Quantum Mechanics

Nature, at small enough scales, does not run on knobs and positions. It runs on amplitudes — complex numbers whose squared magnitudes are probabilities. Getting comfortable with that shift is the whole job. Once you have it, everything from the hydrogen spectrum to qubits to the photons in your phone's camera snaps into focus.

1. Why quantum mechanics exists

By 1900, classical physics was so complete that the University of Munich told young Max Planck not to study it because "all the important problems have been solved." Four problems, left unsolved, turned out to be the bitter ones. Each one, taken seriously, broke classical physics open.

Blackbody radiation. Heat a chunk of iron in a forge. It glows — first dull red, then orange, then yellow-white. The spectrum of the glow depends only on temperature, not material. Classical physics predicted that hot objects should radiate infinite energy at high frequencies (the "ultraviolet catastrophe"). Planck, in 1900, fit the observed spectrum by assuming — reluctantly, and as "an act of despair" — that light comes off in discrete chunks of energy $E = h f$, where $h$ is a new constant he had to invent.

The photoelectric effect. Shine light on a metal plate. Above a certain frequency threshold, electrons pop off. Below that threshold, nothing happens no matter how bright the light. Classical electromagnetism predicted that bright enough light should always eject electrons. Einstein, in 1905 (the same year as special relativity), explained the data by taking Planck's chunks literally: each photon is a particle with energy $h f$, and you need a single photon above threshold to kick out an electron. He won his Nobel for this, not for relativity.

Atomic spectra. When you run an electric current through hydrogen gas, it emits light at very specific wavelengths — a few sharp lines, not a continuous spectrum. Every element has its own fingerprint of spectral lines. Classical mechanics had no way to explain why atoms should emit only at discrete frequencies. Bohr, in 1913, guessed that electrons live on discrete energy levels and jump between them, emitting a photon whose energy equals the gap. It was a guess, but the numerical fit for hydrogen was perfect.

Electron diffraction. In 1927, Davisson and Germer bounced electrons off a nickel crystal and found interference patterns — electrons were behaving like waves. That clinched a weird hypothesis de Broglie had made in 1924: every particle has a wavelength $\lambda = h/p$, where $p$ is its momentum. Particles were waves. Waves were particles. The old categories did not survive.

Between 1925 and 1927, Heisenberg, Schrödinger, Born, Dirac, and Pauli put together a new theory — not a patchwork, an actual theory — that took the amplitude-and-probability picture seriously. By 1928 it had explained the hydrogen spectrum to astonishing precision. By the 1940s it had become the most accurately tested theory in physics. Today's prediction of the electron's magnetic moment from quantum electrodynamics agrees with experiment to about 12 decimal places.

Why should you care in 2026? Three immediate reasons:

• Semiconductors and lasers. Every transistor in every phone, every LED, every laser pointer works because electrons in solids live in bands with energy gaps that only quantum mechanics explains. A modern economy is built on band structures.
• Chemistry and materials. The shape of a protein, the strength of steel, the color of a dye — all of it is electron wave functions doing their thing. Drug design and battery engineering are applied quantum mechanics.
• Quantum computing. Qubits, entanglement, and superposition are the raw material of an industry that has gone from academic curiosity to Google, IBM, and dozens of startups building devices you can run code on.

This page will teach you the conceptual core. We'll use a little linear algebra and a little calculus; we'll assume you know what complex numbers are; and we'll keep the philosophy to a minimum. The philosophy arguments are real, but the math is the same regardless of which camp you sit in.

2. Vocabulary cheat sheet

3. State vectors and Dirac notation

Classical mechanics describes a particle by its position and momentum — six real numbers if it's in three dimensions. Quantum mechanics describes a system by a state vector, a unit-length vector in a complex vector space. Dirac introduced a notation that is a little weird the first time and a lot cleaner every time after that. A state is written

Every ket $|\psi\rangle$ has a partner called a bra, written $\langle\psi|$. The bra is the Hermitian conjugate: transpose the vector and complex-conjugate the entries. Bras and kets combine to form the inner product:

States can be superposed. Given two states $|0\rangle$ and $|1\rangle$, any combination

Two more pieces of notation. An outer product $|\psi\rangle\langle\phi|$ is a linear operator — feed it a ket, get back a ket scaled by $\langle\phi|$ applied to the input. A projection operator $|\phi\rangle\langle\phi|$ projects onto the direction of $|\phi\rangle$. The identity operator on any Hilbert space can be written $\sum_n |n\rangle\langle n|$ for any orthonormal basis $\{|n\rangle\}$; this is the completeness relation and it is how sums and integrals get inserted in calculations.

