Non-relativistic QM review

The non-relativistic formalism is not a museum piece — it is a working tool wherever particle speeds stay well below c. Its most literal modern descendant is non-relativistic QCD (NRQCD) and potential NRQCD, effective field theories in which a heavy quark–antiquark pair such as charmonium (J/ψ) or bottomonium (Υ) is treated as a two-body Schrödinger problem with a QCD potential; lattice and NRQCD predictions of quarkonium spectra and decay rates rest directly on the machinery above. The operator-and-flux language reappears every time an experiment reports a cross section: the measured event rate is divided by the incident flux j = nv derived in Derivation B, now promoted to its relativistic form. The angular-momentum algebra of equation (6) is the literal blueprint for isospin and the SU(2) of the electroweak interaction, so the |ℓ, m⟩ labelling recurs as the (I, I₃) of hadron multiplets. And the de Broglie wavelength of Example 1 is the resolution limit of every electron microscope and the reason electron and neutron diffraction map crystal and magnetic structure — a 100 eV electron's 0.12 nm wavelength is what lets it "see" individual atomic planes.

The whole scheme starts from a picture of a free particle as an infinite, monochromatic wave. The plane wave ψ = N e^{i(p·x − Et)} has a definite momentum, encoded in how fast its phase winds in space, and a definite energy, encoded in how fast it winds in time. Acting with the spatial gradient pulls a factor ip out of the exponent, and acting with the time derivative pulls out −iE — that is the entire content of the operator dictionary p̂ = −i∇ and Ê = i∂ₜ. The first figure follows one derivative at a time, turning a picture of a wave into the equation of motion, and pins the abstraction to a real length scale: a 100 eV electron has a de Broglie wavelength of about 0.12 nm, the atomic spacing that makes electron diffraction work.

The second idea is that ψ is not just a bookkeeping device but a thing that carries a conserved substance. Born's rule makes ρ = ψ*ψ a probability density, and the same manipulation that Maxwell's charge conservation uses — combine the field equation with its complex conjugate — produces a current j and a continuity law ∂ₜρ + ∇·j = 0. Probability that leaves a region must flow out through its boundary; none is created or destroyed. For the plane wave this current is exactly a number density times a velocity, j = nv, and that is the through-line to particle physics: when the wavefunction is later made relativistic, this same j becomes the incident flux dividing every cross section. The second figure shows the flux picture that survives the whole way up.

The starting postulate is the free-particle plane wave, and the two operator identifications are not independent guesses — they are forced by demanding that each derivative reproduce the corresponding factor in the exponent. In natural units ℏ = 1 throughout.

where: ψ — the complex free-particle wavefunction; N — normalisation constant; p — momentum eigenvalue; E — energy eigenvalue; ℏ = 1. Reading: −i∇ acting on ψ returns pψ and i∂ₜ acting on ψ returns Eψ, so on plane waves the operators reproduce their classical values as eigenvalues. Substituting these operators into the non-relativistic energy relation E = p²/2m + V promotes it to an operator equation that any state, not just a plane wave, must satisfy — the equation of motion of the theory.

where: Ĥ — the Hamiltonian operator (total energy); the first term is kinetic, V(x) is the potential. Reading: the Hamiltonian generates time evolution; knowing Ĥ and the state now fixes the state at every later instant. For V = 0 the plane wave (1) is an exact solution with E = p²/2m. Born interprets |ψ|² as a probability density. Because total probability must stay equal to one, the loss of density from any region has to be accounted for by a flow across its boundary — a local conservation law with a current, obtained purely from (2) and its complex conjugate.

where: ρ — probability density; j — probability current density; the third equation is the continuity equation. Reading: ρ and j are not independent postulates — Derivation B builds j so that the continuity equation holds identically whenever ψ obeys the Schrödinger equation. Evaluated on the plane wave (1), this current collapses to a strikingly classical object: a number density carried along at the particle velocity. This is the form that survives, essentially unchanged, into the relativistic flux factor.

where: n = |N|² — particle number density (not a single-particle probability); v = p/m — the group velocity of the wave. Reading: the plane wave is best pictured as a uniform beam of density n streaming at v, giving a flux nv — the language of scattering, not of a bound electron. Dynamics of averages is governed by commutation with Ĥ. Differentiating the expectation value ⟨Â⟩ = ∫ψ*Âψ d³x and using (2) gives Ehrenfest's theorem, the bridge between quantum evolution and classical mechanics and the exact criterion for a conserved quantity.

