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Quantum Mechanics | Math Infinitum

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Quantum Mechanics

Physics · Course 05

Quantum Mechanics

A full university course in quantum mechanics, from the first amplitudes to fields. States and measurement, wave mechanics and the abstract formalism, the oscillator, angular momentum, spin and hydrogen, approximation methods, entanglement and open systems, scattering, path integrals, the Dirac equation and second quantisation — each built from the mathematics up and tied to what an experiment would show.

i ℏ \frac{\partial ψ}{\partial t} = H^ψ

∣ ψ ⟩ = n \sum c_{n} ∣ n ⟩

P (a) = ∣ ⟨ a ∣ ψ ⟩ ∣^{2}

[x^, p^] = i ℏ

E_{n} = ℏ ω (n + \frac{1}{2})

ρ^= k \sum p_{k} ∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣

K = \int D x e^{\frac{i S [ x ]}{ℏ}}

H^∣ E ⟩ = E ∣ E ⟩

Course map

27 chapters

01Quantum ideas and the state spaceWhy the classical picture fails, and what replaces it: amplitudes that add before they are squared, states that superpose, and the complex vector space every quantum state lives in. The chapter ends with Dirac’s notation and the expansion of a state in a basis, which every later chapter speaks.
02OperatorsEverything that can be done to a quantum state is a linear operator. Adjoints, eigenvalues, the Hermitian operators that will stand for measurable quantities and the unitary ones that will move states in time, projectors, the spectral decomposition that writes an operator through its eigenvalues, and the commutator that measures how badly two operators fail to commute.
03Observables and measurementThe postulates that connect the mathematics to an experiment: which numbers a measurement can return, with what probabilities, and what it leaves the state as. Expectation values and spreads, the uncertainty relation proved from the Cauchy–Schwarz inequality, what happens when measurements are repeated, and which observables can be known together.
04Position, momentum and wavefunctionsA particle on a line needs a basis with a continuum of states. Position states and the wavefunction they define, probability density and normalisation, the position and momentum operators, the canonical commutation relation, and the Fourier transform that turns a state’s position picture into its momentum picture.

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05Time evolutionHow a quantum state changes when nobody measures it. The Schrödinger equation, the Hamiltonian, stationary states and the beating of their superpositions, the evolution operator, the Heisenberg picture in which operators move instead of states, and what stays constant. The chapter runs beside the one on position and momentum, and meets it for the particle in a potential, the probability current and Ehrenfest’s theorem.↗

06Motion in one dimensionThe time-independent Schrödinger equation on a line, and what it gives in the problems every later chapter borrows from: the infinite and the finite square well, the delta-function well, the potential step, barriers that are tunnelled through and wells that let some energies pass untouched. Along the way, why one-dimensional bound states never share an energy and why the next one up always has one more node.↗

07The harmonic oscillatorThe one system met everywhere, from molecules to light, solved twice: by the algebra of ladder operators, which gives the whole spectrum without solving a differential equation, and by the wavefunctions with their Hermite polynomials. Zero-point energy, uncertainty in the number states, motion in time, and the coherent states that move exactly as a classical oscillator does.↗

08SymmetryA symmetry is a change of point of view that no experiment can detect. In quantum mechanics it is a unitary operator, a continuous one is generated by an observable, and whenever it commutes with the Hamiltonian that observable is conserved. Translations and momentum, rotations and angular momentum, parity, time reversal, and the degeneracy that symmetry forces on a spectrum.↗

09Angular momentumThe generators of rotations, and the three commutation relations that decide everything about them. Ladder operators, the quantum numbers $j$ and $m$ with the half-integers they allow, the matrices of angular momentum, the orbital case in which only whole numbers survive, spherical harmonics, and what a rotation does to a state of given $j$.↗

10SpinThe angular momentum a particle carries with nothing to rotate. The Stern–Gerlach experiment, the two-dimensional state space of spin one-half and its picture on a sphere, the Pauli matrices, measurement along any axis, spinors that change sign under a full turn, precession in a magnetic field, and the discovery that every two-level system is a spin in disguise.↗

11Three dimensions and central potentialsWhere the one-dimensional techniques meet angular momentum. The Schrödinger equation in space, separation of variables, the central potentials whose rotational symmetry splits every problem into spherical harmonics and a radial equation, the effective potential with its centrifugal barrier, and the reduction of two bodies to one.↗

12The hydrogen atomThe problem that made quantum mechanics believable, solved exactly. Atomic units, the radial equation and the series that must stop, the spectrum with its lines, the degeneracy that is bigger than rotations explain, the radial wavefunctions and the orbitals built from them, expectation values, and the ions and isotopes that scale the same answer.↗

