Why advanced quantum mechanics: from textbook problems to modern quantum technology

Quantum Mechanics I and II, PHYS-372 and PHYS-373, built the standard machinery of the subject: wave functions and operators, the hydrogen atom solved exactly, spin and the addition of angular momenta, identical particles, perturbation theory, the variational and WKB methods, Fermi's golden rule and the partial-wave theory of scattering. Those tools explain the periodic table, atomic spectra and the chemical bond. They were essentially complete by 1930, and for half a century most physicists treated the foundations of the theory as settled and turned to its applications.

This course takes the theory further in two directions. The first is technical. Time-dependent problems and scattering problems were treated in PHYS-373 at the level of first-order formulas; here they are rebuilt on the time-evolution operator, the Dyson series and the Lippmann–Schwinger equation, which organize every order of perturbation theory at once and make clear which results are exact. The second direction is conceptual, and it is the reason the subject has changed so much since the 1980s. Real quantum systems are rarely in pure states and are never perfectly isolated. They are entangled with their surroundings, they lose coherence, and they can be measured, copied (or not) and used to carry information. The density matrix, the theory of open systems and the theory of entanglement turn those facts from embarrassments into tools. Together they are the physics behind quantum computers, quantum cryptography and quantum sensors.

The plan of the course

The twelve lessons fall into three groups. Lessons 2 to 4 deal with dynamics. Lesson 2 sets out Dirac notation properly and introduces the Schrödinger, Heisenberg and interaction pictures, three equivalent ways of dividing the time dependence between states and operators. Lesson 3 develops time-dependent perturbation theory to all orders through the Dyson series, then turns to slow changes: the adiabatic theorem, Landau–Zener transitions at avoided crossings and the geometric Berry phase. Lesson 4 applies these ideas to the interaction of atoms with light, deriving the Einstein coefficients and the selection rules that decide which transitions occur.

Lessons 5 and 6 rebuild scattering theory. Lesson 5 introduces the Lippmann–Schwinger equation, the Born series and the optical theorem, which ties the forward scattering amplitude to the total cross-section. Lesson 6 looks at what happens at very low energy, where the whole effect of a potential is summarized by one number, the scattering length, and at resonances, including the Feshbach resonances that let experimenters tune the interactions in ultracold gases with a magnetic field.

Lessons 7 to 12 form the third group, on states, information and technology. Lesson 7 introduces the density matrix, mixed states, the Bloch ball and the reduced density matrix of a subsystem. Lesson 8 treats open quantum systems: decoherence, the Lindblad master equation and the relaxation times \(T_1\) and \(T_2\) that limit every qubit. Lesson 9 is about entanglement, Bell's theorem and the CHSH inequality, and the experiments, culminating in the loophole-free tests of 2015, that rule out local hidden variables. Lessons 10 and 11 build the circuit model of quantum computation, from gates and the no-cloning theorem to teleportation, the Deutsch–Jozsa and Grover algorithms and quantum error correction. Lesson 12 surveys the technologies of 2026: superconducting, trapped-ion, neutral-atom and photonic qubits, quantum key distribution and quantum sensing.

Why the size of Hilbert space matters

One fact runs through the whole course. A single two-level system, a qubit, needs two complex amplitudes. Two qubits need four, because the general state is a superposition of \(|00\rangle\), \(|01\rangle\), \(|10\rangle\) and \(|11\rangle\). In general \(n\) qubits need \(2^n\) amplitudes. This exponential growth is what makes many-body quantum mechanics hard to simulate: storing the state of 50 qubits on a classical computer takes about 18 petabytes, more than the memory of the largest supercomputers, and every step of a simulation must update all those numbers. Richard Feynman drew the conclusion in a 1981 lecture published in 1982: if nature is quantum mechanical, the natural way to simulate it is with a machine that is itself quantum mechanical. That remark is often taken as the starting point of quantum computing.

The same exponential space allows correlations that no classical system can have. In the Bell state \((|00\rangle + |11\rangle)/\sqrt2\) the two qubits always give the same result when both are measured along \(z\), and also when both are measured along \(x\). A classical pair of coins can be correlated in one basis, but a product state like \(|{+}\rangle|{+}\rangle\) agrees in the \(x\) basis and only half the time in the \(z\) basis. Entanglement is the name for correlations of the first kind, which cannot be written as a product of states of the parts. Lesson 9 shows that they cannot even be reproduced by any local model with hidden variables.

import cmath, math

# Hilbert space size: n two-level systems need 2**n complex amplitudes
print("Qubits  amplitudes        memory at 16 bytes each")
for n in (1, 2, 10, 30, 50):
    amps = 2 ** n
    mem = amps * 16
    unit = next(u for u, s in (("PB", 1e15), ("TB", 1e12), ("GB", 1e9), ("MB", 1e6), ("kB", 1e3), ("B", 1)) if mem >= s)
    scale = {"PB": 1e15, "TB": 1e12, "GB": 1e9, "MB": 1e6, "kB": 1e3, "B": 1}[unit]
    print(f"  {n:3d}   {amps:>18,d}   {mem / scale:10.3g} {unit}")
print()

