# Quantum Mechanics

Quantum mechanics trades the certainty of a trajectory for the
bookkeeping of an **amplitude**: a complex-valued wave whose
squared magnitude gives the probability of what you will find. A particle
no longer _has_ a position until you look — it has a distribution.


_Figure 001 — A matter wavepacket: a localized bump of probability drifting and
spreading, built from a band of momenta around ħk.
_

_Figure 002 — Single quanta land as points, yet their tally builds an interference
pattern — |ψ₁ + ψ₂|².
_

The double slit is the whole subject in miniature. Fire electrons one at
a time and each strikes the screen as a single dot, particle-like; let
thousands accumulate and the dots organize into fringes, wave-like. The
**superposition** of two paths interferes, and asking which
slit it took destroys the pattern.


The **Schrödinger equation** is the law of motion for the
amplitude. Given a potential, it evolves the wavefunction deterministically
in time — the randomness lives only in what a measurement extracts, not in
how the state itself develops.


_Figure 003 — Confine a particle and its energy quantizes: standing-wave eigenstates on
discrete levels.
_

Confinement forces **quantization**. Bound in a well, only
the standing waves that fit survive, and their energies become a discrete
ladder. This is why atoms have sharp spectral lines and why the world at
small scales is granular rather than continuous.


Observables — position, momentum, energy, spin — become **operators**,
and the only values a measurement can return are their eigenvalues. When
two operators fail to commute, as position and momentum do, no state can
have a definite value of both: that is the uncertainty principle.


_Figure 004 — A qubit as a point on the Bloch sphere, precessing about the z-axis under
its Hamiltonian.
_

The simplest quantum system is a **two-level** one — a spin, a
qubit — and its entire state space is the surface of a sphere. Angular
momentum and spin, quantized and intrinsically non-classical, run through
everything from magnetic resonance to the structure of the periodic table.


_Figure 005 — Measurement collapses a superposition onto one eigenstate, with the
Born-rule probabilities.
_

Measurement is where the theory meets the world. A superposition carries
several outcomes at once, each weighted by an amplitude; observing it
_collapses_ the state onto a single eigenstate, at random, with
probability given by the Born rule.


Because exact solutions are rare, the working physicist leans on
**approximation** — perturbation theory, the variational
method, and the algebraic tricks of the harmonic oscillator — to make real
atoms, molecules, and solids tractable.


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## Contents

### 1. Origins of the Quantum

1. [Blackbody Radiation and the Planck Quantum](/quantum-mechanics/old-quantum-theory/blackbody-radiation-and-the-planck-quantum)
2. [The Photoelectric Effect and the Photon](/quantum-mechanics/old-quantum-theory/the-photoelectric-effect-and-the-photon)
3. [X-Rays and the Compton Effect](/quantum-mechanics/old-quantum-theory/x-rays-and-the-compton-effect)
4. [The Old Quantum Theory: Bohr, Sommerfeld, and Correspondence](/quantum-mechanics/old-quantum-theory/the-old-quantum-theory-bohr-and-sommerfeld)

### 2. The Wave Nature of Matter

1. [De Broglie Waves and Electron Diffraction](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction)
2. [Wave Packets and the Probabilistic Wave Function](/quantum-mechanics/matter-waves/wave-packets-and-the-probability-interpretation)
3. [The Uncertainty Principle and Wave-Particle Duality](/quantum-mechanics/matter-waves/the-uncertainty-principle)

### 3. Wave Mechanics in One Dimension

1. [The Schrödinger Equation in One Dimension](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
2. [The Free Particle and Wave-Packet Dynamics](/quantum-mechanics/wave-mechanics-1d/the-free-particle-and-wave-packet-dynamics)
3. [Particle in Infinite and Finite Square Wells](/quantum-mechanics/wave-mechanics-1d/particle-in-infinite-and-finite-square-wells)
4. [Operators, Expectation Values, and the Harmonic Oscillator](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator)
5. [The Dirac-Delta Potential: A Single Bound State and Scattering](/quantum-mechanics/wave-mechanics-1d/the-dirac-delta-potential)
6. [Barrier Penetration and Quantum Tunneling](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling)

