Solve it first. Abstract it once you have earned the abstraction.

Topics and mapped lessons

• Where the classical account fails ↗

  Line spectra are discrete, heat capacities freeze out, cavity radiance turns over, and a classical orbiting electron radiates into the nucleus in about ten picoseconds. Order-of-magnitude estimates beside data. Each failure has a patch; only the pattern forces a theory.

• Blackbody radiation and the Planck hypothesis ↗

  Counting cavity modes and giving each kT yields Rayleigh-Jeans and a divergence; restricting a mode to energies n h nu and summing a geometric series yields Planck's law, with h fixed by fitting radiance data. Planck quantised the walls, not the field.

• The photoelectric effect and the photon hypothesis ↗

  Threshold frequency, prompt emission, and intensity that sets rate rather than energy. Technique: fit eVₛ = h ν - φ, so Vₛ against ν has slope h/e and intercept −φ/e. The data force quantised absorption, not yet a quantised field.

• Compton scattering and photon momentum ↗

  Photon and electron as a relativistic two-body collision: conserve energy and momentum, eliminate the recoil angle, and get δ-λ = (h/mₑ c)(1 - cos θ), h/mₑ c = 2.43 pm. It assumes a free electron at rest; bound ones return the unshifted line.

• de Broglie matter waves and electron diffraction ↗

  λ = h/p carried from photons to matter as h/√(2 m e V) for an accelerated electron, then tested through the Bragg condition against Davisson-Germer and Thomson ring angles, later neutrons and C60. Non-relativistic, and single scattering assumed.

• Wave-particle duality and the double slit ↗

  Electrons arrive one at a time as localised detections yet accumulate into fringes that any which-path measurement destroys. Method: add amplitudes, then square - never add probabilities. Complementarity is read off the experiments, not derived.

• The Born interpretation: P(x, t) = |ψ(x, t)|² ↗

  psi is complex and never measured directly; |ψ(x, t)|² is a probability density, so a probability is its integral over an interval and psi has units of inverse root length. It predicts frequencies over identical preparations, never the outcome of one run.

• Normalising psi and reading probabilities off it ↗

  The constant left free by a linear equation is fixed by requiring the integral of |ψ|² over all x to equal 1, evaluated with Gaussian and polynomial integrals. Only square-integrable psi normalise; a plane wave does not, and needs a box or delta convention.

• The requirements on ψ(x, t), and the reason for each ↗

  Single-valued because one point carries one probability; continuous because a jump would make ψ-double-prime the derivative of a δ, which no potential in the equation can balance; square-integrable so it normalises; ψ-prime continuous wherever V is finite, since ψ-double-prime is finite there. Infinite walls and delta wells relax the last, scattering states the third.

Learning outcomes

• Name four quantitative failures of classical physics and, for each, the measurement it contradicts and the scale at which it fails.
• Extract h from a stopping-potential fit and h/mₑ c from a Compton shift, and say what each experiment establishes and what it merely permits.
• Apply λ = h/√(2 m e V) to an accelerated electron and predict the diffraction angles a known lattice spacing produces.
• Read P(x, t) = |ψ(x, t)|² as a density: normalise ψ and obtain a probability by integrating |ψ|² over a stated interval.
• Justify each requirement on ψ - single-valued, continuous, square-integrable, psi' continuous where V is finite - from the Born rule or the Schrodinger equation, and name the cases that relax the last two.

Prerequisite thread

• University Physics I-III or equivalent, including waves, interference and electromagnetism
• A modern-physics course: relativity, photons, atomic spectra and the Bohr model
• Multivariable calculus and ordinary differential equations
• Complex numbers in modulus-argument form, plus Gaussian and improper integrals

Laboratory directions

Problem practice

Getting a number out of each experiment before interpreting it, and writing every probability as an integral of |ψ|² over an interval

Capstone: Normalise a piecewise ψ(x) that is continuous but kinked, compute P(0 < x < a/2) from it, then name the requirement the kink violates, the potential that would have to sit there to permit it, and why a tail falling as 1/√(x) cannot be normalised at all while a 1/x tail can.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• The time-dependent Schrodinger equation ↗

  i ℏ dPsi/dt = H-hat Ψ: first order in time, so one Ψ(x,0) fixes every later time, and linear, so superpositions propagate. Plane waves with E = ℏ omega only motivate it - it is postulated, not derived, and it is non-relativistic.

• The Hamiltonian operator ↗

  H-hat = p-hat²/2m + V(x), built by putting −i ℏ d/dx in place of p in the classical energy, giving -(ℏ²/2m) d2/dx2 + V(x). One spinless particle in a real, local, velocity-independent V; magnetic forces need minimal coupling instead.

