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 $$|\psi'\rangle = \begin{pmatrix}1/\sqrt{3}\\0\\\sqrt{2/3}\end{pmatrix}$$.

Compatible and Incompatible Observables

Created: 2024-05-30 Thu 21:21

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