Quantum Mechanics
Quantum mechanics describes the behavior of matter and energy at atomic and subatomic scales. It replaced classical mechanics for small systems after a series of experimental crises in the early 20th century — blackbody radiation (Planck, 1900), the photoelectric effect (Einstein, 1905), the Bohr model (1913), and the full wave mechanics of Schrodinger and Heisenberg (1925-26). The theory is probabilistic, non-local, and profoundly counterintuitive, yet it is the most precisely tested theory in all of science. This skill covers the foundational concepts, mathematical framework, and key applications of non-relativistic quantum mechanics, with pointers to relativistic extensions.
Agent affinity: feynman (quantum mechanics, Opus)
Concept IDs: phys-quantum-basics, phys-nuclear-physics, phys-special-relativity
Quantum Mechanics at a Glance
| # | Topic | Key equations | Core idea | |---|---|---|---| | 1 | Wave-particle duality | E = hf, p = h/lambda | Matter and light have both wave and particle properties | | 2 | Photoelectric effect | KE_max = hf - phi | Light comes in quanta (photons) | | 3 | de Broglie wavelength | lambda = h/p = h/(mv) | Particles have associated wavelengths | | 4 | Uncertainty principle | Delta x Delta p >= hbar/2 | Conjugate variables cannot both be precisely known | | 5 | Schrodinger equation | H psi = E psi | Wave equation for quantum states | | 6 | Wave functions | P = |psi|^2 | Probability from amplitude squared | | 7 | Hydrogen atom | E_n = -13.6 eV / n^2 | Quantized energy levels | | 8 | Spin & Pauli exclusion | s = 1/2 for electrons | No two identical fermions in same state | | 9 | Quantum tunneling | T ~ exp(-2 kappa L) | Particles penetrate classically forbidden barriers | | 10 | Path integrals | K = sum over all paths of exp(iS/hbar) | All histories contribute | | 11 | Entanglement | Bell inequality violation | Non-local correlations without signaling | | 12 | Measurement problem | Collapse vs. decoherence vs. many-worlds | Interpretation remains open |
Topic 1 — Wave-Particle Duality
The central mystery. Light behaves as a wave in interference and diffraction experiments (Young's double slit, 1801), yet it behaves as a particle in the photoelectric effect and Compton scattering. Matter behaves as particles in everyday life, yet electrons and even large molecules produce interference patterns. Wave-particle duality is not a defect of the theory — it is a fundamental feature of nature.
Planck's relation: E = hf, where h = 6.626 * 10^-34 J s is Planck's constant and f is frequency.
de Broglie's relation: lambda = h/p, where p is momentum. This connects the wave and particle descriptions.
Complementarity (Bohr). Wave and particle aspects are complementary — an experiment reveals one or the other, never both simultaneously. The double-slit experiment with which-path detectors demonstrates this: detecting which slit the particle passes through destroys the interference pattern.
Topic 2 — The Photoelectric Effect
Experimental facts (Hertz, 1887; Lenard, 1902):
- Light below a threshold frequency produces no electrons, regardless of intensity.
- Above threshold, electrons appear instantly (no delay).
- Increasing intensity increases the number of electrons, not their maximum energy.
- Maximum kinetic energy increases linearly with frequency.
Einstein's explanation (1905): Light consists of photons, each carrying energy E = hf. A photon ejects an electron only if hf >= phi (the work function). The maximum kinetic energy of the ejected electron is KE_max = hf - phi.
Worked example. Ultraviolet light of wavelength 200 nm strikes a sodium surface (phi = 2.28 eV). Find the maximum kinetic energy of emitted electrons.
Solution. Photon energy: E = hc/lambda = (6.626e-34)(3e8)/(200e-9) = 9.94e-19 J = 6.21 eV. KE_max = 6.21 - 2.28 = 3.93 eV.
Stopping voltage. The maximum KE can be measured by applying a retarding voltage: eV_stop = KE_max, so V_stop = 3.93 V.
Topic 3 — de Broglie Wavelength
de Broglie's hypothesis (1924): Every particle with momentum p has an associated wavelength lambda = h/p. This was experimentally confirmed by Davisson and Germer (1927), who observed electron diffraction from nickel crystals.