• Separation of variables and the constant E ↗

  Try Ψ(x, t) = ψ(x) f(t) and divide by it: each side then depends on one variable alone, so both equal a constant, and f(t) = exp(−iEt/ℏ) follows. The trick needs V independent of t, and it returns special solutions, not the general one.

• The time-independent Schrodinger equation ↗

  -(ℏ²/2m) psi'' + V(x) ψ = E ψ, an eigenvalue problem read through curvature: psi''/ψ = 2m(V - E)/ℏ² bends ψ toward the axis where E > V and away from it where E < V. Set in one dimension with a static real V; that E comes out real is a result proved from normalisability, not an assumption.

• Stationary states, and superpositions that are not ↗

  In a separable solution the phase exp(−iEt/ℏ) cancels from |Ψ|², so the density and every expectation value of a time-independent quantity stand still. Two levels together make |Ψ|² slosh at angular frequency (E2 - E1)/ℏ; writing a general Psi as such a sum assumes the ψₙ are complete.

• Probability density and probability current ↗

  ρ(x, t) = |Ψ(x, t)|², a density of dimension one over length, and j(x, t) = (ℏ/m) Im(Ψ* dPsi/dx), of dimension one over time - the rate at which probability flows past x. Both are built from Psi alone, and j vanishes wherever psi is real, so a bound stationary state carries no current.

• The continuity equation and conservation of probability ↗

  Differentiate ρ, substitute the equation and its conjugate, and the terms collect into d ρ/dt + d j/dx = 0. Integrating that holds the norm fixed - a theorem, not an assertion - and it needs V real and Psi vanishing at the limits.

• Admissibility and matching conditions on psi ↗

  psi is continuous everywhere; psi' is continuous wherever V is finite, jumps at an infinite wall where ψ = 0, and breaks by a set amount at a δ well. The rules come from integrating the equation across the step. A bound state must also be square-integrable - step and plane-wave solutions are not, and are made physical by packets in a later unit.

• How boundary conditions quantise the energy ↗

  The ODE solves for every E; only isolated values leave a solution meeting both boundaries and staying normalisable. ψ(0) = ψ(L) = 0 gives Eₙ = n² π² ℏ² / 2mL². Only E below V's asymptotic value can be discrete.

Learning outcomes

• State i ℏ dPsi/dt = H-hat Ψ, identify H-hat inside it, and say which of its features are postulated rather than derived.
• Separate a static-V problem into ψ(x) exp(−iEt/ℏ), and explain why the separation constant is the energy.
• Solve -(ℏ²/2m) psi'' + V ψ = E ψ for a piecewise-constant V, matching ψ at every boundary and ψ' only where V is finite.
• Derive d ρ/dt + d j/dx = 0 from the equation and its conjugate, and show the norm is constant when V is real.
• Show that normalisability and the boundary conditions, not the differential equation, select the discrete energies.

Prerequisite thread

• Unit 1's wavefunction, the Born rule, and normalisation of psi on a stated interval
• Second-order linear ODEs, and separation of variables for partial differential equations
• Complex numbers in modulus-argument form, complex conjugation, and Euler's formula
• Partial derivatives, integration by parts, and improper integrals over the whole line

Laboratory directions

Problem practice

Naming V, the interval and the boundary conditions before integrating, and checking that every claimed bound state is normalisable.

Capstone: Take the infinite square well: get ψₙ and Eₙ from ψ(0) = ψ(L) = 0, form the normalised superposition (ψ₁ + ψ₂)/√(2), find the angular frequency (E₂ - E₁)/ℏ at which |Ψ|² sloshes, then compute ρ and j for that state and check d ρ/dt + d j/dx = 0 point by point.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• Operators: linear instructions that act on a wavefunction ↗

  An operator is an instruction - multiply by x, differentiate - and only linear ones qualify, tested by A(af + bg) = aAf + bAg. Functions then behave like vectors and operators like matrices, but this space is infinite-dimensional, so no finite matrix is exact.

• The position operator x-hat and the momentum operator p-hat ↗

  x-hat acts by multiplication, x-hat ψ = x ψ, and is Hermitian simply because x is real; p-hat = −i ℏ d/dx is read off exp(i k x), which it returns multiplied by ℏ k. That plane wave is not normalisable, so it motivates the rule rather than deriving it; the i is what makes p-hat Hermitian, since d/dx on its own is anti-Hermitian.

• Kinetic energy, potential energy, and the Hamiltonian operator ↗

  Putting p-hat into p²/2m gives T-hat = -(ℏ²/2m) d²/dx², so H-hat = T-hat + V(x) turns the time-independent Schrodinger equation into H-hat ψ = E ψ. The substitution is unambiguous only because V depends on x alone, never on p, and H-hat is Hermitian only because V(x) is real.

