ECE 305: Introduction to Quantum Mechanics

Sources and References

Equivalent UW courses — PHYS 234 (Quantum Physics 1), PHYS 334 (Quantum Physics 2), NE 232 (Quantum Mechanics for Nanoengineering) Primary textbook — McIntyre, David H. Quantum Mechanics: A Paradigms Approach, Pearson (required) Supplementary references — Griffiths, Introduction to Quantum Mechanics (the standard PHYS 234 / NE 232 text); Shankar, Principles of Quantum Mechanics for deeper operator / Dirac-notation material; Oregon State SPINS simulation software for Stern-Gerlach experiments.

Equivalent UW Courses

ECE 305 is a one-term introduction that spans material PHYS splits across PHYS 234 and (the first third of) PHYS 334. NE 232 is the nanoengineering version, which uses a very similar syllabus and textbook (Griffiths) but with a condensed-matter slant. PHYS 234 covers wave-particle duality, the Schrodinger equation, 1D bound and scattering states, and the harmonic oscillator; PHYS 334 continues into angular momentum, the hydrogen atom, spin, and perturbation theory. ECE 305 packages a subset of both into one term by following McIntyre’s “spins-first” order, which is a very different pedagogical path than the wavefunction-first route used in PHYS 234.

What This Course Adds Beyond the Equivalents

• Spins-first pedagogy. Quantum mechanics is built up starting from Stern-Gerlach experiments and two-level spin-1/2 systems, so students meet state vectors, operators, and measurement as finite-dimensional linear algebra before wavefunctions ever appear. This is McIntyre’s approach and contrasts sharply with the Griffiths / PHYS 234 route (which starts from Schrodinger and postpones spin).
• Quantum computing preview in the final weeks, tying two-level systems back to qubits and gates — an engineering-motivated application PHYS 234 does not discuss.
• Omits, relative to PHYS 234 + PHYS 334: detailed 1D scattering and tunneling problems (only briefly seen through the finite square well), variational and WKB methods, and full time-independent perturbation theory (mentioned in the calendar description but shallow in practice). The identical-particles / fermion-boson statistics chapter of PHYS 334 is not covered.

Topic Summary

Stern-Gerlach and Quantum State Vectors

The Stern-Gerlach experiment is used to argue that a spin-1/2 system has two orthogonal basis states. The abstract state vector \( |\psi\rangle \) and its decomposition in a chosen basis are introduced before any wavefunction language.

Linear Algebra and Matrix Notation

Inner products, orthonormal bases, matrix representations of operators, Hermitian and unitary matrices, eigenvalues and eigenvectors. Worked out for spin-1/2 (the Pauli matrices) and demonstrated with the SPINS simulation software.

Operators, Measurement, Expectation Values

Observables as Hermitian operators; measurement postulate: an eigenvalue \( a_n \) of an operator \( \hat A \) is obtained with probability \( |\langle a_n|\psi\rangle|^2 \), and the post-measurement state is \( |a_n\rangle \). Expectation value

\[ \langle A \rangle = \langle \psi | \hat A | \psi \rangle \]

and its time derivative for a prescribed Hamiltonian.

Commutation Relations and the Uncertainty Principle

Commutators \( [\hat A,\hat B] \), compatibility of observables, and the generalised uncertainty relation

\[ \Delta A\,\Delta B \ge \tfrac{1}{2}\,|\langle [\hat A,\hat B] \rangle| \]

with position-momentum as the canonical example.

The Schrodinger Equation

Time-dependent Schrodinger equation \( i\hbar\,\partial_t|\psi\rangle = \hat H|\psi\rangle \). Stationary states as energy eigenstates and the separation-of-variables route to the time-independent equation. Spin precession in a magnetic field as a first worked example of time evolution.

Wavefunctions and the Energy Eigenvalue Equation

Transition from the abstract formulation to wavefunctions \( \psi(x) = \langle x|\psi\rangle \). The time-independent Schrodinger equation as a second-order ODE, and the boundary / normalisation conditions that quantise the spectrum.

Infinite and Finite Square Wells

Particle in a box: analytic eigenvalues \( E_n = n^2\pi^2\hbar^2/(2mL^2) \) and sinusoidal eigenfunctions. Finite well: transcendental matching conditions, penetration into the classically forbidden region.

Harmonic Oscillator

The ubiquitous quadratic potential, solved via either the analytic Hermite-polynomial route or creation / annihilation ladder operators. Zero-point energy and the equal-spacing spectrum motivate later applications to phonons and photons.

Angular Momentum and the Hydrogen Atom

Orbital angular-momentum operators \( \hat L^2,\hat L_z \) and their shared eigenstates (spherical harmonics). The hydrogen radial equation, Coulomb potential, and energy levels \( E_n = -13.6\,\text{eV}/n^2 \). Orbital quantum numbers \( n, \ell, m \) and their degeneracies.

Applications: Quantum Computing

Two-level systems as qubits; single-qubit gates as unitary rotations on the Bloch sphere; brief mention of superposition, entanglement, and measurement as the physical substrate for quantum information processing.