How physics describes matter at microscopic and subatomic scales.

Topics and mapped lessons

• Complex amplitudes, Euler's formula, and the phase that survives ↗

  Modulus-argument form, exp(i θ) as an eigenfunction of d/d theta with eigenvalue i, and |z|² = z*z as the only bridge from amplitude to probability. A global phase is unobservable, a relative phase inside a superposition interferes; unlike a phasor, no real part is taken at the end.

• Probability distributions, moments, and expectation values ↗

  Discrete distributions and continuous densities, normalisation, mean, variance and moments as integrals against a weight. The Born rule later supplies the weight |ψ|²; the machinery here is classical and predicts ensemble statistics, never one outcome.

• Linear vector spaces over C and their inner products ↗

  Vector-space axioms over C, an inner product conjugate-linear in its first slot in the physics convention, the induced norm, and Cauchy-Schwarz, which later becomes the Robertson uncertainty bound. Square-integrable functions form an infinite-dimensional space assumed complete here.

• Orthogonality, normalisation, and expansion on a basis ↗

  Gram-Schmidt on a non-orthogonal or degenerate set, ⟨ψ|ψ⟩ = 1 fixing the scale, coefficients cₙ = ⟨n|ψ⟩ read as projections, and Parseval's identity making the probabilities sum to one. Completeness is assumed: an incomplete set loses probability silently.

• Dirac notation: kets, bras, projectors, and the identity ↗

  Kets as vectors, bras as the dual functionals Riesz pairs with them, outer products |n⟩⟨n| as projectors, and a resolution of the identity inserted to change representation or read a matrix element. Plane waves are not square-integrable, so they are no ket at all: they enter as distributions normalised to a delta.

• Matrices, eigenvalues, and diagonalisation ↗

  An operator becomes a matrix once a basis is chosen; a similarity transform moves it to another. The characteristic polynomial gives eigenvalues for small n, worked on the Pauli matrices; past 3x3 use numpy.linalg.eigh when the matrix is Hermitian and eig otherwise, and a defective matrix has no eigenbasis.

• Hermitian operators, the spectral theorem, and A|ψ⟩ = a|ψ⟩ ↗

  Adjoints, real eigenvalues, eigenvectors of distinct eigenvalues orthogonal and degenerate ones orthogonalisable, a spectral theorem giving an eigenbasis: observables are Hermitian operators, measured values their eigenvalues, and commuting ones share a basis. In infinite dimensions all of this depends on the domain.

• Fourier series, Fourier transforms, and conjugate variables ↗

  Fourier series as expansion on an interval's exponential basis, the transform as its continuous limit with δ-function orthonormality, Plancherel's theorem. The FFT assumes sampled periodic data, so aliasing above Nyquist and leakage are artefacts.

• Differential equations as eigenvalue problems ↗

  Second-order ODEs in Sturm-Liouville form, boundary conditions selecting a discrete spectrum, and that problem discretised into a matrix for numpy.linalg.eigh. Self-adjointness depends on those boundary conditions; a finite grid pushes the high eigenvalues below their true values.

Learning outcomes

• Manipulate complex amplitudes in polar form, and separate a global phase, which is unobservable, from a relative phase, which is not.
• Verify the inner-product axioms on a candidate space, normalise a state to ⟨ψ|ψ⟩ = 1, and orthogonalise a degenerate set by Gram-Schmidt.
• Write vectors, duals, projectors and matrix elements in Dirac notation, and insert a resolution of the identity to change representation.
• Diagonalise a Hermitian matrix by hand for n = 2 and with numpy.linalg.eigh beyond it, reading A|ψ⟩ = a|ψ⟩ as the measurement statement.
• Expand a function on a Fourier basis, transform between conjugate variables, and state the width bound that follows from the transform alone.

