Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: wavefunctions and the delta function, densities and normalisation, the position and momentum operators, their commutator and Heisenberg’s relation, the Fourier transform to momentum space and the Gaussian packet. No worked example sits above the answers.

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ψ (x) = ⟨ x ∣ ψ ⟩

\int ∣ ψ (x) ∣^{2} d x = 1

p^= - i ℏ \frac{\partial}{\partial x}

[x^, p^] = i ℏ

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Intuition

Bring the chapter together: wavefunctions and the delta function, densities and normalisation, the position and momentum operators, their commutator and Heisenberg’s relation, the Fourier transform to momentum space and the Gaussian packet. No worked example sits above the answers.

A translator moves between two languages without losing the meaning. These questions move between position and momentum.

Before you practise

A state on a line is its wavefunction $\psi(x)=\braket{x|\psi}$ , with density $|\psi|^{2}$ integrating to one. Position multiplies, momentum differentiates, and they satisfy $[\hat{x},\hat{p}]=i\hbar$ ; the momentum-space wavefunction is the Fourier transform.

\hat{x}\to x,\qquad \hat{p}\to-i\hbar\frac{d}{dx},\qquad \Delta x\,\Delta p\ge\frac{\hbar}{2}

Reminders

$\int\delta(x-a)f(x)\,dx=f(a)$ .
$\langle p\rangle=0$ for a real wavefunction; a phase $e^{ik_{0}x}$ adds $\hbar k_{0}$ .
$[f(\hat{x}),\hat{p}]=i\hbar f'(\hat{x})$ .
Gaussian packet: $\Delta x=\sigma$ , $\Delta p=\hbar/2\sigma$ .

Applications