Midterm oral quiz and exam topics

The oral *quiz* will only cover topics from Chapter 1. The midterm exam will cover topics from Chapters 1 and 2.

From Chapter 1

  • Normalizing state vectors. That any state vector describing a physical quantum mechanical state is normalized.
  • Math of complex numbers: Calculating $|a|^2$ as $a^* a$; using Euler's relation $e^{i\theta}=\cos \theta + i\sin\theta$; Expressing any complex number as either the sum of a real and imaginary part, or as a vector in the complex plane, described as magnitude times phase angle, $Ae^{i\theta}$, along the lines of polar coordinates.
  • Calculating the probability of a measurement outcome as $|\langle \psi_\text{out}\ket{\psi_\text{in}}|^2$.
  • Expressing $\ket \psi$ as a superposition of other states.
  • Column vector representations of a state $\Leftrightarrow$ ket vector representations of a state; Matrix multiplication: Evaluating the "dot" or "inner product" of bra-ket. Checking if two column vectors are orthogonal (inner product = 0). Knowing the column vector representation of a ket, how to make the corresponding bra vector; Hermitian conjugate = transpose and complex conjugate of a matrix.
  • Wave function "collapse" as a result of measurement. How to estimate outcomes of different sequences of analyzers (along the lines of that Spins Simulator).
  • Difference between a superposition state and a mixed state.

From Chapter 2

  • Matrix multiplication. Inner ('dot') and outer products of column and row vectors.
  • The eigenvalue equation, and the characteristic equation which you can solve to find the eigenvalues: $$\text{det}\Big( \mym M - \lambda{\mathbf 1} \Big) = 0.$$
  • Be able to test whether a particular column vector is or is not an eigenvector of a matrix.
  • You should be able to use CoCalc to find eigenvectors.
  • The Hermitian transform, or "dagger" which is the transpose and complex conjugate of a matrix.
  • Because observations / measurements on a system are always real, this turns out to imply that the matrices which represent observables are Hermitian or "self-adjoint" matrices.
  • The eigenvalues $\{a_1,a_2...\}$ of a Hermitian operator / matrix $\hat A$ are real. And the corresponding, normalized eigenvectors--labelled according to their eigenvalues as $\{\ket{ a_1},\ket{a_2},...\}$--are orthonormal: $$\innerp{a_i}{a_j}=\delta_{ij}.$$
  • Measurement: Repeated measurement of an observable $\hat A$ on identically prepared states $\ket \psi$ will return an expection value, equal to the probability-weighted possible eigenvalue, $a_i$, outcomes of the operator: $$\langle A \rangle =\bra\psi\hat A\ket \psi = \sum_i a_i\P _{a_i}.$$
  • Measurement - wavefunction collapse: After measuring a state $\psi$ and finding the (eigen)value $a_i$, the new state is the corresponding eigenstate $\ket{\psi_\text{out}}=\ket{a_i}$.
  • Measurement spread: The quantum nature of measurement results in a spread of measurements: $$\Delta A=\sqrt{\langle ( A-\overline A)^2\rangle}=\sqrt{\langle A^2 \rangle - \langle A\rangle^2}$$
  • The commutator $$[\hat A, \hat B]=\hat A\hat B-\hat B\hat A.$$ How to calculate from the matrix representations of the operators. Connection with the uncertainty relationship of two observables (2.98).
  • More operators: $\hat S^2$, $\hat S_{\uv n}=...$
  • Projection operators: These are really useful for showing certain relationships. But I won't examine you on these.
  • (More coming)