Prerequisite thread

• University Physics I-IV, including the wave function and the Schrodinger equation
• Linear algebra: vector spaces, bases, matrices, determinants, eigenvalue problems
• Multivariable calculus and ordinary differential equations with boundary conditions
• Python with NumPy, SciPy and Matplotlib, or the willingness to learn it in week one

Laboratory directions

Problem practice

Naming the space, the basis and the operator before computing, and keeping the conjugate on every inner product and expansion coefficient

Capstone: Write a two-state Hamiltonian as a 2x2 Hermitian matrix, diagonalise it by hand and in NumPy, expand one given state on both the original basis and the eigenbasis, and show the probabilities sum to one in each.

• 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

• Four quantitative failures of the classical account ↗

  Larmor's 16 ps collapse of a classical hydrogen orbit, Ritz combination differences in line spectra, heat capacities freezing out, and Sackur-Tetrode fixing phase-space cells at h-cubed. Each admits a patch; only a shared resolution argues for one theory.

• Cavity modes, equipartition, and the Planck law ↗

  Modes as Dirichlet eigenfunctions of the cavity Laplacian, counted by Weyl's law; equipartition then diverges. Restricting a mode to n h nu and summing a geometric series gives the Bose-Einstein occupancy - yet Planck quantised the walls, not the field.

• The photoelectric effect and what it does not prove ↗

  A threshold frequency, prompt emission, and a stopping-potential slope of h over e - all three follow from first-order perturbation theory with a classical vector potential, so the effect quantises the atom, not the field. Antibunching, g-two of zero below one, does that.

• Compton scattering and photon momentum ↗

  Four-momentum conservation, contracted with itself, gives a shift of h over m-sub-e c times one minus cos θ and the 2.426 pm Compton wavelength; Klein-Nishina reduces to Thomson at low energy. The shift alone also follows from a Doppler treatment of a recoiling electron, so it fixes h over m-sub-e c, not h; Bothe-Geiger coincidences force quantum-by-quantum momentum transfer. Bound electrons break the free-electron premise.

• λ = h/p and momentum eigenfunctions ↗

  Lorentz-invariant phase sets k-μ equal to p-μ over ℏ, so λ = h/p, the relativistic form a half-percent correction at 10 kV. Then e to the i k x is an eigenfunction of p-hat = minus i ℏ d by dx with eigenvalue ℏ k. It is not in L-squared, so it is δ-normalised - bra p ket p-prime equals delta of p minus p-prime - and sharp momentum is never occupied.

• Electron diffraction, from Davisson-Germer to molecules ↗

  Constructive scattering when momentum transfer matches a reciprocal-lattice vector - the Fourier transform of the lattice - fits 54 eV electrons at 0.167 nm, then neutrons, atoms and C-sixty. Single scattering is assumed; slow electrons need dynamical theory.

• Duality made quantitative on a two-state space ↗

  Path alternatives span a two-state Hilbert space, so ρ = half of I plus a-dot-σ: visibility V is the off-diagonal coherence and predictability P the σ-z component, with P-squared plus V-squared at most one. Englert's distinguishability D is instead the trace distance of the marker's two conditional states, and D-squared plus V-squared equals one on any pure joint state. Both are theorems about that space, not about kicks.

• The Born rule and the probability interpretation ↗

  Psi's squared modulus as a density on L-squared, normalisation to one, overlap-squared as an outcome probability, and a continuity current that conserves the norm for real V. Gleason forbids rival measures in dimension three or more but says nothing about a qubit; the rule predicts frequencies, not outcomes.

• Quantum states, Dirac notation, and mixtures ↗

  A state as a ray in a complex separable Hilbert space: kets and inner products, a resolution of the identity in any orthonormal basis, global phase unphysical where relative phase is not, and density operators for mixtures. Coherences alone separate the two.

