Quantum Mechanics · Lesson 11

Chapter practice

Bring the chapter together: the Schrödinger equation and its conservation of probability, the Hamiltonian and its levels, stationary states and the beats between them, the evolution operator, the Heisenberg picture, conserved quantities, the free particle, the probability current and Ehrenfest’s theorem. No worked example sits above the answers.

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i ℏ \frac{d}{d t} ∣ ψ ⟩ = H^∣ ψ ⟩

H^∣ E ⟩ = E ∣ E ⟩

U^(t) = e^{- \frac{i H ^ t}{ℏ}}

ψ (t) = e^{- \frac{i E t}{ℏ}} ψ (0)

Chapter navigation

Note

Prerequisites

Complete The Ehrenfest theorem first.

Intuition

Bring the chapter together: the Schrödinger equation and its conservation of probability, the Hamiltonian and its levels, stationary states and the beats between them, the evolution operator, the Heisenberg picture, conserved quantities, the free particle, the probability current and Ehrenfest’s theorem. No worked example sits above the answers.

A clockmaker knows each gear and how they turn together. These questions ask for the whole mechanism of quantum time.

Before you practise

States evolve by $\hat{U}(t)=e^{-i\hat{H}t/\hbar}$ : energy eigenstates only turn their phase, superpositions beat at Bohr frequencies, averages change at the rate of their commutator with $\hat{H}$ , and probability flows with the current $j$ .

i\hbar\frac{d}{dt}\ket{\psi}=\hat{H}\ket{\psi},\qquad \frac{d}{dt}\langle\hat{A}\rangle=\frac{i}{\hbar}\langle[\hat{H},\hat{A}]\rangle

Reminders

$\ket{\psi(t)}=\sum c_{n}e^{-iE_{n}t/\hbar}\ket{E_{n}}$ ; beats at $(E_{m}-E_{n})/\hbar$ .
$\hat{A}_{H}=\hat{U}^{\dagger}\hat{A}\hat{U}$ , with .
Free particle: $\omega=\hbar k^{2}/2m$ , packets move at $p/m$ and spread.
$j=\tfrac{\hbar}{m}\operatorname{Im}(\psi^{*}\psi')$ ; .

Applications

Practice

Try it

Levels at 2 eV and 5 eV are superposed. With $\hbar\approx6.58\times10^{-16}$ eV s, what is the Bohr angular frequency, in units of $10^{15}$ radians per second? Give two decimal places.