Quantum Mechanics · Lesson 09

Chapter practice

Bring the chapter together: which results a measurement can give and with what probabilities, the state it leaves, averages and spreads, the uncertainty relation, sequences of measurements, compatible observables and degenerate results. No worked example sits above the answers.

Foundations8 min readRevision 1

Read freely. Sign in when you want to save your place.

Sign in to save progress

A^^{†} = A^

P (a) = ∣ ⟨ a ∣ ψ ⟩ ∣^{2}

⟨ A^⟩ = ⟨ ψ ∣ A^∣ ψ ⟩

Δ A Δ B \geq \frac{1}{2} ∣ ⟨[A^, B^]⟩ ∣

Chapter navigation

Intuition

Bring the chapter together: which results a measurement can give and with what probabilities, the state it leaves, averages and spreads, the uncertainty relation, sequences of measurements, compatible observables and degenerate results. No worked example sits above the answers.

A referee who knows the rules does not reread them at every whistle. These questions are for knowing the postulates that well.

Before you practise

An observable is Hermitian; its eigenvalues are the possible results. The probability of a result is $\bra{\psi}\hat{P}_{a}\ket{\psi}$ , and afterwards the state is its normalised projection. Averages are brackets, spreads are square roots of variances, and non-commuting observables cannot both be sharp.

\langle\hat{A}\rangle=\bra{\psi}\hat{A}\ket{\psi},\qquad \Delta A\,\Delta B\ge\tfrac12\bigl|\langle[\hat{A},\hat{B}]\rangle\bigr|

Reminders

$(\Delta A)^{2}=\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle^{2}$ , zero exactly in eigenvectors.
Sequences multiply: $P(a,\text{ then }b)=P(a)\,|\braket{b|a}|^{2}$ .
Commuting observables share eigenvectors and can be sharp together.
For eigenvalues $\pm1$ : $\langle\hat{A}\rangle=P(+1)-P(-1)$ and .

Applications

Practice

Try it

A qubit is in $\tfrac{1}{\sqrt3}\ket{0}+\sqrt{\tfrac23}\,\ket{1}$ . What is the probability that $Z=\operatorname{diag}(1,-1)$ gives $-1$ ? Give three decimal places.