State
The wavefunction ψ encodes probability amplitudes. Different measurements can use different representations of the same state.
01 • QUANTUM MECHANICS
Not trajectories, but states.
In quantum theory, a system is described by a state that predicts probabilities for measurement outcomes.
iℏ ∂ψ/∂t = Ĥψ
The Schrödinger equation: the quantum analogue of a law of motion that governs the evolution of the wavefunction.
Observables
Physical quantities are represented by operators. Possible measurement results are tied to their eigenvalues.
Measurement
Outcome probabilities are given by the Born rule, while the specific experimental outcome is random.
UNCERTAINTY PRINCIPLE
Δx · Δp ≥ ℏ/2
You cannot make the position and momentum distributions arbitrarily narrow at the same time. This is not an instrument flaw, but a feature of quantum description.
more precise x
more precise p
FROM MODEL TO TECHNOLOGY
Where does quantum theory live?
Lasers
Controlled transitions between energy levels create coherent light.
Transistors
Quantum properties of band structure underpin modern microelectronics.
Quantum sensors
Ultra-sensitive measurements use coherence, spins and interference.
INTERACTIVE TOOL
Uncertainty calculator
For position uncertainty Δx, the uncertainty principle gives a lower bound for Δp: Δp ≥ ℏ/(2Δx).
Δp ≥ 5.273e-25 kg·m/s
Uses ℏ = 1.054571817 × 10⁻³⁴ J·s. This is an ideal lower bound, not a measurement prediction.
Seeing a quantum state
NIST/JILA uses imaging techniques to study atoms prepared in quantum superposition and their measurement.
Marti / JILASource ↗