Worked example. Find the de Broglie wavelength of an electron accelerated through 100 V.
Solution. Kinetic energy: KE = eV = 1.6e-19 100 = 1.6e-17 J. Momentum: p = sqrt(2mKE) = sqrt(2 9.11e-31 * 1.6e-17) = sqrt(2.916e-47) = 5.4e-24 kg m/s. Wavelength: lambda = h/p = 6.626e-34 / 5.4e-24 = 1.23e-10 m = 0.123 nm.
This is comparable to atomic spacings, which is why electrons can be diffracted by crystal lattices. For a baseball (0.15 kg at 40 m/s), lambda = 1.1e-34 m — immeasurably small, which is why baseballs don't diffract.
Topic 4 — The Heisenberg Uncertainty Principle
Statement: Delta x Delta p >= hbar/2, where hbar = h/(2 pi) = 1.055 * 10^-34 J s.
Energy-time form: Delta E Delta t >= hbar/2.
What it means. The uncertainty principle is not about measurement limitations or experimental clumsiness. It is a fundamental property of wave-like systems: a wave packet that is localized in position must be spread in wavelength (and hence momentum), and vice versa. This is a mathematical consequence of Fourier analysis.
Worked example. An electron is confined to a region of width 10^-10 m (approximately an atomic diameter). What is the minimum uncertainty in its momentum and the corresponding minimum kinetic energy?
Solution. Delta p >= hbar/(2 Delta x) = 1.055e-34 / (2 1e-10) = 5.28e-25 kg m/s. Minimum KE ~ (Delta p)^2/(2m) = (5.28e-25)^2/(2 9.11e-31) = 2.79e-49 / 1.82e-30 = 1.53e-19 J = 0.96 eV.
This is on the order of atomic binding energies — the uncertainty principle explains why electrons in atoms have characteristic energy scales of a few eV.
What it does NOT mean. The uncertainty principle does not say "observation disturbs the system" (though it can). It says that the state itself does not possess simultaneously precise position and momentum. This is a statement about nature, not about instruments.
Topic 5 — The Schrodinger Equation
Time-dependent Schrodinger equation:
i hbar d psi/dt = H psi
where H = -(hbar^2)/(2m) nabla^2 + V(x) is the Hamiltonian operator.
Time-independent Schrodinger equation (for stationary states with definite energy):
H psi = E psi, or equivalently: -(hbar^2)/(2m) d^2 psi/dx^2 + V(x) psi = E psi
Particle in a box (infinite square well). A particle confined between x = 0 and x = L with infinite potential walls.
Solutions: psi_n(x) = sqrt(2/L) sin(n pi x / L), E_n = n^2 pi^2 hbar^2 / (2mL^2), n = 1, 2, 3, ...
Worked example. An electron in a 1D box of width L = 1 nm. Find the energy of the ground state (n = 1).
Solution. E_1 = pi^2 hbar^2 / (2mL^2) = pi^2 (1.055e-34)^2 / (2 9.11e-31 (1e-9)^2) = 1.097e-67 * 9.87 / (1.822e-48) = 6.02e-20 J = 0.376 eV.
Key features: Energy is quantized (only discrete E_n allowed). The ground state energy is nonzero (zero-point energy). The wave function has n-1 nodes. Higher energy states oscillate more rapidly.
Topic 6 — Wave Functions and Probability
Born's rule (1926): The probability of finding a particle in the interval [x, x + dx] is P(x) dx = |psi(x)|^2 dx. The wave function psi is a probability amplitude; its modulus squared is the probability density.
Normalization: integral from -infinity to +infinity of |psi|^2 dx = 1. The particle must be found somewhere.
Expectation values: <x> = integral of x |psi|^2 dx. <p> = integral of psi* (-i hbar d/dx) psi dx. These give the average result of measuring position or momentum over many identically prepared systems.
Superposition. If psi_1 and psi_2 are valid states, then c_1 psi_1 + c_2 psi_2 is also a valid state. Measurement collapses the superposition: the probability of finding the system in sta