Learning outcomes

• Derive Planck's law from the cavity Laplacian's mode spectrum and a geometric sum over n h ν, recovering the Rayleigh-Jeans and Wien forms as limits.
• Extract h from the h-over-e stopping-potential slope and h over m-sub-e c from the Compton shift, and separate the quantisation each experiment establishes from the one it merely permits.
• Apply λ = h/p to electron-diffraction geometry, and treat e to the i k x as an eigenfunction of minus i ℏ d by dx with eigenvalue ℏ k, outside L-squared until δ-normalised to bra p ket p-prime equals δ of p minus p-prime.
• Write a two-path state as a density matrix in the Pauli basis and distinguish predictability P, the σ-z component bounded by P-squared plus V-squared at most one, from Englert's distinguishability D, the marker's trace distance, for which D-squared plus V-squared equals one on any pure joint state.
• Normalise a state in a Hilbert space, compute overlap-squared probabilities, and say why Gleason's theorem leaves Born's rule no alternative in dimension three or more - and why it is silent on a single qubit.

Prerequisite thread

• A first pass at wave functions and the Schrodinger equation, as in University Physics IV
• Inner-product spaces, Hermitian matrices, eigenbases, and the spectral theorem
• Fourier transforms, the Dirac δ, Gaussian integrals, and complex exponentials
• Relativistic four-momentum, Boltzmann factors, and NumPy or SciPy curve fitting

Laboratory directions

Problem practice

Naming the operator, its eigenvalue and its eigenbasis before integrating, and stating which quantisation each experiment actually forces

Capstone: Take one data set - stopping potentials, a Compton spectrum, or diffraction rings - extract h or h over m-sub-e c from it, then state which quantisation that experiment establishes, which it merely permits, what a semiclassical model predicts instead, and which further experiment closes the gap.

• 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 state vector and its position representation ↗

  A state is a ray in the Hilbert space L2(R), and ψ(x) = ⟨x|ψ⟩ is its component along the position basis. The inner product ⟨φ|ψ⟩ = ∫ φ* ψ dx supplies norms and overlaps. One spinless non-relativistic particle; |x⟩ is not in the space.

• Born's rule, probability density, and normalisation ↗

  P(x) = |ψ(x)|² is a density, so P(a < x < b) is its integral and ⟨ψ|ψ⟩ = 1 fixes the dimensions of ψ at inverse square root of length. Only square-integrable states normalise: a plane wave needs a box or delta convention instead.

• Observables as self-adjoint (Hermitian) operators and their spectra ↗

  An observable is an operator equal to its own adjoint on a stated domain, which forces a real spectrum and a resolution of the identity. An orthonormal eigenbasis comes only from the discrete part: x and p have purely continuous spectra and no eigenvectors in L2. Self-adjointness rests on the domain and the boundary terms, not the formula, so −i ℏ d/dx on a half-line stays symmetric yet admits no self-adjoint extension.

• Expectation values, variances, and moments ↗

  ⟨A⟩ = ⟨ψ|A|ψ⟩ = ∫ ψ* A ψ dx, with ⟨x⟩ = ∫ x |ψ|² dx and variance ⟨A²⟩ - ⟨A⟩². These are ensemble means over identically prepared systems, so no single run returns one, and moments diverge for slowly decaying psi.

• Superposition and expansion in an eigenbasis ↗

  Linearity writes any psi as a sum of cₙ |n⟩ with cₙ = ⟨n|ψ⟩ and, for a normalised state in a discrete basis, the squared moduli summing to one, the resolution of the identity doing the work. Completeness is assumed rather than proved, and a degenerate eigenvalue fixes a subspace, not a basis.

• Position and momentum representations ↗

  φ(p) = ⟨p|ψ⟩ is the Fourier transform of ψ in the symmetric 2 π ℏ convention: a unitary change of basis, with Plancherel making one normalisation serve both. p = −i ℏ d/dx generates translations, |p⟩ is δ-normalised rather than square-integrable, and [x, p] = i ℏ admits no finite matrices.

