Wave, particle, and the world seen through probability

Dots that arrive one at a time still build up an interference pattern; observe the path and the fringes vanish into particles. de Broglie wavelength λ = h/p.

The wavefunction ψ itself is unseen, but its square |ψ|² is the probability density of where you find the particle. The total area normalizes to 1.

A wave trapped in a box allows only standing waves that vanish at both ends; only integer half-wavelengths fit, so energy is discrete, Eₙ = n²π²ℏ²/2mL².

Squeeze position and momentum spreads; a narrow wave packet is a sum of many wavenumbers, so the product cannot fall below Δx·Δp ≥ ℏ/2.

Even inside a wall it classically cannot cross, the wavefunction is not zero but decays exponentially, so it leaks past with a transmission probability T.

Before measurement, states coexist as α|↑⟩ + β|↓⟩; measuring collapses it to one outcome, with probabilities |α|² and |β|².

An orbital is a probability cloud, with ψ split into a radial part R(r) and an angular part Y(θ,φ); only n, l, mₗ set the shape. Here we view it as a 2D cross-section.

Trap a particle in a soft parabolic valley instead of a hard box and the energy ladder becomes evenly spaced by ℏω, not crowded like n². The ground state is not zero. Eₙ=(n+½)ℏω.

When the walls are finite rather than infinitely high, only a few bound levels fit and the wavefunction leaks exponentially into the walls. A deeper well holds more levels.

A single plane wave moves at the phase velocity, but a real particle is a packet of many wavenumbers that travels at the group velocity v_g=ℏk/m and spreads. For matter waves v_g=2v_p.

A single energy eigenstate only turns its phase e^(−iEt/ℏ) so |ψ|² stays still, but mix two energies and the probability sloshes at the beat frequency ω=(E₂−E₁)/ℏ.

Observables like position, momentum, and energy are each an operator. A measurement returns one eigenvalue, and the average over many measurements is the expectation value ⟨A⟩=∫ψ*Âψ dx.

Because an atom's energy levels are discrete, an electron jumping between them emits exactly one color of photon, E=hf=Eᵢ−E_f. So each element shows its own line spectrum.

Send a beam of neutral atoms through a non-uniform magnetic field and, against classical expectation, it splits into exactly two spots, not a continuous band. Spin is quantized as S_z=±½ℏ.