Quantum Mechanics II
The hydrogen atom, spin, perturbation theory and identical particles, developed as the second half of a full course in quantum mechanics: what exactly solvable models leave out and the hierarchy of corrections set by the fine-structure constant; spin one-half with spinors, Pauli matrices, the Bloch sphere, tilted Stern–Gerlach measurements, Larmor precession and the sign change of a spinor under a full turn; the addition of angular momenta, triplet and singlet states, Clebsch–Gordan coefficients and LS and jj coupling; identical particles, bosons and fermions, the spin–statistics connection, the Pauli exclusion principle, Slater determinants and the exchange force; helium's singlet and triplet spectra from direct and exchange integrals, the central-field approximation, the periodic table and Hund's rules, and the free-electron gas with its Fermi energy and degeneracy pressure in metals and white dwarfs; time-independent perturbation theory to second order, degenerate perturbation theory and good quantum numbers, the anharmonic oscillator and the linear Stark effect; the fine structure of hydrogen from relativity and spin–orbit coupling, the Lamb shift, the Zeeman effect from the Landé regime to Paschen–Back, and the hyperfine splitting behind the 21 cm line; the variational principle applied to helium with a screened charge and to the hydrogen molecule ion and the chemical bond; the WKB approximation, connection formulas, semiclassical quantization and Gamow's theory of alpha decay; time-dependent perturbation theory, Rabi oscillations, Fermi's golden rule, spontaneous emission and dipole selection rules; scattering theory with partial waves, phase shifts, the optical theorem and the Born approximation for Yukawa and Rutherford scattering; and a capstone on two-level systems, magnetic resonance and spin echoes, entanglement and Bell–CHSH tests, and qubits, with runnable Python examples throughout.
PHYS-373 Year 3 3 credits 12 lessons · 4 h 20 min ▶ Video lessons coming soon
Offered by HSM University · Physics
What you'll learn
- Identify the physics omitted from exactly solvable models such as the PHYS-372 hydrogen atom, estimate its size with the fine-structure constant, and choose a suitable approximation method.
- Represent spin one-half with spinors and Pauli matrices, compute measurement probabilities along any axis on the Bloch sphere, and describe Larmor precession in a magnetic field.
- Add angular momenta, construct triplet, singlet and other coupled states with Clebsch–Gordan coefficients, and evaluate operators such as \(\mathbf L\cdot\mathbf S\) and \(\mathbf S_1\cdot\mathbf S_2\) in the coupled basis.
- Symmetrize and antisymmetrize many-particle wave functions, apply the Pauli exclusion principle and Slater determinants, and explain the exchange force.
- Explain the singlet and triplet spectra of helium with direct and exchange integrals, build atomic configurations and terms, and compute the Fermi energy and degeneracy pressure of a free-electron gas.
- Apply non-degenerate perturbation theory to first and second order and degenerate perturbation theory with good quantum numbers, including the Stark effect.
- Derive the fine structure of hydrogen, the weak- and strong-field Zeeman effect with Landé factors, and the hyperfine splitting behind the 21 cm line.
- Use the variational principle to bound ground-state energies, including the helium atom with a screened charge and the hydrogen molecule ion in the LCAO approximation.
- Apply the WKB approximation to tunneling and bound states, and use Gamow's theory to explain alpha decay and the Geiger–Nuttall law.
- Compute transition amplitudes with time-dependent perturbation theory, solve the driven two-level system, apply Fermi's golden rule, and state and justify the electric dipole selection rules.
- Describe scattering with amplitudes and cross-sections, compute phase shifts and partial-wave cross-sections, apply the optical theorem, and use the first Born approximation.
- Analyze two-level systems on the Bloch sphere, explain magnetic resonance, relaxation and spin echoes, and describe entanglement, Bell–CHSH tests and qubits.
Course content
12 lessons · 4 h 20 min total · final exam
- 1 Why more quantum mechanics: from ideal models to real atoms 23 min · quiz
- 2 Spin one-half: Pauli matrices, Stern–Gerlach and spinors 23 min · quiz
- 3 Addition of angular momenta and Clebsch–Gordan coefficients 21 min · quiz
- 4 Identical particles: bosons, fermions and the Pauli exclusion principle 21 min · quiz
- 5 Atoms and solids: helium, the periodic table and free-electron gases 22 min · quiz
- 6 Time-independent perturbation theory: non-degenerate and degenerate 21 min · quiz
- 7 The hydrogen atom revisited: fine structure, the Zeeman effect and hyperfine splitting 22 min · quiz
- 8 The variational principle: helium ground state and the hydrogen molecule ion 21 min · quiz
- 9 The WKB approximation: tunneling and alpha decay 21 min · quiz
- 10 Time-dependent perturbation theory: transitions, Fermi's golden rule and selection rules 21 min · quiz
- 11 Scattering theory: partial waves and the Born approximation 22 min · quiz
- 12 Capstone: spin, entanglement and two-level systems 22 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
The hydrogen atom, spin, perturbation theory and identical particles, developed as the second half of a full course in quantum mechanics: what exactly solvable models leave out and the hierarchy of corrections set by the fine-structure constant; spin one-half with spinors, Pauli matrices, the Bloch sphere, tilted Stern–Gerlach measurements, Larmor precession and the sign change of a spinor under a full turn; the addition of angular momenta, triplet and singlet states, Clebsch–Gordan coefficients and LS and jj coupling; identical particles, bosons and fermions, the spin–statistics connection, the Pauli exclusion principle, Slater determinants and the exchange force; helium's singlet and triplet spectra from direct and exchange integrals, the central-field approximation, the periodic table and Hund's rules, and the free-electron gas with its Fermi energy and degeneracy pressure in metals and white dwarfs; time-independent perturbation theory to second order, degenerate perturbation theory and good quantum numbers, the anharmonic oscillator and the linear Stark effect; the fine structure of hydrogen from relativity and spin–orbit coupling, the Lamb shift, the Zeeman effect from the Landé regime to Paschen–Back, and the hyperfine splitting behind the 21 cm line; the variational principle applied to helium with a screened charge and to the hydrogen molecule ion and the chemical bond; the WKB approximation, connection formulas, semiclassical quantization and Gamow's theory of alpha decay; time-dependent perturbation theory, Rabi oscillations, Fermi's golden rule, spontaneous emission and dipole selection rules; scattering theory with partial waves, phase shifts, the optical theorem and the Born approximation for Yukawa and Rutherford scattering; and a capstone on two-level systems, magnetic resonance and spin echoes, entanglement and Bell–CHSH tests, and qubits, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.