• Measurement outcomes and projection operators ↗

  Outcomes lie in the spectrum, with probability |⟨a|ψ⟩|², or ⟨ψ|P|ψ⟩ for the projector onto a degenerate eigenspace, and an interval integral of the spectral measure for a continuous spectrum. The rule is a postulate returning ensemble statistics, never one run's result.

• Collapse, repeatability, and superposition versus mixture ↗

  The Luders rule sends psi to P psi over its norm, so an immediate repeat reproduces the outcome. That projection is a postulate for computing what comes next, not an observed mechanism, and interpretations divide over whether anything collapses. A superposition differs from a mixture only in the off-diagonal terms of the density matrix; decoherence suppresses those but selects no outcome.

• Discretised states and operators as matrices ↗

  Sampling psi on a grid makes it a vector, x a diagonal matrix and p a finite-difference or FFT matrix, so every expectation value becomes a vector-matrix-vector product. In finite dimensions Hermitian and self-adjoint coincide, so the grid hides the domain question; box truncation and spacing set the error and make [x, p] = i ℏ impossible to satisfy exactly.

Learning outcomes

• Write a state as |ψ⟩ with position components ψ(x) = ⟨x|ψ⟩, and say what the L2 inner product, the norm, and a global phase each do.
• Normalise ψ, compute P(a < x ⟨b) from |ψ|², and state why a plane wave admits only delta or box normalisation.
• Evaluate ⟨A⟩ = ⟨ψ|A|ψ⟩ and its variance for x, p and H in either representation, and test self-adjointness from the stated domain and the boundary terms.
• Transform between ψ(x) and φ(p) by Fourier transform, and use Plancherel's theorem to show one normalisation serves both bases.
• Apply Born's rule with projectors for a discrete eigenvalue, for a degenerate eigenspace, and for an interval of continuous spectrum, and separate a superposition from a mixture by its interference term.

Prerequisite thread

• Complex inner-product spaces, orthonormal bases, adjoints, and the spectral theorem
• Dirac bra-ket notation and finite-dimensional state vectors from Units 1 and 2
• Fourier transforms, the Dirac δ, Gaussian integrals, and integration by parts
• NumPy practice with array slicing, matrix products, numpy.linalg.eigh, and the FFT

Laboratory directions

Problem practice

Naming the basis, the domain, and the normalisation convention before integrating, and writing every probability as the squared modulus of an inner product

Capstone: Take a two-term superposition of infinite-well eigenstates with a complex relative phase: normalise it, get P(0 < x < L/2), ⟨x⟩, ⟨p⟩ and both variances by hand, reproduce each from a discretised matrix calculation on the same box, then say which of them the interference term controls, which survive unchanged when averaged over the outcomes of an energy measurement, and what domain −i ℏ d/dx needs before ⟨p⟩ in a box means anything.

• 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 Hamiltonian as a self-adjoint operator ↗

  H-hat = p-hat squared over 2m plus V(x-hat) by canonical quantisation, kinetic energy as minus ℏ squared over 2m times the Laplacian. Ordering is ambiguous wherever x and p mix, and symmetry is not self-adjointness until the domain is fixed.

• The time-dependent equation in Dirac notation ↗

  i ℏ d|Ψ⟩/dt = H-hat|Ψ⟩ as a linear first-order law on Hilbert space: one state at one time fixes every other, superposition survives, and hermiticity conserves the norm. Postulated, non-relativistic, spinless, closed system only.

• Unitary evolution and the propagator exp(−iHt/ℏ) ↗

  For a time-independent H, integrating gives U(t) = exp(minus i H t over ℏ), evaluated by spectral decomposition or by expm on a truncated matrix. U is unitary because H is self-adjoint, and it is the further fact that H commutes with U that holds the mean of H fixed: a time-dependent H is still unitary yet need not conserve energy, and needs a time-ordered series.