### 4. The Formalism of Quantum Mechanics

1. [Hilbert Space and Dirac Bra–Ket Notation](/quantum-mechanics/formalism/hilbert-space-and-dirac-notation)
2. [Observables, Hermitian Operators, and the Spectral Theorem](/quantum-mechanics/formalism/observables-hermitian-operators-and-eigenvalues)
3. [The Postulates and Quantum Measurement](/quantum-mechanics/formalism/the-postulates-and-quantum-measurement)
4. [Position, Momentum, and Continuous Spectra](/quantum-mechanics/formalism/position-momentum-and-continuous-spectra)
5. [Commutators and the Generalized Uncertainty Principle](/quantum-mechanics/formalism/commutators-and-the-generalized-uncertainty-principle)
6. [Time Evolution, Propagators, and the Heisenberg Picture](/quantum-mechanics/formalism/time-evolution-schrodinger-and-heisenberg-pictures)

### 5. The Oscillator Algebraically, and Symmetry

1. [Ladder Operators and the Number States](/quantum-mechanics/oscillator-and-symmetry/ladder-operators-and-the-number-states)
2. [Coherent and Squeezed States](/quantum-mechanics/oscillator-and-symmetry/coherent-and-squeezed-states)
3. [Symmetries, Generators, and Conservation Laws](/quantum-mechanics/oscillator-and-symmetry/symmetries-generators-and-conservation-laws)
4. [Parity, Time Reversal, and Discrete Symmetries](/quantum-mechanics/oscillator-and-symmetry/parity-time-reversal-and-discrete-symmetries)

### 6. Angular Momentum

1. [Orbital Angular Momentum and Spherical Harmonics](/quantum-mechanics/angular-momentum/orbital-angular-momentum-and-spherical-harmonics)
2. [The Angular-Momentum Algebra and Ladder Operators](/quantum-mechanics/angular-momentum/the-angular-momentum-algebra)
3. [Addition of Angular Momenta and Clebsch–Gordan Coefficients](/quantum-mechanics/angular-momentum/addition-of-angular-momenta-and-clebsch-gordan)

### 7. Central Potentials

1. [The Schrödinger Equation in Three Dimensions](/quantum-mechanics/central-potentials/the-schrodinger-equation-in-three-dimensions)
2. [The Hydrogen Atom](/quantum-mechanics/central-potentials/the-hydrogen-atom)
3. [The Isotropic Oscillator and Hidden Symmetry](/quantum-mechanics/central-potentials/the-isotropic-oscillator-and-hidden-symmetry)

### 8. Spin

1. [Spin-½, the Pauli Matrices, and Stern–Gerlach](/quantum-mechanics/spin/spin-half-pauli-matrices-and-stern-gerlach)
2. [Spin in a Magnetic Field: Precession and Resonance](/quantum-mechanics/spin/spin-in-a-magnetic-field-precession-and-resonance)
3. [Two-Level Systems and the Bloch Sphere](/quantum-mechanics/spin/two-level-systems-and-the-bloch-sphere)

### 9. Identical Particles

1. [Identical Particles and Exchange Symmetry](/quantum-mechanics/identical-particles/identical-particles-and-exchange-symmetry)
2. [The Pauli Principle, Atoms, and the Periodic Table](/quantum-mechanics/identical-particles/the-pauli-principle-atoms-and-the-periodic-table)

### 10. Approximation Methods for Bound States

1. [Time-Independent Perturbation Theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
2. [Fine Structure and the Real Hydrogen Atom](/quantum-mechanics/approximation-methods/fine-structure-and-the-real-hydrogen-atom)
3. [The Zeeman and Stark Effects](/quantum-mechanics/approximation-methods/the-zeeman-and-stark-effects)
4. [The Variational Method](/quantum-mechanics/approximation-methods/the-variational-method)
5. [The WKB Approximation](/quantum-mechanics/approximation-methods/the-wkb-approximation)