Quantum Mechanics I

Why quantum mechanics is forced on us, and how to calculate with it

About this course

The postulates of quantum mechanics built from Stern–Gerlach and two-state systems, then applied to the oscillator, angular momentum, spin and hydrogen, with approximation methods, identical particles and Bell's inequality.

What you will be able to do

  • Represent two-state systems as amplitudes and on the Bloch sphere, and compute measurement probabilities
  • Use commutators to decide compatibility and derive uncertainty relations
  • Solve one-dimensional bound-state and tunnelling problems, and the harmonic oscillator by ladder operators
  • Add angular momenta and solve the hydrogen atom
  • Apply perturbation theory, the variational method and WKB to estimate energies

What this course covers

  • Amplitude addition rule and interference term
  • Single-particle double-slit interference
  • Which-path tagging and the visibility–information trade-off
  • Photoelectric effect and quantised energy exchange
  • Photon indivisibility via coincidence measurement
  • Three failed pictures of a quantum particle
  • Conservation of total arrivals and the lecture's central result
  • State, basis, and wave function distinguished
  • Continuous labels and the Dirac delta distribution
  • Momentum as generator of spatial translation
  • Momentum eigenstates, normalization, and the Fourier transform as change of basis
  • Width-product invariance under rescaling (transform theorem)
  • The Heisenberg uncertainty relation as one physical substitution
  • Sharp edges, infinite momentum spread, and the square-pulse example
  • Operators, matrices, and observables distinguished
  • Basis-independence of eigenvalues and self-adjointness
  • The measurement assumption and its falsifiability
  • Perpendicularity of distinct-reading definite states forced by certainty
  • Spectral decomposition and the self-adjoint operator derived
  • Expected value as a consequence, not a postulate
  • Real-expectation-value condition and its fragility over the reals
  • The perturbation expansion setup
  • The lowest-level theorem and level repulsion
  • First- and second-order energy corrections
  • The level-gap ratio as the true expansion parameter
  • Degenerate perturbation theory: choosing the right starting states
  • Good quantum numbers and the hydrogen fine structure
  • Convergence, series radius, and the limits of the method
  • Rotation geometry and the three commutator relations
  • Compatible and incompatible observables for angular momentum
  • Uncertainty floor and angular momentum states
  • Ladder operators and the closed ladder
  • The angular momentum spectrum: λ = L(L+1), integer and half-integer L
  • Spherical harmonics as a representation of the algebra
  • Exclusion of half-integer orbital angular momentum by single-valuedness
  • Physical tests and the classical limit
  • The variational theorem (upper-bound proof)
  • Calibrating and limiting the variational method
  • Symmetry and excited-state bounds
  • WKB approximation: derivation and the 1/√P amplitude
  • WKB quantisation rule and the quarter-wave correction
  • Comparison on the quartic potential: the central result
  • Helium ground state and the meaning of each number
  • Why half-integer angular momentum was excluded
  • Building the spin-½ matrices from ladder step weights
  • The Bloch sphere as exact physical content
  • Spin as conserved quantity, not orbital motion
  • Classical g-factor and the factor-of-two discrepancy
  • Rotation operator and the 720-degree periodicity of spin states
  • Observing the sign change via neutron interferometry
  • Stern-Gerlach Device: Force from Field Gradient
  • Classical Prediction vs. Observed Two-Spot Result
  • Cascaded Devices and the Quarter-Beam Result
  • Angle Dependence: Cosine-Squared-Half-Angle Law
  • Non-Commutation of Spin Measurements
  • Refuting the 'Violent Apparatus' (Kick) Story
  • Central Result: Questions Without Joint Answers
  • Two inherited results: equation of motion and position basis
  • Projecting the equation of motion onto position basis
  • Choosing the Hamiltonian: three packed assumptions
  • The Schrödinger equation as the central result
  • Role of the imaginary unit i: probability and reversibility
  • Free-particle dispersion: group velocity and phase velocity
  • Limits of the chosen Hamiltonian and the classical limit
  • The projection postulate: statement and falsifiability
  • Density matrix: representing pure states and mixtures
  • Superposition versus mixture: the tilted-magnet test
  • The ninety-degree experiment and its laboratory threshold
  • Testing three interpretive claims by computation
  • Conflict between the projection postulate and the linear rule
  • Indistinguishability of mixtures and the four responses to the conflict
  • Three distinct sources of measurement spread
  • Compatible observables and the commutator criterion
  • The commutator as a state-dependent observable
  • Spread (standard deviation) of an observable
  • Derivation of the generalised uncertainty bound
  • The famous uncertainty relation as a weakened corollary
  • States where the famous bound is trivially zero
  • Amplitude arithmetic vs. probability arithmetic
  • A probability pair is not a quantum state
  • Why quantum state numbers must be complex
  • Why the Born rule uses the square (not another power)
  • What the lecture derives, assumes, and leaves open
  • Overall phase vs. relative phase
  • Superposition vs. ignorance: the central result
  • Identical particles and exchange symmetry
  • Bosons, fermions, and the spin-statistics theorem
  • The exclusion principle as arithmetic, not a rule
  • Exchange interaction: separation without a force
  • Helium exchange splitting: prediction and bench test
  • Fermi energy and degeneracy pressure in a metal
  • Switching the sign off: why matter has surfaces
  • Inherited angular momentum algebra for the total
  • Product-state basis and dimension of the combined space
  • Compatible observables in the combined space
  • The staircase argument and the range of total angular momentum
  • Dimensional closure as proof of the multiplet list
  • Explicit construction of two-spin-½ states: triplet and singlet
  • Physical prediction: hyperfine splitting ratio as a counting result
  • Separation of variables and stationary states
  • Continuity conditions derived by integration
  • Discreteness as a counting problem
  • Transcendental matching conditions and parity
  • Graphical solution and the no-threshold theorem
  • Quantum tunnelling and the exponential transmission formula
  • Dimension dependence of binding and stellar fusion
  • Commuting operators and separation of the radial equation
  • The centrifugal barrier and the L = 0 special case
  • Series solution, termination, and the principal quantum number
  • The hydrogen energy spectrum and state count
  • Limits of the Bohr model: right answer, wrong argument
  • L-degeneracy as a hidden symmetry: the Runge-Lenz vector
  • Comparison with experiment and the systematic discrepancy
  • Why two free parameters don't fix a surface
  • The Bloch sphere derived from three measurement averages
  • Probability as a dot product; orthogonal means opposite
  • The factor-of-two angle doubling between state space and sphere
  • Every operation on a two-level system is a rotation
  • Reading probability directly from a chord on the sphere
  • Limits of the Bloch sphere: mixed states and entanglement
  • Why the harmonic oscillator is universal
  • Ladder operators and the Hamiltonian factorization
  • Raising and lowering commutation relations
  • The energy floor and the zero-point energy
  • The complete energy spectrum and ladder weights
  • Ground-state wavefunction from a first-order equation
  • Zero-point energy as the uncertainty-principle floor
  • Linearity and unitarity of the time-evolution map
  • The Hamiltonian as generator of time evolution
  • The Schrödinger equation as the equation of motion
  • Closed-form matrix exponential for two-level systems
  • Stationary states and the energy-gap theorem
  • Rabi oscillations: computing the ceiling and period
  • Purity conservation and the measurement-versus-evolution tension
  • Entanglement as a measurable number
  • Quantum correlation of two spin measurements
  • The Bell inequality: derivation and ceiling of two
  • Local realism and its three assumptions
  • Quantum prediction at optimal settings: 2√2
  • No-signalling theorem
  • Experimental verdict and what it does and does not settle

