Quantum Mechanics

A Spins-First, Experimental Approach¶

Quantum mechanics is not classical mechanics with a layer of uncertainty added. It is a different framework for connecting preparations, transformations, and measurements. This book begins where that difference is easiest to see: a beam of atoms passing through Stern–Gerlach analyzers.

Starting with a two-state system lets us confront the essential ideas before differential equations obscure them. States are vectors, alternatives combine through complex amplitudes, observables are operators, and time evolution is unitary. Wavefunctions will later appear as the coordinates of a state in a continuous position basis—not as the definition of a quantum state.

Who this book is for¶

The intended reader has completed introductory mechanics and electromagnetism, is comfortable with calculus and complex numbers, and has seen basic matrix algebra. The necessary linear algebra is reviewed when it first becomes useful.

The text emphasizes four habits:

1. Begin with a physical preparation and a clearly specified measurement.

2. Keep the state, its coordinates, and the physical system distinct.

3. Move deliberately among diagrams, bra–ket notation, and matrices.

4. Check every result against normalization, dimensions, and limiting cases.

Book structure¶

The book develops one connected route from two-state experiments to modern applications.

Part I: quantum states and dynamics¶

• Stern–Gerlach experiments establish the experimental rules that a theory must explain.

• States and probability amplitudes introduce kets, bases, inner products, and interference.

• Operators and measurement turn measurement outcomes into a general mathematical framework.

• Time evolution develops unitary dynamics and spin precession.

Part II: composite and continuous systems¶

• Composite systems and entanglement introduce tensor products, reduced states, Bell correlations, and two-qubit gates.

• Position and momentum representations reinterpret wavefunctions as continuous-basis coordinates and develop Fourier duality.

• One-dimensional wave mechanics treats bound states, scattering, tunneling, and the harmonic oscillator.

• Three-dimensional wave mechanics develops separability, central potentials, and hydrogen.

Part III: structure and applications¶

• Angular momentum develops rotation algebra, spherical harmonics, and addition of angular momenta.

• Approximation methods introduce perturbation theory, variational estimates, WKB, and driven transitions.

• Identical particles connect exchange symmetry to exclusion, occupation numbers, and the structure of matter.

• Modern quantum applications apply the framework to information processing, error correction, sensing, clocks, and simulation.

Conventions¶

States are normalized kets such as $|\psi\rangle$. Operators carry hats, such as $\hat S_z$. Unless stated otherwise, spin operators have units of angular momentum and their eigenvalues contain $\hbar$. Probabilities are dimensionless and lie between zero and one.

This is an original open textbook inspired by the spins-first teaching tradition. It does not reproduce the prose, figures, or problems of commercial texts.

Downloadable editions¶

The website may contain changes made after these downloadable editions were exported.

• PDF with worked solutions — exported 2026-10-01, revision 0762ee9.

• PDF with exercises only — exported 2026-10-01, revision 0762ee9.

• Word edition — exported 2026-10-01, revision 0762ee9.