4. Operators, eigenvalues, and measurement

An observable — something you can measure — is represented by a Hermitian operator on the Hilbert space. Hermitian means $\hat A^\dagger = \hat A$: the operator equals its own conjugate transpose. Hermitian operators have real eigenvalues, which is exactly the property you want for things you can read off a voltmeter.

The measurement rules (collectively the Born rules) are these. Let $\hat A$ be an observable with eigenstates $|a_n\rangle$ and corresponding eigenvalues $a_n$: $\hat A |a_n\rangle = a_n |a_n\rangle$. Suppose the system is in state $|\psi\rangle$. Expand in the eigenbasis:

Now measure $\hat A$. The rule is:

• The measurement yields one of the eigenvalues $a_n$ — never anything else.
• The probability of getting $a_n$ is $|c_n|^2 = |\langle a_n | \psi \rangle|^2$.
• After measurement, the state of the system collapses to $|a_n\rangle$ — the eigenstate corresponding to the outcome.

The expected value of $\hat A$ in state $|\psi\rangle$ is

Two observables commute if $\hat A \hat B = \hat B \hat A$, and they share an eigenbasis. Measuring one does not disturb the other. Two observables that don't commute do not share eigenstates; measuring one changes the state so that subsequent measurements of the other give different outcomes. The canonical non-commuting pair is position and momentum:

5. The Schrödinger equation

Between measurements, the state vector evolves deterministically in time. The equation of motion was written down by Erwin Schrödinger in late 1925 and published in early 1926:

In the position representation, writing $\psi(x, t) = \langle x | \psi(t) \rangle$, the Schrödinger equation for a particle of mass $m$ in a potential $V(x)$ becomes a partial differential equation:

If $\hat H$ is time-independent, you can separate variables: $\psi(x, t) = \psi(x) e^{-iEt/\hbar}$. Plugging in gives the time-independent Schrödinger equation:

6. Canonical systems

Particle in a box (infinite square well)

The simplest bound-state problem. A particle lives in a one-dimensional region $0 \le x \le L$ with infinite potential walls: $V(x) = 0$ inside, $\infty$ outside. The wave function must vanish at $x = 0$ and $x = L$ (no penetration into infinite walls). Solve $-(\hbar^2/2m) \psi'' = E \psi$ with these boundary conditions, and you get sine waves that fit an integer number of half-wavelengths in the box.

The ground-state energy is not zero. Even in its lowest state, the particle has kinetic energy $E_1 = \pi^2 \hbar^2 / (2 m L^2)$. This is the zero-point energy, and it is a direct consequence of the uncertainty principle: a particle confined to a region of size $L$ has uncertain momentum of order $\hbar/L$, and therefore kinetic energy of order $\hbar^2/(2 m L^2)$.

Harmonic oscillator

The second canonical system. A particle in the potential $V(x) = \tfrac{1}{2} m \omega^2 x^2$ (a spring, essentially). The time-independent Schrödinger equation here can be solved by an algebraic trick (ladder operators) that is one of the cleanest calculations in physics. The result is

Hydrogen atom

The big result — the original reason anyone believed quantum mechanics. A single electron in the Coulomb potential of a proton, $V(r) = -k e^2/r$ where $k = 1/(4\pi\epsilon_0)$, has bound states with energies

The full solution has three quantum numbers: $n$ (energy), $\ell$ (angular momentum magnitude), and $m$ (angular momentum projection). With spin added, $n, \ell, m, m_s$ account for every state in the atom — and, via the Pauli exclusion principle, for the entire periodic table.

7. Interactive: particle in a box

Slide the quantum number $n$ to see the wave function $\psi_n(x)$ (cyan) and its probability density $|\psi_n(x)|^2$ (violet). Count the antinodes: state $n$ has exactly $n$ of them. The energy $E_n$ grows as $n^2$, and the probability density gets wavy with more and more peaks — until, for large $n$, it starts to look roughly uniform, which is the correspondence principle in action (classical particles in a box spend equal time everywhere).

8. The uncertainty principle

Heisenberg's uncertainty relation is not a statement about measurement error. It is a statement about what kinds of states exist. For any state $|\psi\rangle$, define

Heisenberg's relation then says:

A consequence: the electron in a hydrogen atom cannot just "fall into the nucleus." If it did, $\Delta x$ would vanish, forcing $\Delta p$ (and hence kinetic energy) to infinity. The balance between kinetic uncertainty energy and the attractive Coulomb energy determines the size of the atom — the Bohr radius $a_0 \approx 5.29 \times 10^{-11}$ m. Atoms are big because of the uncertainty principle.

9. Spin and two-level systems (qubits)

Classical angular momentum is a continuous three-vector. Quantum angular momentum — spin — has a discrete spectrum. An electron has spin $1/2$, which means its spin angular momentum along any chosen axis takes one of exactly two values: $+\hbar/2$ ("up") or $-\hbar/2$ ("down"). Not "anywhere in between." Just those two.

