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Quantum Mechanics

For students. For researchers.

Quantum Mechanics

From physical ideas
to precise predictions.

A connected reference for understanding the theory, working through the mathematics, and putting quantum models to the test.

Explore the Library Find your starting point

18 subject volumes · Learning paths · Computational labs

Superposition in motion
One quantum state, over one periodSeventeen slices show the position probability density of an equal superposition of the first two harmonic-oscillator energy states. Position q runs left to right, time τ runs diagonally toward the back, and density rises vertically. The density changes shape and returns to its initial form after one period. The blue highlighted slice is selected with the phase control.

Two energy states. One evolving probability density.

Explore the harmonic oscillator
The model behind the figure

A closed, one-dimensional harmonic oscillator, initially in the equal superposition ∣ψ(0)⟩=(∣0⟩+∣1⟩)/2|\psi(0)\rangle=(|0\rangle+|1\rangle)/\sqrt2. The curves show probability density, not a particle trajectory.

ρ(q,τ)=e−q2π(12+q2+2qcos⁡τ)\rho(q,\tau)=\frac{e^{-q^2}}{\sqrt\pi}\left(\frac12+q^2+\sqrt2q\cos\tau\right)

q=x/ℏ/(mω),τ=ωtq=x/\sqrt{\hbar/(m\omega)},\quad\tau=\omega t. The density integrates to one over the real line; the figure shows −4 ≤ q ≤ 4 on a fixed scale.

The subject, connected

The Library18 volumes

Start with a question. Find its home.
Follow the connections as far as you need.

Tools and Core Concepts

Build the language. Meet the evidence. Learn how quantum predictions are made.

Dynamics and General Methods

Follow evolution, control approximations, and move from one particle to many.

Systems and Applications

Connect the theory to atoms, materials, information, and relativistic systems.

Mathematical Structure and Foundations

Examine the mathematical structure and the questions behind its interpretation.

Read in the order your question needs.Open the full catalog

A model worth knowing

One oscillator.
Many connections.

Begin with an exactly solvable system. Understand its quantum states, translate the equation into a calculation, and test the result.

New to wavefunctions? Start with the foundations.

Follow the 4-step learning path
  1. The physicsMeet the harmonic oscillatorEnergy levels, wavefunctions, and the ladder-operator construction.
  2. The methodTurn derivatives into a matrixBuild a finite-difference representation and understand its errors.
  3. The checkKnow when the answer is reliableSeparate grid-spacing error from the effect of a finite spatial domain.
  4. The investigationCompute the spectrumRun the downloadable lab and compare its results with exact energies.

Choose your next question

A first encounter.
A deeper understanding.

Come for a course, a calculation, or a question that has stayed with you.

Choose a learning path
Starting outA first course in quantum mechanicsGoing furtherA route through graduate quantum mechanicsAt the frontierWhat is known. What remains open.