Quantum mechanics from wave-particle duality through entanglement and measurement. Covers the photoelectric effect, de Broglie wavelength, Heisenberg uncertainty principle, the Schrodinger equation (time-dependent and time-independent), wave functions and probability, quantum states and superposition, the hydrogen atom (energy levels, quantum numbers), spin, Pauli exclusion principle, quantum tunneling, Feynman path integrals (conceptual), QED basics, entanglement (Bell's theorem), the measurement problem, and Dirac notation. Use when analyzing phenomena at atomic or subatomic scales where classical physics breaks down.
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
| # | 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 |
| 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 |
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.
Experimental facts (Hertz, 1887; Lenard, 1902):
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.
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.
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.
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.
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 state psi_n is |c_n|^2.
Energy levels: E_n = -13.6 eV / n^2, n = 1, 2, 3, ...
Quantum numbers:
Degeneracy. For hydrogen, all states with the same n have the same energy (2n^2 states per level, including spin). In multi-electron atoms, this degeneracy is broken by electron-electron interactions.
Worked example. Find the wavelength of light emitted when a hydrogen atom transitions from n = 3 to n = 2 (the H-alpha line).
Solution. Delta E = -13.6(1/9 - 1/4) = -13.6(-5/36) = 1.889 eV = 3.025e-19 J. lambda = hc/Delta E = (6.626e-34)(3e8)/(3.025e-19) = 6.57e-7 m = 657 nm. This is red light — the prominent red line in the hydrogen emission spectrum, first measured by Balmer in 1885.
Spectral series: Lyman (to n=1, UV), Balmer (to n=2, visible), Paschen (to n=3, IR), Brackett (to n=4, IR).
Spin. Electrons possess an intrinsic angular momentum called spin, with quantum number s = 1/2. The spin component along any axis has two values: m_s = +1/2 ("spin up") or m_s = -1/2 ("spin down"). Spin has no classical analog — it is not the electron "spinning" on an axis.
The Stern-Gerlach experiment (1922). A beam of silver atoms passing through an inhomogeneous magnetic field splits into exactly two beams, demonstrating the quantization of angular momentum and the existence of spin.
Pauli exclusion principle. No two identical fermions (particles with half-integer spin, including electrons, protons, neutrons) can occupy the same quantum state simultaneously. This principle explains:
Worked example. How many electrons can occupy the n = 3 shell of an atom?
Solution. For n = 3: l = 0, 1, 2. For each l, m_l ranges from -l to +l, giving (2l+1) values. Each (n, l, m_l) state holds 2 electrons (spin up and down). Total: 2(1 + 3 + 5) = 2(9) = 18 electrons.
The phenomenon. A quantum particle can penetrate a potential energy barrier even when its kinetic energy is less than the barrier height. Classically, this is forbidden — a ball rolling toward a hill with insufficient energy always rolls back. Quantum mechanically, the wave function decays exponentially inside the barrier but does not reach zero, so there is a nonzero probability of appearing on the other side.
Transmission coefficient (rectangular barrier): T ~ exp(-2 kappa L), where kappa = sqrt(2m(V_0 - E)) / hbar, L is the barrier width, and V_0 - E is the energy deficit.
Worked example. An electron (E = 5 eV) encounters a barrier of height V_0 = 10 eV and width L = 0.5 nm. Estimate the tunneling probability.
Solution. kappa = sqrt(2 * 9.11e-31 * 5 * 1.6e-19) / 1.055e-34 = sqrt(1.457e-48) / 1.055e-34 = 1.207e-24 / 1.055e-34 = 1.144e10 m^-1.
2 kappa L = 2 * 1.144e10 * 5e-10 = 11.44.
T ~ exp(-11.44) = 1.07 * 10^-5, or about 1 in 100,000.
Applications. Tunneling is not exotic — it is essential to:
The path integral formulation (Feynman, 1948). Instead of solving the Schrodinger equation directly, compute the propagator K(b, a) as a sum over all possible paths from point a to point b:
K(b, a) = sum over all paths of exp(i S[path] / hbar)
where S is the classical action (integral of the Lagrangian along the path).
Key insights:
Why it matters. The path integral formulation is mathematically equivalent to the Schrodinger equation but provides deeper physical intuition and extends naturally to quantum field theory and QED. Feynman diagrams arise from this framework.
Entanglement. Two particles are entangled when their quantum state cannot be written as a product of individual states. Measuring one particle instantaneously determines the state of the other, regardless of distance.
The EPR argument (Einstein, Podolsky, Rosen, 1935). If quantum mechanics is complete, then measuring one particle instantaneously affects a distant particle (non-locality). Einstein called this "spooky action at a distance" and argued that quantum mechanics must be incomplete — there must be hidden variables determining the outcomes in advance.
Bell's theorem (1964). John Bell derived an inequality that any local hidden variable theory must satisfy. Quantum mechanics predicts violations of this inequality.
Experimental verdict (Aspect, 1982; and many since). Bell's inequality is violated — nature is non-local in the quantum sense. No local hidden variable theory can reproduce quantum mechanics.
No-signaling theorem. Despite non-local correlations, entanglement cannot be used to send information faster than light. The correlations are only visible when you compare measurements from both particles, which requires classical communication.
The problem. The Schrodinger equation is deterministic and linear — superpositions evolve into superpositions. Yet measurement always yields a definite outcome. How does a deterministic wave equation produce probabilistic results? This is the measurement problem.
Major interpretations:
Current status. All interpretations make identical experimental predictions for all known experiments. The measurement problem remains open.
Kets and bras. A quantum state is written as |psi> (a "ket"). The dual vector is <psi| (a "bra"). The inner product is <phi|psi> (a "bracket").
Observables as operators. An observable A acts on kets: A|psi> = a|psi> means |psi> is an eigenstate of A with eigenvalue a.
Completeness: sum over n of |n><n| = I (the identity), where {|n>} is a complete set of eigenstates. This allows expanding any state as |psi> = sum of c_n |n>, where c_n = <n|psi>.
Why Dirac notation matters. It abstracts away the specific representation (position, momentum, energy) and makes the linear-algebraic structure of quantum mechanics transparent. It is the standard language of graduate-level and research quantum mechanics.
| Mistake | Why it fails | Fix |
|---|---|---|
| Treating wave functions as physical waves | psi is a probability amplitude, not a physical displacement | Always interpret |
| Confusing uncertainty with measurement error | The uncertainty principle is about the state, not the instrument | Uncertainty is intrinsic to quantum states |
| Adding probabilities instead of amplitudes | Quantum interference requires adding amplitudes first, then squaring | P = |
| Assuming measurement reveals a pre-existing value | Measurement produces outcomes; the value did not necessarily exist before | Use the Born rule and state preparation language |
| Using non-relativistic QM for high-energy particles | Schrodinger equation fails when v approaches c | Use Dirac equation or QFT for relativistic particles |
| Ignoring normalization | Unnormalized wave functions give meaningless probabilities | Always verify integral of |