The state space for spin-$1/2$ is $\mathbb{C}^2$. You can write any state as

The observables for spin are the Pauli matrices:

Worked example — measuring a superposed qubit

A qubit is in state $|\psi\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. What happens when you measure $\sigma_z$? The outcomes are $+1$ (the $|0\rangle$ eigenvalue) with probability $|1/\sqrt{2}|^2 = 1/2$, and $-1$ with probability $1/2$. A 50-50 coin flip. Measure $\sigma_x$ instead, and the state $|\psi\rangle$ is already the $+1$ eigenstate of $\sigma_x$ — you get $+1$ with certainty. Same state, different basis, different outcomes. This is why the choice of measurement axis is part of the physics.

10. Entanglement and Bell's theorem

Two qubits live in $\mathbb{C}^2 \otimes \mathbb{C}^2 = \mathbb{C}^4$, spanned by $|00\rangle, |01\rangle, |10\rangle, |11\rangle$. A general two-qubit state is some complex combination of those four. Most of them cannot be written as a tensor product of single-qubit states. Those states are called entangled. The most famous one is

Bell, in 1964, derived an inequality that any "local hidden variables" theory must satisfy. Then he showed that quantum mechanics violates the inequality for certain entangled states. Experiments starting with Alain Aspect in 1982 and culminating in loophole-free tests around 2015 have repeatedly verified the quantum prediction. No local classical story reproduces the data. The universe is non-classical in a provable, experimentally decided way. Aspect, Clauser, and Zeilinger shared the 2022 Nobel Prize in Physics for this line of work.

Entanglement is not "faster-than-light communication." You cannot use it to send a signal. But it is a genuine correlation that cannot be explained by any pre-measurement labeling. It is the physical resource that makes quantum computing, quantum cryptography, and quantum teleportation work. Most quantum information theory is, at the end of the day, bookkeeping about who has access to which part of an entangled state.

See also

11. Quantum mechanics in code

A couple of short primitives: construct the Hamiltonian of a 1D particle in a box, diagonalize it with NumPy, and verify the $n^2$ spectrum. Then a qubit simulation — build a state, apply a gate, compute expectation values.

import numpy as np

# ---------- Particle in a box via finite-difference Hamiltonian ----------
def box_spectrum(N=400, L=1.0, hbar=1.0, m=1.0):
    dx = L / (N + 1)
    main = np.full(N, -2.0)
    off  = np.full(N - 1, 1.0)
    T = np.diag(main) + np.diag(off, 1) + np.diag(off, -1)
    H = -(hbar * hbar) / (2 * m * dx * dx) * T
    eigvals, eigvecs = np.linalg.eigh(H)
    return eigvals, eigvecs, dx

evals, _, dx = box_spectrum()
# Exact: E_n = n^2 pi^2 / 2 for hbar=m=L=1
for n in range(5):
    exact = (n + 1) ** 2 * np.pi ** 2 / 2
    print(f"n={n+1}: numeric={evals[n]:.4f}, exact={exact:.4f}")

# ---------- Qubit state, Pauli operators, expectation values ----------
sx = np.array([[0, 1], [1, 0]], dtype=complex)
sy = np.array([[0, -1j], [1j, 0]], dtype=complex)
sz = np.array([[1, 0], [0, -1]], dtype=complex)

# Equal superposition: (|0> + |1>) / sqrt(2)
psi = (1 / np.sqrt(2)) * np.array([1, 1], dtype=complex)
print(f"<sz> = {np.real(psi.conj() @ sz @ psi):.3f}")  # 0
print(f"<sx> = {np.real(psi.conj() @ sx @ psi):.3f}")  # 1

# Bell state |Phi+> = (|00> + |11>) / sqrt(2)
bell = np.zeros(4, dtype=complex)
bell[0] = bell[3] = 1 / np.sqrt(2)
I = np.eye(2, dtype=complex)
Z1 = np.kron(sz, I)
Z2 = np.kron(I, sz)
print(f"<Z1 Z2> = {np.real(bell.conj() @ (Z1 @ Z2) @ bell):.3f}")  # 1

import math

# Hydrogen-level emission wavelengths (Balmer series), no NumPy needed.
R_eV = 13.605693                   # Rydberg constant in eV
hc_eV_nm = 1239.84193              # h*c in eV*nm

def balmer_wavelength_nm(n):
    # Transition from n -> 2
    dE = R_eV * (1 / 4 - 1 / (n * n))
    return hc_eV_nm / dE

for n in range(3, 7):
    print(f"Balmer n={n}->2: {balmer_wavelength_nm(n):7.2f} nm")
# Should print ~656, 486, 434, 410 nm

12. Cheat sheet