• Separation of variables and stationary states ↗

  A static V separates Psi into ψ(x) times exp(minus i E t over ℏ), leaving the eigenvalue problem H|n⟩ = Eₙ|n⟩. The density and every expectation value then stand still; in a superposition only relative phases move, the density beating at angular frequency (Eₘ - Eₙ)/ℏ, so at frequency (Eₘ - Eₙ)/h.

• The time-independent equation in the position basis ↗

  Minus ℏ squared over 2m times ψ-double-prime plus V ψ = E ψ, read as a Sturm-Liouville eigenvalue problem: psi''/ψ = 2m(V - E)/ℏ squared bends ψ toward the axis where E > V and away from it where E < V. One dimension, local real V.

• Admissibility, matching conditions, and the operator's domain ↗

  Square-integrability, continuity of psi everywhere, continuity of ψ-prime wherever V is bounded, ψ = 0 at an infinite wall, and at V = minus α δ(x) the jump ψ-prime(0+) minus ψ-prime(0-) = minus 2 m α ψ(0) over ℏ squared, whose sign is what allows the one bound state at E = minus m α squared over 2 ℏ squared. These conditions define H's domain, not decorate it.

• How boundary conditions force a discrete spectrum ↗

  Integrating from both ends, the decaying solutions match only at isolated E: normalisability, not the ODE, selects eigenvalues, and the infinite well returns Eₙ = n-squared π-squared ℏ-squared over 2mL-squared. The criterion is energetic, not geometric: only E below the asymptotic value of V can be discrete, and above it the spectrum is continuous, which is why an unconfined finite well still holds bound states.

• Discrete and continuous spectra, degeneracy, completeness ↗

  Bound states orthonormal to a Kronecker δ, scattering states to a Dirac δ, and a resolution of the identity summing over the discrete part and integrating over the continuum. One-dimensional bound states are non-degenerate by a Wronskian argument; continuum kets sit outside the Hilbert space.

• Matrix Hamiltonians and numerical eigenvalues ↗

  A three-point Laplacian on a grid, or truncation in a basis, turns H into a real symmetric matrix for scipy.linalg.eigh, but the two errors point opposite ways. Rayleigh-Ritz basis truncation is variational and bounds levels from above; the grid replaces k squared by (4 over h squared) sin squared of kh/2, an order h squared underestimate. A finite box separately fakes the continuum into discrete levels.

Learning outcomes

• Construct H-hat by canonical quantisation, and separate a merely symmetric operator from a self-adjoint one by naming its domain.
• Solve i ℏ d|Ψ⟩/dt = H|Ψ⟩ for a static H as |Ψ(t)⟩ = exp(−iHt/ℏ)|Ψ(0)⟩, and show unitarity preserves the norm and every inner product while [H, U] = 0 is what fixes the mean energy.
• Separate a static Hamiltonian into H|n⟩ = Eₙ|n⟩, expand a state in that eigenbasis, and predict the angular beat frequency (Eₘ - Eₙ)/ℏ of a two-level superposition.
• Impose square-integrability and the matching of psi and ψ-prime to select the eigenvalues the differential equation alone leaves continuous.
• Diagonalise a discretised Hamiltonian with scipy.linalg.eigh, test the eigenvalues for convergence in grid spacing and box size, and say which way each truncation biases them.

Prerequisite thread

• Dirac notation, inner products, and completeness in a separable Hilbert space
• Hermitian and unitary operators: the spectral theorem and the matrix exponential
• Second-order linear ODEs, Sturm-Liouville problems, Fourier series and transforms
• NumPy and SciPy: dense and tridiagonal eigensolvers, and grid-convergence testing

Laboratory directions

Problem practice

Naming the operator, its domain, and its eigenbasis before integrating anything, and checking that every claimed spectrum is normalisable

Capstone: Take a finite square well: solve the transcendental matching condition for its bound-state energies, diagonalise the same H on a grid, compare the two spectra, then state which levels the box boundary invented, which way the grid biased the genuine ones, and how the bound-state count changes as the well deepens.

• 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