Syllabus

  1. Why Quantum Mechanics

    1. L1 · The Experiments That Left No Choice
    2. L2 · Stern–Gerlach, and Measurements That Refuse to Commute
    3. L3 · Superposition, Amplitudes and Probability
    4. L4 · The Bloch Sphere

  2. The Formalism

    1. L5 · States, Operators and the Hermitian Requirement
    2. L6 · Measurement, Projection and What Collapse Does Not Say
    3. L7 · Commutators, Compatibility and Uncertainty
    4. L8 · Time Evolution, Unitarity and Rabi Oscillation

  3. Wave Mechanics in One Dimension

    1. L9 · Continuous Bases and the Fourier Pair
    2. L10 · The Schrödinger Equation, Derived
    3. L11 · Bound States and Tunnelling in One Dimension
    4. L12 · The Harmonic Oscillator by Ladder Operators

  4. Angular Momentum and Hydrogen

    1. L13 · Orbital Angular Momentum and the Spherical Harmonics
    2. L14 · Spin, and Why It Is Not a Little Sphere
    3. L15 · Adding Angular Momenta
    4. L16 · Hydrogen, Solved

  5. Approximations, Identical Particles and Entanglement

    1. L17 · Perturbation Theory and Fine Structure
    2. L18 · The Variational Method and WKB
    3. L19 · Identical Particles, and Why Matter Is Rigid
    4. L20 · Entanglement and Bell's Inequality