where: [Ĥ, Â] = Ĥ − ÂĤ is the commutator; the last term vanishes for operators without explicit time dependence. Reading: if  has no explicit t-dependence and commutes with Ĥ, then ⟨Â⟩ is constant in time — this is Noether's theorem in operator language. Finally, the algebra of commutators sets which observables are simultaneously knowable and supplies the labels for states. The canonical relation controls uncertainty; the angular-momentum relations supply the multiplet structure reused throughout particle physics.

where: δ_{ij} — Kronecker delta; ε_{ijk} — Levi-Civita symbol; L̂ = x̂ × p̂ — orbital angular momentum. Reading: [x̂, p̂ₓ] = i forces the uncertainty bound ΔxΔp ≥ 1/2; because L̂² commutes with L̂_z but the L̂ᵢ do not commute with each other, states are labelled |ℓ, m⟩ with L̂²|ℓ, m⟩ = ℓ(ℓ + 1)|ℓ, m⟩ and L̂_z|ℓ, m⟩ = m|ℓ, m⟩.

1 · The de Broglie wavelength of a 100 eV electron (confidence builder)

An electron is accelerated through 100 V, giving it kinetic energy E_k = 100 eV. Find its momentum, de Broglie wavelength, and speed, and confirm that non-relativistic mechanics is justified.

Step 1. The electron is non-relativistic (to be checked), so E_k = p²/2m and p = √(2mE_k). Working with pc = √(2mc²·E_k) avoids unit juggling: with mc² = 511 000 eV, pc = √(2 × 511 000 × 100) eV = √(1.022 × 10⁸) eV = 10.1 keV.

Step 2. The de Broglie wavelength is λ = 2π/p = hc/(pc). Using hc = 1239.8 eV·nm, λ = 1239.8 / 10 109 = 0.123 nm. This is a fraction of a nanometre — comparable to atomic spacings in a crystal, which is precisely why 100 eV electrons diffract off crystal surfaces (the basis of low-energy electron diffraction, LEED).

Step 3. The speed follows from v/c = pc/(mc²) = 10 109 / 511 000 = 0.0198, so v = 5.9 × 10⁶ m/s = 0.020c. The relativistic correction is of order (v/c)² ≈ 4 × 10⁻⁴, equivalently γ − 1 = E_k/mc² = 2 × 10⁻⁴, so treating the electron non-relativistically is accurate to about 0.02%. λ = 0.123 nm at v = 0.020c, comfortably non-relativistic.

2 · The probability current of a plane wave is a flux (fully worked)

Take the free plane-wave state ψ = N e^{i(p·x − Et)}. Compute its probability density and current explicitly, and interpret the result for the electron of Example 1.

Step 1. Density: ρ = ψ*ψ = |N|² e^{−i(p·x−Et)} e^{i(p·x−Et)} = |N|², a constant everywhere. The state has no localisation — it is spread uniformly over all space, which is why it must be read as a beam, not a single located particle.

Step 2. Current: with ∇ψ = ipψ we get ψ∇ψ = i<strong>p</strong>|N|² and ψ∇ψ = −ip|N|². Then j = (1/2im)(ψ∇ψ − ψ∇ψ) = (1/2im)(2ip|N|²) = |N|²p/m.

Step 3. Interpret: write n = |N|² and v = p/m, so j = nv. This is a number density times a velocity — a flux. For the 100 eV electron, v = 5.9 × 10⁶ m/s, so a beam of density n carries a flux j = n × 5.9 × 10⁶ particles per m² per second. In a scattering calculation this j is exactly what a measured event rate is divided by to define a cross section. j = |N|²p/m = n v — the incident flux factor of scattering theory.

3 · The angular-momentum algebra from the canonical commutator

Using only [x̂ᵢ, p̂ⱼ] = iδ_{ij} (coordinates mutually commute, momenta mutually commute), show that [L̂ₓ, L̂_y] = iL̂_z, verifying the SU(2) template of equation (6).

