Chapter 12

Relativistic quantum mechanics

Klein–Gordon and Dirac, and why spin is an inevitable consequence of relativity.

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12.1Why we need relativistic quantum mechanicsWhy is gold golden? Why is mercury liquid? Because their inner electrons move at sixty percent of the speed of light. The Schrödinger equation is first order in time and second order in space — congenitally at odds with relativity. This chapter rebuilds the equation.
12.2The Klein–Gordon equationTranslate E²=p²c²+m²c⁴ directly into a wave equation: covariance achieved, at the price of a probability density that can go negative. Track the negative probability to its source and you find it is reporting something true — the existence of antiparticles.
12.3The Dirac equationTake the square root of E²=p²c²+m²c⁴: what numbers cannot do, matrices can. The instant the equation is written down, spin 1/2 walks in uninvited — not an extra postulate, but the price of making relativity and quantum mechanics compatible.
12.4Antiparticles and the negative-energy seaWith no floor to the spectrum, what keeps atoms stable? Dirac's answer was to fill the vacuum: the negative-energy sea. A hole in the sea is a positron — written on paper in 1931, found in a cloud chamber in 1932.
12.5Spin and relativity, inseparably joinedPut the Dirac equation in an electromagnetic field and take the low-speed limit: g=2 drops out by itself. Expand one order further and the fine-structure trio — relativistic correction, spin-orbit coupling (Thomas 1/2 included), Darwin term — arrives automatically, complete. Spin is relativity's gift to quantum mechanics.

Start with 12.1 Why we need relativistic quantum mechanics →

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
$\hbar c=197.3\ \text{eV·nm},\qquad k_BT\big|_{300\text{K}}=25.9\ \text{meV}$