• Eigenvalue equations: solving A-hat ψ = a ψ ↗

  Act on a trial function and ask whether it returns multiplied: exp(i k x) is an eigenfunction of p-hat with eigenvalue ℏ k, and sin(n π x / a), handed over here as a function to test rather than as a system already solved, is one of H-hat inside a box. Boundary conditions and normalisability select the eigenvalues; x-hat and p-hat keep no normalisable one.

• Hermitian operators, and why their eigenvalues are real ↗

  Test integral f*(A-hat g) dx = integral (A-hat f)* g dx, done for p-hat by integrating by parts. Setting f and g both equal to an eigenfunction forces a* = a, so measured values are real. The boundary term must vanish, so Hermiticity is a claim about an operator together with a domain, never about a formula alone.

• Orthogonality of eigenfunctions belonging to different eigenvalues ↗

  Sandwiching a Hermitian A-hat between two eigenfunctions and moving it each way gives (aₘ - aₙ) integral ψₘ* ψₙ dx = 0, so distinct eigenvalues force orthogonality. Degenerate ones prove nothing: there orthogonality is arranged by Gram-Schmidt, not inherited.

• Completeness: expanding any state on an eigenbasis ↗

  For a discrete spectrum, write psi as a sum of cₙ ψₙ and get cₙ = integral ψₙ* ψ dx by Fourier trick, orthonormality doing the work, so sum |cₙ|² = 1 and |cₙ|² is the probability of measuring aₙ; for p-hat, whose spectrum is continuous, that sum becomes a Fourier integral, named here and used later. Completeness is asserted, not proved, and holds in mean square only. That a measurement yielding aₙ leaves the system described by ψₙ is stated as a postulate; whether anything physically collapses is an interpretive question this course flags rather than settles.

• Expectation values, variance, and standard deviation ↗

  ⟨A⟩ = integral ψ* A-hat ψ dx, the operator between the factors so it acts on ψ first, equal to sum aₙ |cₙ|² in an eigenbasis; σ² = ⟨A²⟩ - ⟨A⟩² needs A-hat applied twice, not ⟨A⟩ squared. Both are means over an ensemble of identically prepared systems, never the outcome of one measurement, and sigma is the spread of that ensemble rather than an instrument error.

• Commutators, [x-hat, p-hat] = i ℏ, and compatibility ↗

  [A-hat, B-hat] = A-hat B-hat - B-hat A-hat, evaluated on a test function f, where the product rule turns [x-hat, p-hat] f into i ℏ f. Commuting Hermitian operators admit a complete common eigenbasis and so are compatible; because i ℏ is a nonzero multiple of the identity, x-hat and p-hat share not even one. The quantitative uncertainty bound waits for Unit 4.

Learning outcomes

• Apply x-hat, p-hat = −i ℏ d/dx and H-hat = -(ℏ²/2m) d²/dx² + V(x) to a given ψ in the right order.
• Decide whether a trial function is an eigenfunction of a stated operator, read off the eigenvalue, and say which boundary condition selects it.
• Prove p-hat and H-hat Hermitian by integration by parts, naming the vanishing boundary term and real V(x) as the conditions actually used, and derive real eigenvalues and orthogonality from that proof.
• Expand a state on a discrete eigenbasis, extract cₙ by Fourier's trick, and read |cₙ|² as the probability of measuring aₙ.
• Compute ⟨A⟩ and σ_A as integrals, and evaluate [A-hat, B-hat] on a test function to obtain [x-hat, p-hat] = i ℏ.

Prerequisite thread

• Units 1-2: the wavefunction, normalisation, the Born rule, and the Schrodinger equation
• Integration by parts, improper integrals over the whole line, and partial derivatives
• Complex conjugates, modulus squared, and Euler's formula for exp(i k x)
• Vectors, dot products and matrix products; eigenvectors are developed here, not assumed

Laboratory directions

Problem practice

Writing the operator between ψ-star and psi before integrating, and testing every claimed eigenfunction against the boundary conditions

Capstone: Take as supplied data the infinite-well eigenfunctions ψₙ(x) = √(2/a) sin(n π x / a) on 0 < x < a, with H-hat ψₙ = Eₙ ψₙ and Eₙ = n² π² ℏ² / (2 m a²); the well itself is solved in a later unit, so here they are only a ready-made orthonormal eigenbasis to practise on. Build ψ = c₁ ψ₁ + c₂ ψ₂ with a complex relative phase, normalise it, then compute ⟨x⟩, ⟨p⟩, ⟨H⟩ and σₓ by hand. Say which of them the interference term moves, which the relative phase leaves untouched, and check your ⟨H⟩ against sum Eₙ |cₙ|².