Step 1. Write the components: L̂ₓ = ŷp̂_z − ẑp̂_y and L̂_y = ẑp̂ₓ − x̂p̂_z. Expand the commutator into four pieces: [L̂ₓ, L̂_y] = [ŷp̂_z, ẑp̂ₓ] − [ŷp̂_z, x̂p̂_z] − [ẑp̂_y, ẑp̂ₓ] + [ẑp̂_y, x̂p̂_z].

Step 2. Evaluate each using [ẑ, p̂_z] = i and dropping terms of commuting operators. The middle two vanish: [ŷp̂_z, x̂p̂_z] = 0 and [ẑp̂_y, ẑp̂ₓ] = 0 (no conjugate pair meets). The surviving pieces are [ŷp̂_z, ẑp̂ₓ] = ŷp̂ₓ[p̂_z, ẑ] = −iŷp̂ₓ and [ẑp̂_y, x̂p̂_z] = x̂p̂_y[ẑ, p̂_z] = ix̂p̂_y.

Step 3. Combine with the signs from Step 1: [L̂ₓ, L̂_y] = −iŷp̂ₓ + ix̂p̂_y = i(x̂p̂_y − ŷp̂ₓ) = iL̂_z. The algebra closes on itself, cyclically. This is the SU(2) Lie algebra; replacing L̂ by isospin Î reproduces [Îₓ, Î_y] = iÎ_z, so the proton–neutron doublet and the pion triplet are organised by the very same commutators derived here from [x̂, p̂] = i.

Non-relativistic quantum mechanics has two fathers — Heisenberg's matrix mechanics and Schrödinger's wave mechanics — but it is the wave equation, and the wavefunction it acts on, that this article is built from, so the portrait is Schrödinger's.

Texts. Thomson, Modern Particle Physics, §2.3 (non-relativistic quantum mechanics) — the direct source and the particle-physics framing used here · Griffiths, Introduction to Quantum Mechanics, 3rd ed., Chs. 1–4 (wavefunction, formalism, angular momentum) · Sakurai & Napolitano, Modern Quantum Mechanics, Chs. 1–3 (operator and commutator structure) · Shankar, Principles of Quantum Mechanics, Chs. 4–12 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §2.1 (where this review's flux and operator language is made relativistic).

Provenance. The plane-wave conventions and ℏ = 1 bookkeeping are recalled by name from aq-pp-natural-units and aq-pp-four-vector-notation; the operator correspondence, continuity equation, and angular-momentum algebra are re-derived in full above. A verifier should confirm: (1) −i∇ψ = pψ and i∂ₜψ = Eψ on the plane wave, yielding i∂ₜψ = (−∇²/2m + V)ψ; (2) ∂ₜρ + ∇·j = 0 identically from the Schrödinger equation and its conjugate, with j = |N|²p/m = nv for a plane wave; (3) d⟨Â⟩/dt = i⟨[Ĥ, Â]⟩ and [Ĥ, L̂²] = [Ĥ, L̂_z] = 0 for a central potential; (4) [L̂ₓ, L̂_y] = iL̂_z from [x̂ᵢ, p̂ⱼ] = iδ_{ij}; (5) the 100 eV electron numbers pc = 10.1 keV, λ = 0.123 nm, v = 0.020c. All standard values (mc² = 511 keV, hc = 1239.8 eV·nm) are as tabulated.

Prerequisites. Natural Units · Four-Vector Notation. Leads to. Klein–Gordon Equation · Dirac Equation · Symmetries in Quantum Mechanics · Isospin · Two-Flavour Neutrino Oscillations.

:::faq[What does [x̂, p̂ₓ] = i actually mean physically?] It is the mathematical root of the uncertainty principle. Because position and momentum operators do not commute, they cannot share a common set of eigenstates, so no state can have both a definite position and a definite momentum at once. Quantitatively, the commutator sets the bound ΔxΔp ≥ 1/2 (that is ℏ/2 in ordinary units): the more sharply localised a wavefunction is in space, the broader the spread of momenta needed to build it out of plane waves, and vice versa. This is not a limitation of measurement apparatus but a property of the states themselves — a narrow wave packet is genuinely a superposition of many momenta. The same non-commutation, applied to angular-momentum components, is why you can know L² and L_z but never Lₓ and L_y together.

Ask the tutor: ask me to push any plane-wave result here — the current, the commutator algebra, or the de Broglie number — one step further into its relativistic successor, and watch where each piece lands in Klein–Gordon, Dirac, or a cross-section calculation.