Advanced Quantum Mechanics
Time-dependent perturbation theory, scattering and an introduction to quantum information, developed as a course beyond the standard two-semester sequence: the modern postulates with density operators, tensor products and general measurements, and why the exponential size of many-qubit Hilbert space matters; Dirac notation, the unitary time-evolution operator and the Schrödinger, Heisenberg and interaction pictures, with conservation laws and the Mandelstam–Tamm relation; the Dyson series and time-ordered exponentials, second-order and virtual transitions, harmonic perturbations, the adiabatic theorem, Landau–Zener transitions at avoided crossings and Berry's geometric phase; semiclassical radiation theory, the dipole approximation, Einstein's A and B coefficients and the lifetime of hydrogen 2p, the Wigner–Eckart theorem and dipole selection rules, natural line shapes and population inversion; the Lippmann–Schwinger equation, outgoing Green's functions, the Born series and its convergence, and the optical theorem; low-energy scattering with the scattering length and effective range, shallow bound states and universality, Breit–Wigner resonances and the Feshbach resonances used to tune ultracold gases; density matrices, mixed states, the Bloch ball, partial traces, the Schmidt decomposition and von Neumann entropy; open quantum systems, decoherence, Kraus operators, the Lindblad master equation, the relaxation times T1 and T2, Ramsey fringes and spin echoes; Bell states, local hidden variables, the CHSH inequality and Tsirelson bound, Werner states and the loophole-free experiments of 2015; qubits, universal gate sets, the no-cloning theorem, teleportation and superdense coding; the Deutsch–Jozsa, Grover and Shor algorithms, quantum error correction with syndromes, the threshold theorem and the surface code; and a capstone on superconducting, trapped-ion, neutral-atom, photonic and spin qubits, BB84 quantum key distribution and quantum sensing, with runnable Python examples throughout.
PHYS-472 Year 4 3 credits 12 lessons · 4 h 21 min ▶ Video lessons coming soon
Offered by HSM University · Physics
This is the Canadian version of this course. See the American version.
What you'll learn
- Explain how advanced quantum mechanics extends the standard theory to all-orders dynamics, scattering, mixed and open systems and quantum information, and estimate the resources needed to describe many qubits.
- Use Dirac notation and the time-evolution operator, and move between the Schrödinger, Heisenberg and interaction pictures, deriving Heisenberg equations of motion and conservation laws.
- Expand time evolution in the Dyson series, recognize when perturbation theory fails, and apply the adiabatic theorem, the Landau–Zener formula and Berry's geometric phase.
- Derive the Einstein coefficients for absorption, stimulated and spontaneous emission, compute a radiative lifetime, and justify the electric dipole selection rules from symmetry.
- Formulate scattering with the Lippmann–Schwinger equation, generate the Born series, judge its convergence, and apply the optical theorem.
- Describe low-energy scattering by the scattering length and effective range, relate them to shallow bound states, and analyse resonances and Feshbach resonances.
- Represent pure and mixed states by density matrices and Bloch vectors, compute reduced density matrices and the von Neumann entropy, and quantify entanglement in pure bipartite states.
- Model decoherence with quantum channels and the Lindblad equation, and extract and interpret the relaxation times \(T_1\) and \(T_2\) from simulated experiments.
- Derive the CHSH inequality for local hidden variables, show that quantum mechanics violates it up to the Tsirelson bound, and describe the experiments that tested it.
- Build quantum circuits from standard gates, prove the no-cloning theorem, and explain and simulate quantum teleportation.
- Explain the Deutsch–Jozsa, Grover and Shor algorithms and the principles of quantum error correction, including syndromes, the threshold and surface-code scaling.
- Compare qubit platforms against the DiVincenzo criteria, analyse the security of BB84 key distribution, and explain the limits of quantum sensing.
Course content
12 lessons · 4 h 21 min total · final exam
- 1 Why advanced quantum mechanics: from textbook problems to modern quantum technology 24 min · quiz
- 2 Dirac notation, pictures of quantum dynamics and the time-evolution operator 23 min · quiz
- 3 Time-dependent perturbation theory revisited: the Dyson series, harmonic perturbations and the adiabatic theorem 25 min · quiz
- 4 Interaction of atoms with light: semiclassical radiation theory, Einstein coefficients and selection rules 23 min · quiz
- 5 Scattering theory: the Lippmann–Schwinger equation, the Born series and the optical theorem 22 min · quiz
- 6 Resonances, bound states and low-energy scattering 20 min · quiz
- 7 Density matrices: pure and mixed states, reduced density matrices and the Bloch ball 22 min · quiz
- 8 Open quantum systems: decoherence, the Lindblad equation and T1 and T2 21 min · quiz
- 9 Entanglement: Bell states, Bell inequalities and nonlocality 20 min · quiz
- 10 Quantum information: qubits, gates, circuits and no-cloning 20 min · quiz
- 11 Quantum algorithms and error correction: Deutsch–Jozsa, Grover, Shor and the surface code 20 min · quiz
- 12 Capstone: quantum technologies in 2026 — superconducting, trapped-ion and photonic qubits, quantum key distribution and sensing 21 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Time-dependent perturbation theory, scattering and an introduction to quantum information, developed as a course beyond the standard two-semester sequence: the modern postulates with density operators, tensor products and general measurements, and why the exponential size of many-qubit Hilbert space matters; Dirac notation, the unitary time-evolution operator and the Schrödinger, Heisenberg and interaction pictures, with conservation laws and the Mandelstam–Tamm relation; the Dyson series and time-ordered exponentials, second-order and virtual transitions, harmonic perturbations, the adiabatic theorem, Landau–Zener transitions at avoided crossings and Berry's geometric phase; semiclassical radiation theory, the dipole approximation, Einstein's A and B coefficients and the lifetime of hydrogen 2p, the Wigner–Eckart theorem and dipole selection rules, natural line shapes and population inversion; the Lippmann–Schwinger equation, outgoing Green's functions, the Born series and its convergence, and the optical theorem; low-energy scattering with the scattering length and effective range, shallow bound states and universality, Breit–Wigner resonances and the Feshbach resonances used to tune ultracold gases; density matrices, mixed states, the Bloch ball, partial traces, the Schmidt decomposition and von Neumann entropy; open quantum systems, decoherence, Kraus operators, the Lindblad master equation, the relaxation times T1 and T2, Ramsey fringes and spin echoes; Bell states, local hidden variables, the CHSH inequality and Tsirelson bound, Werner states and the loophole-free experiments of 2015; qubits, universal gate sets, the no-cloning theorem, teleportation and superdense coding; the Deutsch–Jozsa, Grover and Shor algorithms, quantum error correction with syndromes, the threshold theorem and the surface code; and a capstone on superconducting, trapped-ion, neutral-atom, photonic and spin qubits, BB84 quantum key distribution and quantum sensing, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 4.