# A Bell state and a product state: compare joint statistics in two bases
r = 1 / math.sqrt(2)
bell = [r, 0, 0, r]                      # (|00> + |11>)/sqrt2, order 00, 01, 10, 11
prod = [0.5, 0.5, 0.5, 0.5]              # |+>|+>
H = [[r, r], [r, -r]]                    # Hadamard: measure in the x basis

def apply_both(U, psi):
    out = [0j] * 4
    for a in range(2):
        for b in range(2):
            for c in range(2):
                for d in range(2):
                    out[2 * a + b] += U[a][c] * U[b][d] * psi[2 * c + d]
    return out

def same(psi):                          # probability that the two outcomes agree
    p = [abs(x) ** 2 for x in psi]
    return p[0] + p[3]

for name, psi in (("Bell (|00>+|11>)/sqrt2", bell), ("product |+>|+>", prod)):
    print(f"{name:24s} P(agree) z basis = {same(psi):.3f}, x basis = {same(apply_both(H, psi)):.3f}")
print()

# Coherence time budget: how many gates fit before decoherence?
print("Platform (typical 2025 values)   T2         gate time   gates per T2")
for name, T2, tg in (("superconducting transmon", 100e-6, 50e-9), ("trapped ion", 1.0, 100e-6), ("neutral atom", 1.0, 1e-6)):
    print(f"  {name:28s} {T2:9.2e} s  {tg:9.1e} s  {T2 / tg:10.0f}")

Qubits  amplitudes        memory at 16 bytes each
    1                    2           32 B
    2                    4           64 B
   10                1,024         16.4 kB
   30        1,073,741,824         17.2 GB
   50   1,125,899,906,842,624           18 PB

Bell (|00>+|11>)/sqrt2   P(agree) z basis = 1.000, x basis = 1.000
product |+>|+>           P(agree) z basis = 0.500, x basis = 1.000

Platform (typical 2025 values)   T2         gate time   gates per T2
  superconducting transmon      1.00e-04 s    5.0e-08 s        2000
  trapped ion                   1.00e+00 s    1.0e-04 s       10000
  neutral atom                  1.00e+00 s    1.0e-06 s     1000000

The last part of the output previews lesson 8. A qubit is useful only while it keeps its phase coherence, for a time called \(T_2\). Dividing \(T_2\) by the time a gate takes gives a rough count of the operations possible before the quantum information is lost. The numbers are typical of 2025 hardware and vary widely between devices, but they show why every platform has its strengths. Superconducting circuits are fast but decohere quickly; trapped ions are slow but long-lived. For any of them, long calculations need error correction, the subject of lesson 11.

The postulates in modern form

It is worth restating the postulates of quantum mechanics in the form this course uses. They differ from the PHYS-372 version mainly by allowing mixed states and general measurements.

Nothing here contradicts the earlier courses. A pure state \(|\psi\rangle\) corresponds to the density operator \(\rho = |\psi\rangle\langle\psi|\), and for it \(\operatorname{tr}(\rho P_a) = \langle\psi|P_a|\psi\rangle\). The extension matters because the subsystems of entangled systems, and systems in contact with an environment, cannot be described by state vectors of their own. The mathematical background is the theory of Hilbert spaces and self-adjoint operators from MATH-445 Functional Analysis. There the spectral theorem justifies the decomposition of an observable into projections, and the distinction between bounded and unbounded operators explains why position and momentum need more care than spin.

A short history of the second quantum revolution

The foundations returned to the centre of physics in stages.

Canada has played a large part in this story, from BB84 itself to the Institute for Quantum Computing in Waterloo, the Perimeter Institute for Theoretical Physics and companies such as D-Wave in Burnaby, Xanadu in Toronto and Photonic in Vancouver. In the United States the major industrial efforts include IBM and Google, the trapped-ion companies IonQ and Quantinuum, and the national laboratories and NIST. Lesson 12 returns to these programs.

What this course assumes

The course assumes PHYS-373 and the mathematics behind it. From PHYS-372 and PHYS-373 you should be comfortable with the Schrödinger equation in one and three dimensions, the hydrogen atom, spin one-half and the Pauli matrices, the addition of angular momenta, first-order time-dependent perturbation theory and Fermi's golden rule, and partial-wave scattering. From linear algebra you need eigenvalues and eigenvectors of Hermitian matrices, unitary matrices and tensor products; MATH-322 covers all of these. Complex exponentials, Fourier transforms and simple differential equations appear throughout, and a few lessons use ideas from statistical mechanics, such as the Boltzmann distribution and entropy from PHYS-361. Where a result from an earlier course is needed, the lesson restates it briefly and points to where it was derived, so that each lesson can be read on its own. Readers who want more depth on the functional analysis behind the formalism will find it in MATH-445, and readers interested in the algorithms of lessons 10 and 11 from the computing side will find the classical background in CMPUT-304.

How to read this course

Each lesson combines derivations with small numerical experiments in standard-library Python. The programs use complex arithmetic from the cmath module and write out the few matrix operations they need, typically for two to eight dimensions. They are short enough to read in full, and each one checks a formula from the text: a matrix exponential against an analytic precession, a numerically integrated Landau–Zener sweep against the exact formula, a simulated Bell test against the Tsirelson bound. Reading the code is part of the lesson, and changing the parameters is the quickest way to build intuition.

We use units with \(\hbar = 1\) in most of the code and much of the theory, restoring \(\hbar\) where numbers in SI units are wanted. Kets \(|\psi\rangle\), bras \(\langle\phi|\) and brackets \(\langle\phi|\psi\rangle\) appear from lesson 2 on; for qubits we write \(|0\rangle\) and \(|1\rangle\) for the eigenstates of \(\sigma_z\) with eigenvalues \(+1\) and \(-1\), and \(|{\pm}\rangle = (|0\rangle \pm |1\rangle)/\sqrt2\).