Chapter 01

History and physical pictures

From black-body radiation to delayed choice: build the intuition before the formalism.

Sections: 7
Finalised: 7/7
Simulations: 1
Estimated time: 2 hours

1.1The crisis of classical physicsBy 1900 physics looked finished. Three unremarkable experimental facts tore up the foundations.
1.2Black-body radiation and Planck's quantum hypothesisPlanck got the right formula first and went looking for a justification afterwards. The justification he found, he refused to take seriously for over a decade.
1.3The photoelectric effect and Einstein's light quantaFour experimental facts, none of which the classical wave picture can explain. Einstein changed one sentence and all four fell at once.
1.4The Bohr model and the correspondence principleA model its own author admitted was an unprincipled patchwork — and it gave the hydrogen spectrum to four significant figures. Both what it got right and what it got wrong are worth seeing clearly.
1.5de Broglie and wave–particle dualityA doctoral thesis his supervisor could not evaluate, built on a one-line formula. It turned Bohr's unmotivated rule into something close to obvious.
1.6The double slit, told in full◈ 1 sim(s)One experiment, four versions, a hundred and eighty years apart. Feynman said it contains the only mystery of quantum mechanics — and he was not being rhetorical.
1.7Where the uncertainty principle came fromHeisenberg's microscope convinced a generation of physicists. The argument gives the right formula for the wrong reason — and seeing why matters more than memorising the formula.

Start with 1.1 The crisis of classical physics →

Formula reference

Basic equations

Time-dependent Schrödinger equationsource
$\ii\hbar\frac{\partial}{\partial t}\Psi(x,t)=\hat H\,\Psi(x,t)$
The equation of motion of quantum mechanics; first order in time
Time-independent Schrödinger equationsource
$-\frac{\hbar^2}{2m}\frac{\dd^2\psi}{\dd x^2}+V(x)\psi=E\psi$
An eigenvalue problem; solve for E and ψ
Hamiltonian operatorsource
$\hat H=\frac{\hat p^2}{2m}+V(\hat x),\qquad \hat p=-\ii\hbar\frac{\partial}{\partial x}$
Time evolution of a stationary statesource
$\Psi(x,t)=\psi(x)\,\ee^{-\ii Et/\hbar}$
Only an overall phase is added; |Ψ|² does not change with time

Probability and expectation values

Born rulesource
$P(a\le x\le b)=\int_a^b|\Psi(x,t)|^2\,\dd x$
Normalisation conditionsource
$\int_{-\infty}^{\infty}|\Psi(x,t)|^2\,\dd x=1$
Probability current densitysource
$j=\frac{\hbar}{2m\ii}\left(\Psi^*\frac{\partial\Psi}{\partial x}-\Psi\frac{\partial\Psi^*}{\partial x}\right)$
Satisfies the continuity equation ∂ρ/∂t + ∂j/∂x = 0
Expectation valuesource
$\langle \hat A\rangle=\int \Psi^*\hat A\,\Psi\,\dd x$
Uncertainty principlesource
$\sigma_x\sigma_p\ge\frac{\hbar}{2}$
Ehrenfest's theoremsource
$\frac{\dd\langle x\rangle}{\dd t}=\frac{\langle p\rangle}{m},\qquad \frac{\dd\langle p\rangle}{\dd t}=\left\langle-\frac{\partial V}{\partial x}\right\rangle$
Expectation values obey Newton's equations

Exact solutions in one dimension

Infinite square wellsource
$\psi_n=\sqrt{\tfrac{2}{L}}\sin\frac{n\pi x}{L},\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$
Harmonic-oscillator levelssource
$E_n=\left(n+\tfrac12\right)\hbar\omega,\qquad n=0,1,2,\dots$
Harmonic-oscillator ground statesource
$\psi_0=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\ee^{-m\omega x^2/2\hbar}$
Ladder operatorssource
$\hat a_\pm=\frac{1}{\sqrt{2\hbar m\omega}}(\mp \ii\hat p+m\omega\hat x),\qquad [\hat a_-,\hat a_+]=1$
Gaussian wave packetsource
$\Psi(x,0)=(2\pi\sigma^2)^{-1/4}\ee^{-(x-x_0)^2/4\sigma^2}\ee^{\ii k_0x}$
Free-particle dispersion relationsource
$\omega(k)=\frac{\hbar k^2}{2m},\qquad v_g=\frac{\dd\omega}{\dd k}=\frac{\hbar k}{m}$
The group velocity is twice the phase velocity
Rectangular-barrier transmission (E < V₀)source
$T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1},\quad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}$
Thick-barrier approximationsource
$T\approx\frac{16E(V_0-E)}{V_0^2}\ee^{-2\kappa a}$
For κa ≫ 1; the exponential factor dominates

Constants and conversions

Reduced Planck constant
$\hbar=1.054\,571\times10^{-34}\ \text{J·s}=6.582\times10^{-16}\ \text{eV·s}$
Electron mass
$m_e=9.109\times10^{-31}\ \text{kg}=511\ \text{keV}/c^2$
Bohr radius
$a_0=\frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}=0.529\ \text{Å}$
Hydrogen ground-state energy
$E_1=-13.606\ \text{eV}$
Fine-structure constant
$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}$
Handy conversions
$\hbar c=197.3\ \text{eV·nm},\qquad k_BT\big|_{300\text{K}}=25.9\ \text{meV}$