13The variational method and WKBAlmost nothing can be solved exactly, so this chapter is about getting good answers anyway. The variational principle turns any guess into an upper bound on the ground state energy; the WKB approximation reads a slowly varying potential as a wave of slowly varying wavelength, with turning points, Bohr–Sommerfeld quantisation and tunnelling rates falling out of it.↗

14Time-independent perturbation theoryWhen a problem is a solvable one plus something small, the answer is the solvable answer plus corrections, order by order. The first-order energy shift and state correction, the second-order shift that always pushes the ground state down, what to do when levels are degenerate, the two-level problem that shows exactly where the expansion breaks, and the Stark effect in hydrogen.↗

15Time-dependent perturbationsA Hamiltonian that changes in time makes a system jump between levels. The interaction picture, transition amplitudes in first order, the response to a constant and to a periodic push, Fermi’s golden rule for decay into a continuum, and the two extremes of a changing Hamiltonian: the sudden change that leaves the state behind and the slow one that carries it along, with the geometric phase it leaves.↗

16Adding angular momentaTwo angular momenta in one system: the tensor product their states live in, the coupled and uncoupled bases, the rule that says which totals are possible, singlet and triplet, Clebsch–Gordan coefficients, spin–orbit coupling, the selection rules that decide which transitions light can drive, and the Wigner–Eckart theorem that says why they hold.↗

17Density matricesA state known only with probabilities is not a superposition, and needs an object of its own. Pure and mixed states, the density matrix and its rules, purity and the Bloch ball, how a density matrix evolves, von Neumann entropy, and the composite systems whose parts are described by a partial trace and by the Schmidt decomposition.↗

18Entanglement and nonlocalityStates of two systems that are not states of each. The Bell states, the argument of Einstein, Podolsky and Rosen, the correlations of the singlet, Bell’s inequality and the experiments that break it, why entanglement still cannot send a message or be copied, and how much of it a state holds.↗

19Open quantum systemsNo real system is alone. Its entanglement with an environment turns superpositions into mixtures, which is decoherence; the maps that describe what happens to the system alone are quantum channels written with Kraus operators; and in continuous time they obey the Lindblad equation, whose relaxation and dephasing are what every experiment with qubits fights.↗

20Driven two-level systemsThe simplest quantum system pushed from outside, which is how atoms are driven by lasers and qubits by microwaves. Two-level Hamiltonians, avoided crossings, Rabi oscillations, the rotating frame, detuning, and the Landau–Zener formula for a level crossing swept at a finite speed.↗

21Charged particles in electromagnetic fieldsHow a charge enters quantum mechanics: through potentials rather than forces. Minimal coupling, gauge transformations as a change of phase, the Aharonov–Bohm effect of a field the particle never touches, Landau levels, magnetic moments and the Pauli Hamiltonian, the Zeeman effect, and diamagnetism and paramagnetism.↗

22Identical particlesTwo electrons cannot be told apart even in principle, and quantum mechanics takes that literally. The exchange operator, bosons and fermions, symmetric and antisymmetric states, the Pauli principle and Slater determinants, exchange effects, the helium atom, the shell structure of many-electron atoms and the self-consistent field of Hartree and Fock.↗

23ScatteringMost of what is known about forces between particles was learned by firing one at another and counting what comes out. Cross sections, the scattering amplitude, the integral equation and its Born approximation, Coulomb scattering, the partial waves and phase shifts of a central potential, the optical theorem, and scattering at low energy with its resonances.↗

24Path integralsFeynman’s reformulation: the amplitude to go from one place to another is a sum over every path between them, each weighted by the phase its classical action gives. The propagator, time slicing, the sum over histories, the classical limit by stationary phase, the free particle and the oscillator done exactly, and what the method is for.↗

25Relativistic quantum mechanicsWhat changes when particles move near the speed of light. Lorentz symmetry, the Klein–Gordon equation and its trouble with probability, Dirac’s first-order equation with its gamma matrices and four-component spinors, antiparticles, the non-relativistic limit that gives spin its magnetic moment of exactly two, and the fine structure of hydrogen.↗

26Second quantisation and fieldsWhen particles are created and destroyed, or are too many and too identical to label, the state has to say how many occupy each mode instead. Fock space, creation and annihilation operators for bosons and for fermions, operators written through them, the quantised modes of a field, the scalar and the spinor field, and particles as the quanta of fields.↗

27SynthesisThe course looked at whole. What quantum theory says about states, measurement and reality; how symmetries, operators and conservation laws are one idea; how wave mechanics and the abstract formalism are one theory; how one particle becomes many and many become a field; and a review that asks about all of it.↗