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• Observables, Hermitian operators, and real eigenvalues ↗

  Demand ⟨Q⟩ = ⟨Q⟩* for every state, integrate by parts to move the derivative across, and Hermiticity follows - with it a real spectrum and orthogonal eigenfunctions. The boundary terms must vanish, so Hermiticity belongs to the operator plus its domain.

• Eigenvalue equations and determinate states ↗

  Q ψ = q ψ is the statement that every measurement of Q returns q: substitute it into σ-Q squared = ⟨(Q - q)²⟩ and the spread is zero. Only normalisable eigenfunctions count as states, so no determinate state of x or p exists on the whole line.

• Expansion on an eigenbasis and the probability of an eigenvalue ↗

  Project with cₙ = integral ψₙ* ψ dx, then P(qₙ) = |cₙ|², sum |cₙ|² = 1, and ⟨Q⟩ = sum qₙ |cₙ|². Completeness is assumed rather than proved; a degenerate eigenvalue needs its probabilities summed, and a continuous spectrum gives a density.

• Collapse, repeatability, and the state after measurement ↗

  Getting qₙ replaces psi by ψₙ, renormalised, and Schrodinger evolution resumes from there, so an immediate repeat returns qₙ. The projection is a postulate for computing later statistics, not an observed mechanism; a real detector projects onto a band.

• Commutators and the canonical relation [x-hat, p-hat] = i ℏ ↗

  Act [A, B] on a differentiable test function and let the product rule do the work: with p-hat as −i ℏ d/dx this gives [x-hat, p-hat] = i ℏ. The derivation assumes a Cartesian coordinate on the whole line, and no finite matrices can satisfy it.

• Compatible and incompatible observables ↗

  Commuting Hermitian operators share a complete set of eigenfunctions, so both can be sharp together; non-commuting ones have no such complete set, and an A-B-A sequence need not repeat its first result. They may still share isolated eigenfunctions - angular momentum will supply one later - so what fails is the common basis, not every single state; for x-hat and p-hat the commutator is a nonzero constant, so no shared eigenfunction exists at all.

• The general uncertainty relation from the Schwarz inequality ↗

  Set f = (A-hat - ⟨A⟩)psi and g = (B-hat - ⟨B⟩)ψ, apply the Schwarz inequality to their integrals, and keep the imaginary part: σ-A σ-B ≥ (1/2)|⟨[A-hat, B-hat]⟩|. The bound is state-dependent and goes empty wherever that mean commutator vanishes.

• Δ-x Δ-p ≥ ℏ/2 and the price of confinement ↗

  Substituting i ℏ into the general bound gives ℏ/2, and because that commutator is a constant the bound never goes empty; only a Gaussian saturates it, the equality condition reducing to a first-order ODE. It forecasts the confinement energy of the bound states ahead, but states a floor, never a value.

• Why uncertainty is not measurement error ↗

  σ-x is a property of the prepared state, computed from psi before any apparatus is named, and read off an ensemble of identically prepared systems each measured once. A perfect detector would not shrink it, and the error on a mean falls as 1/root-N while sigma does not; separating the two needs a resolution calibration, and error-disturbance relations are separate theorems with their own bounds.

Learning outcomes

• Show that requiring ⟨Q⟩ to be real in every state forces Hermiticity, and deduce real eigenvalues and orthogonality of eigenfunctions belonging to distinct eigenvalues.
• Expand a state on an eigenbasis, compute P(qₙ) = |cₙ|² and ⟨Q⟩ = sum qₙ |cₙ|², and give the renormalised post-measurement wavefunction.
• Evaluate a commutator on a test function, obtain [x-hat, p-hat] = i ℏ, and use it to decide whether two observables admit a complete set of shared eigenfunctions.
• Derive σ-A σ-B ≥ (1/2)|⟨[A-hat, B-hat]⟩| from the Schwarz inequality, and specialise it to Δ-x Δ-p ≥ ℏ/2.
• Separate a state's intrinsic spread from instrument resolution and from the error on a mean, which falls as 1/root-N while sigma does not.

Prerequisite thread

• The wavefunction, Born's rule, normalisation, and expectation values from Units 2-3
• The Schrodinger equation, stationary states, and x-hat, p-hat and H-hat as operators
• Complex integrals, integration by parts on decaying functions, and Gaussian integrals
• Variance, standard deviation, and the standard error on a mean from laboratory work

Laboratory directions

Problem practice

Naming the state a spread belongs to before quoting any bound, and reaching for a commutator before an integral.

Capstone: Take a normalised two-term superposition of two orthonormal eigenfunctions: get each P(qₙ), ⟨Q⟩ and σ-Q by hand, give the state after a measurement returning q₁, then say which of those a merely noisy detector would change and which it would not.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge