Foundations — Why Quantum Mechanics Exists — formula sheet

Blackbody: an idealised object that absorbs all radiation falling on it at every wavelength, and therefore (by Kirchhoff's law, which equates emissivity and absorptivity at every wavelength and temperature) is also the best possible emitter. The standard laboratory realisation is not a black surface but a cavity: a hollow box held at temperature T with a small hole in one wall. Radiation entering the hole bounces around and is essentially certain to be absorbed before it finds its way out again, so the hole behaves as a near-perfect absorber, and the radiation leaking back out of it is a faithful sample of the thermal radiation inside.

Ultraviolet catastrophe: the classical prediction that the spectral energy density of cavity radiation diverges as λ^−4 at short wavelength, so that any object at any non-zero temperature would radiate infinite energy. Its origin is the combination of (i) an unbounded number of short-wavelength modes and (ii) equipartition, which gives every one of those modes the same average energy kT.

Planck's quantum hypothesis: an oscillator of frequency ν cannot take up or give out energy continuously. Its energy is restricted to the discrete set E_n = nhν with n = 0, 1, 2, …, where h is a new universal constant. Energy is exchanged with the radiation field only in whole quanta of size hν.

Einstein's light-quantum hypothesis: light of frequency ν consists of localised quanta each carrying energy E = hν. In the photoelectric process a single quantum is absorbed by a single electron, all or nothing. The electron then needs a minimum energy φ, the work function of the metal, to escape the surface; whatever is left over appears as kinetic energy.

Work function φ: the minimum energy needed to remove an electron from the surface of a metal to a point just outside it, at 0 K. It is a property of the material and of the particular crystal face and its cleanliness, typically 2–6 eV for metals. It is the solid-state analogue of the ionisation energy of an atom, but is smaller, because in a metal the electron is already delocalised and near the top of the conduction band.

Compton wavelength of the electron: λ_C = h/m_ec = 2.426310 pm. It is a fixed length built from three constants, not a property of the radiation. It sets the scale of the Compton shift: the maximum possible displacement, at θ = 180°, is 2λ_C = 4.8526 pm.

Rydberg constant: the single empirical constant in the formula above. For hydrogen R_H = 109677.58 cm^−1 = 1.096776 × 10^7 m^−1; the infinite-nuclear-mass value is R_∞ = 109737.32 cm^−1, and the two differ by the reduced-mass factor μ/m_e = 0.9994557. It is one of the most accurately known constants in all of physics, which is exactly why it is such a demanding target for any theory.

de Broglie wavelength: the wavelength λ = h/p associated with a particle of momentum p. For an electron accelerated from rest through a potential difference V, p = √(2m_eeV) and λ = h/√(2m_eeV) = (1.2264/√V) nm with V in volts — a form worth memorising, since it turns almost every examination question on this topic into one division and one square root.

Wave packet: a superposition of waves with a range of wavenumbers, localised in space. The spread in position Δx and the spread in wavenumber Δk obey the Fourier relation ΔxΔk ≥ ½, which through p = ℏk is ΔxΔp ≥ ℏ/2 — the Heisenberg uncertainty principle. It is a property of waves, not an extra postulate about measurement.

Born interpretation (1926): for a particle described by a wavefunction ψ(x, y, z, t), the quantity |ψ|^2dτ = ψ*ψdτ is the probability of finding the particle in the volume element dτ at that position at that time. ψ itself is a probability amplitude; it may be negative and is in general complex, and it is not itself observable.

Probability current density: j = (ℏ/2mi)(ψ*∇ψ − ψ∇ψ*). Together with ρ = |ψ|^2 it satisfies the continuity equation ∂ρ/∂t + ∇·j = 0, which is the mathematical statement that probability is conserved: it can flow from place to place but is never created or destroyed. This is what guarantees that a normalised wavefunction stays normalised as it evolves.

A physically acceptable wavefunction must be: (i) single-valued — one value of ψ at each point, since |ψ|^2 is a probability and a probability cannot have two values; (ii) continuous — and its first derivative continuous too, except where the potential is infinite; (iii) finite everywhere — an infinite ψ would mean an infinite probability density; (iv) square-integrable — ∫|ψ|^2dτ must converge, so that ψ can be normalised. Functions satisfying these are called well-behaved.

Orthogonality: two functions ψ_m and ψ_n are orthogonal if their overlap integral vanishes, ∫ψ_m*ψ_ndτ = 0. A set that is both normalised and mutually orthogonal is orthonormal: ∫ψ_m*ψ_ndτ = δ_mn, where δ_mn is 1 if m = n and 0 otherwise.

Postulate 1. The state of a quantum-mechanical system is completely specified by a wavefunction Ψ(r, t) which is a single-valued, continuous, finite and square-integrable function of the coordinates and the time. All the information that can be known about the system is contained in Ψ.

Postulate 2. To every observable physical quantity there corresponds a linear Hermitian operator. The operators for position and momentum are x̂ = x× (multiplication) and p̂_x = −iℏ∂/∂x; the operator for any other observable is obtained by writing its classical expression in terms of x and p and replacing each by the corresponding operator.

Postulate 3. The only values that a measurement of the observable A can return are the eigenvalues a_n of its operator Â, that is, the numbers satisfying Âψ_n = a_nψ_n. If the system is in the state φ = ∑c_nψ_n, the probability of obtaining a_n is |c_n|^2.

Postulate 4. The average value of the observable A over many measurements on identically prepared systems is ⟨A⟩ = ∫ψ*Âψdτ / ∫ψ*ψdτ, which reduces to ⟨A⟩ = ∫ψ*Âψdτ for a normalised ψ.

Postulate 5. If a measurement of A on a system in state φ returns the eigenvalue a_n, then immediately after the measurement the system is in the corresponding eigenstate ψ_n. A second measurement made at once returns a_n again with certainty.

Postulate 6. Between measurements the wavefunction evolves according to the time-dependent Schrödinger equation, iℏ ∂Ψ/∂t = ĤΨ.