Lecture 12 — Measurement, probability, and barriers
Graduate Physics Studio · Spring 2026
Today's Topics
01 Wave-Particle Duality
02 The Schrödinger Equation
03 Probability Densities
04 The Uncertainty Principle
05 Quantum Tunneling
Wave-Particle Duality
Louis de Broglie proposed that all matter exhibits both wave-like and particle-like properties. The wavelength associated with a particle depends on its momentum.
Energy-frequency relation
de Broglie wavelength
Wave → Particle
(click to morph)
The Schrödinger Equation
Click → to expand the derivation
The Hamiltonian operator Ĥ encodes the total energy of the system — kinetic plus potential.
Ψ(’r, t) — wave function
encodes all quantum state information
V(’r) — potential energy field
determines bound state solutions
Probability Densities
Particle in a box — quantum number n = 1, 2
Gaussian wave packet — minimum uncertainty state
Atomic Orbital Probability
Probability amplitude |ψ(’r)|² for a hydrogen-like orbital — surface height encodes probability density
The Uncertainty Principle
We cannot simultaneously know both the exact position and momentum of a particle — this is not a limitation of our instruments, but a fundamental property of nature.
Robertson generalization — any two observables
Quantum Tunneling
V₀ Barrier
Particle wave packet
(click to tunnel through)
Transmission probability decays exponentially with barrier width L and the evanescent decay constant κ
Region I: Propagating
ψ = Ae^{ikx} + Be^{-ikx}
Region III: Transmitted
ψ = Fe^{ikx}
Key Takeaways
01
Wave functions encode all measurable information about a quantum system
02
Schrödinger equation governs the time evolution of quantum states
03
Measurement collapses the wave function — Born rule gives the probability
04
Uncertainty is fundamental to nature — not a limitation of instruments
Questions?
PHYS 7210 — Bring questions to studio discussion
Next lecture: Entanglement, Bell inequalities, and nonlocal correlations
Instructor · Graduate Physics Studio · Spring 2026