Quantum Mechanics · Lesson 10

Chapter practice

Bring the chapter together: photon energies and matter waves, complex arithmetic, amplitudes that interfere, superpositions and their phases, and the vectors, brackets and bases that turn all of it into linear algebra. No worked example sits above the answers.

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E = h f

z = x + i y

P = ∣ ψ ∣^{2}

∣ ψ ⟩ = n \sum c_{n} ∣ n ⟩

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Prerequisites

Complete Expanding a state in a basis first.

Intuition

Bring the chapter together: photon energies and matter waves, complex arithmetic, amplitudes that interfere, superpositions and their phases, and the vectors, brackets and bases that turn all of it into linear algebra. No worked example sits above the answers.

A musician practising scales is not learning new notes but learning to find the right ones without looking. These questions are the scales of quantum mechanics.

Before you practise

Every question here comes down to a few rules. A photon carries $hf=hc/\lambda$ . Undistinguished alternatives add amplitudes and distinguished ones add probabilities. A state is a normalised vector, its bra is its conjugate transpose, and in an orthonormal basis its coefficients $c_{n}=\braket{n|\psi}$ have moduli squared that add to one.

P=\bigl|\braket{\phi|\psi}\bigr|^{2},\qquad \sum_{n}\ket{n}\bra{n}=\hat{1}

Reminders

$|z_{1}+z_{2}|^{2}=|z_{1}|^{2}+|z_{2}|^{2}+2\operatorname{Re}(z_{1}^{*}z_{2})$ : the last term is interference.
Conjugate the left side of every bracket: $\braket{\phi|\psi}=\sum b_{n}^{*}c_{n}$ .
A global phase changes nothing; a relative phase changes the state.
Cauchy–Schwarz: $|\braket{\phi|\psi}|\le\lVert\phi\rVert\,\lVert\psi\rVert$ .

Applications