Chapter 04

One-dimensional problems and numerical methods

Parity, phase shifts, transfer matrices — and writing your own Schrödinger solver.

Sections: 5
Finalised: 5/5
Simulations: 2
Estimated time: 3 hours
Read first: Chapter 02 Chapter 03

4.1Symmetry and parity◈ 1 sim(s)When a potential is left-right symmetric, every bound-state wavefunction is either even or odd — not a coincidence, but a free gift delivered by the commutator [H,P] = 0.
4.2Scattering states and phase shifts◈ 1 sim(s)Everything a potential does to an incident wave condenses, in the end, into one angle: how far it pulled the waveform in, or pushed it out.
4.3The transfer matrix methodPackage each slab of potential as a 2×2 matrix, and an entire multilayer structure becomes a matrix product — the double barrier's razor-sharp resonance and the energy bands of a superlattice both grow straight out of the multiplication.
4.4Solving the Schrödinger equation numericallySwap the continuous x axis for a row of grid points and the Schrödinger equation becomes a tridiagonal matrix eigenvalue problem — ten lines of code solve the bound states of any potential.
4.5A numerical solver for custom potentialsPackage last section's method into a general-purpose solver: feed in any potential curve, get back levels and wavefunctions — then commission it on the double well, explaining the ammonia maser along the way.

Start with 4.1 Symmetry and parity →

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