📘 What does quantum theory actually claim?

What you’ll learn

1. Planck's Quantization of EnergyExplain the ultraviolet catastrophe and Planck's resolution via energy quanta.Planck's constant emerged from fitting the Rayleigh-Jeans law to experimental curves at short wavelengths. The resulting E = nhf introduced the first quantum postulate. This discrete energy step set the scale for all subsequent quantum theory.
2. Photoelectric Effect and Photon MomentumDerive photon energy and momentum from experimental thresholds and apply conservation laws.The linear relation between maximum kinetic energy and frequency directly yields h. Momentum transfer p = h/λ follows from relativistic four-vector consistency. These results established the particle nature of light.
3. Matter Waves and de Broglie RelationsConnect de Broglie wavelength to momentum and apply it to diffraction and interference conditions.The wavelength-momentum relation extends wave-particle duality to massive particles. Electron diffraction confirms the relation to within experimental precision. This duality forces a revision of classical trajectories.
4. The Schrdinger EquationState the time-dependent Schrdinger equation and identify the role of the Hamiltonian operator.The equation iħ /t = Ĥ encodes unitary time evolution for closed systems. The Hamiltonian incorporates kinetic and potential energy as differential operators. Solutions must remain square-integrable and single-valued.
5. Stationary States and Eigenvalue ProblemsSolve the eigenvalue equation for bound states and interpret the resulting discrete spectrum.Stationary states carry definite energy and evolve only by a global phase. Boundary conditions quantize allowed energies. Superpositions of these states produce time-dependent interference.
6. Particle in a Finite Well and TunnelingCalculate transmission and reflection coefficients for piecewise-constant potentials.Exponential decay inside classically forbidden regions yields nonzero tunneling probability. Matching and ' at boundaries produces the exact transmission formula. Applications include alpha decay and scanning tunneling microscopy.
7. Quantum Harmonic OscillatorDerive energy eigenvalues and Hermite-polynomial wave functions for the oscillator.Creation and annihilation operators factor the Hamiltonian and generate the ladder of states. Zero-point energy follows directly from the commutation relation [x,p] = iħ. This model underpins vibrational modes in molecules and fields.
8. Angular Momentum AlgebraApply raising and lowering operators to obtain the spectrum of angular momentum.Eigenvalues of L^2 are l(l+1)ħ^2 with l integer or half-integer. Selection rules for matrix elements follow from the algebra. These rules govern atomic transitions and rotational spectra.
9. Hydrogen Atom Wave FunctionsObtain radial and angular solutions and compute expectation values for the hydrogen atom.Associated Laguerre polynomials and spherical harmonics factor the wave function. Fine-structure corrections require relativistic and spin-orbit terms. Expectation values of r scale as n^2 a_0 for large n.
10. Spin, identical particles, and symmetryConstruct spin-1/2 operators from Pauli matrices and compute spin precession in magnetic fields. Apply symmetrization and antisymmetrization to construct allowed states for fermions and bosons.The Pauli matrices satisfy the su(2) algebra and yield eigenvalues ±ħ/2 along any axis. The gyromagnetic ratio g 2 follows from the Dirac equation. Spin-1/2 states transform under 2 rotations by a minus sign. The Pauli exclusion principle follows from antisymmetric wave functions for electrons. Slater determinants enforce antisymmetry for many-electron atoms. Bosonic condensation arises from symm
11. Entanglement and Bell InequalitiesDerive the CHSH bound and show how quantum mechanics exceeds it for entangled states.Maximally entangled states produce correlation functions 22 cos(θ). Local hidden-variable models are limited to 2. Experimental violation rules out local realism under fair-sampling assumptions.
12. Density Operators and DecoherenceTrace over environmental degrees of freedom to obtain the decoherence superoperator.Off-diagonal elements decay on timescales set by the system-bath coupling strength. Pointer states are selected by the interaction Hamiltonian. This process accounts for the emergence of classical probabilities.
13. Measurement and the Born RuleState the Born rule and discuss its status within the measurement problem.The rule assigns probabilities directly from the squared modulus of projection amplitudes. Different interpretations (Copenhagen, many-worlds, QBism) agree on the rule yet differ on ontology. No derivation from unitary dynamics alone has achieved consensus.

Questions this course answers

Grounded in trusted sources

• OpenStax University Physics
• National Institute of Standards and Technology
• CERN
• David J. Griffiths, Introduction to Quantum Mechanics — wells, oscillator, hydrogen, spin
• J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics — angular momentum and identical particles
• R. Shankar, Principles of Quantum Mechanics — measurement and formalism
• MIT OCW 8.04 / 8.05 quantum courses — worked intuition, https://ocw.mit.edu/

Every Wunder lesson is built from real, reputable sources — never invented.

Related courses