Quantum Mechanics: From the Photon to the Hydrogen Atom

From the Photon to the Hydrogen Atom in 12 chapters — from “Light in lumps: blackbody radiation, the photoelectric effect and Compton scattering” to “The hydrogen atom”.

Chapters

  1. Light in lumps: blackbody radiation, the photoelectric effect and Compton scattering
    • The classical wave prediction
    • Planck quantisation and the blackbody spectrum
    • The photon and the photoelectric equation
    • Photon momentum and the Compton shift
    • What the photon does not abolish
    • Problems
  2. Line spectra, the Bohr atom and the old quantum theory
    • Line spectra and the Rydberg formula
    • The nuclear atom and its collapse
    • Bohr quantisation and energy levels
    • The correspondence principle
    • The Franck-Hertz experiment
    • The old quantum theory rule and its limits
    • Problems
  3. Matter waves and wave-particle duality
    • The de Broglie wavelength
    • The Davisson-Germer experiment
    • Standing-wave quantisation of Bohr's orbits
    • Wave-particle duality and the double slit
    • The finite wave train
    • The Heisenberg uncertainty relation
    • Problems
  4. The Schrödinger equation and the Born rule
    • Complex numbers and the wave function
    • The Schrödinger equation
    • The Born rule and the statistical interpretation
    • Normalisation and the conservation of probability
    • Expectation values and the standard deviation
    • The momentum operator
    • Two wave functions worked in full
    • Problems
  5. Stationary states and bound states in one dimension
    • Separation of variables and the time-independent equation
    • Stationary states
    • The infinite square well
    • Completeness and the expansion in eigenfunctions
    • General properties of one-dimensional bound states
    • The finite square well
    • The delta-function well
    • Problems
  6. Free particles, wave packets and tunnelling
    • Scattering states and the free-particle plane wave
    • Wave packets from plane waves
    • Group velocity and packet spreading
    • Scattering at a potential step
    • Tunnelling through a rectangular barrier
    • The alpha-decay estimate
    • Problems
  7. The harmonic oscillator
    • The quantum harmonic oscillator
    • The canonical commutator
    • Ladder operators and the factorised Hamiltonian
    • Oscillator spectrum and zero-point energy
    • Averages by ladder operators
    • Hermite polynomials from a power series
    • The classical limit of the oscillator
    • Problems
  8. Observables, commutators and the uncertainty relations
    • Observables as Hermitian operators
    • The eigenvalue equation of an observable
    • The generalised statistical interpretation
    • Commutator algebra and Ehrenfest's theorem
    • Compatible observables and shared eigenfunctions
    • The generalised uncertainty principle
    • The energy-time uncertainty relation
    • Problems
  9. Spin one-half, the Pauli matrices and two-level systems
    • The Stern-Gerlach experiment
    • Spin one-half and the spinor
    • Two-by-two matrices and the Pauli matrices
    • Spin eigenspinors along an axis
    • Sequential Stern-Gerlach measurements
    • Spin magnetic moment and Larmor precession
    • The two-level system
    • Problems
  10. Orbital angular momentum, the spherical harmonics and rotations
    • Vector product and orbital angular momentum operators
    • Angular momentum commutation relations
    • Ladder operators and angular momentum eigenvalues
    • Angular momentum in spherical coordinates
    • Spherical harmonics
    • Rotations generated by Lz​
    • Problems
  11. The Schrödinger equation in three dimensions and central potentials
    • Cubic box and degeneracy
    • Central potential separation
    • Radial equation and effective potential
    • Infinite spherical well
    • Isotropic harmonic oscillator
    • Rigid rotor and rotational lines
    • Problems
  12. The hydrogen atom
    • Coulomb radial equation in dimensionless form
    • Asymptotic analysis of u
    • Power series quantisation
    • Hydrogen energy levels
    • Hydrogen radial wave functions
    • Hydrogen degeneracy and spin
    • Spectral series and reduced mass
    • Problems