n}^{i} = ⟨ φ_{n}^{i} | ψ ⟩, P (a_{n}) = \sum_{i}^{g_{n}} | c_{n}^{i} |^{2}$ .

Example

$L_{z}^{2}$ , $λ = 0, 1$ . Probability: $| 0 ⟩ \Rightarrow \frac{1}{4}$ , $| 1 ⟩ \Rightarrow \frac{3}{4}$ . States: $| 0 ⟩ \Rightarrow (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), | 1 ⟩ \Rightarrow (\begin{matrix} a \\ 0 \\ b \end{matrix})$ .

For $| ψ ⟩ = (\begin{matrix} \frac{1}{2} \\ \frac{1}{2} \\ \frac{1}{\sqrt{2}} \end{matrix}) = \frac{1}{2} | 0 ⟩ + \frac{1}{2} | 1_{1} ⟩ + \frac{1}{\sqrt{2}} | 1_{2} ⟩$ .

If we measured $1$ , we would have $| ψ^{'} ⟩ = (\begin{matrix} 1 / \sqrt{3} \\ 0 \\ \sqrt{2 / 3} \end{matrix})$ .

Quantum Mechanical Measurements

Example: Stern-Gerlach

Example

Example

Degenerate Eigenvalues

Example

Compatible and Incompatible Observables