ISEGORIA / MATH ENCYCLOPEDIA
Quantum mechanics: amplitudes and spin
Wavefunctions, uncertainty, and a rotatable Bloch sphere.
Before you begin: Complex numbers and waves
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Wavefunction density
Here a particle is trapped between two infinitely high walls. Choose n to see the eigenstate ψₙ on the energy ladder and its probability density, with n − 1 nodes where the particle is never found. Switch to a superposition of two neighbouring levels and play time: each component turns at its own frequency Eₖ/ħ, so the density sloshes from wall to wall. Drag the two orange handles to find the probability of the particle lying between them.
Worked example. For n = 2 the density vanishes at the centre, and the probability of finding the particle in the middle fifth, 0.4L < x < 0.6L, is only about 0.049, although a classical particle would be there a fifth of the time.
Watch out. Measurement and its collapse of the state are not shown. The walls are infinitely high, so ψ vanishes exactly at x = 0 and x = L.
Where is the particle least likely, and does that change in time for an eigenstate?
At the nodes, where |ψₙ|² = 0. For an eigenstate only the overall phase turns, so the density and its nodes never move; only a superposition makes the density change.
2. Uncertainty principle
Every Gaussian wave packet has a position density and a momentum density, and narrowing one widens the other. Drag the width Δx and watch Δp = ħ/(2Δx) respond; the point in the lower plane stays on the curve Δx·Δp = ħ/2, the smallest product quantum mechanics allows. Press Play to let the packet move freely: it spreads in x while its momentum density is unchanged, so the product climbs above the bound.
Worked example. With ħ = 1 and Δx = 0.7, the momentum spread is Δp = 1/(2 × 0.7) ≈ 0.714, and the product is exactly 0.5.
Watch out. The widths are standard deviations. Only a Gaussian without a position-dependent phase reaches the bound; every other state, including this packet after it has spread, gives a larger product.
Why does a narrow packet spread faster?
Its momentum spread Δp = ħ/(2Δx₀) is large, so its components run apart at speeds differing by about Δp/m; the spreading time is 2mΔx₀²/ħ.
3. Bloch sphere
Every pure state of a qubit is a point on a sphere. Drag the gold tip, or set the polar angle θ and the azimuth φ; drag anywhere else to turn the sphere. On the right, the amplitudes α and β are drawn as arrows in the complex plane, and the bars give the probability of each outcome when the qubit is measured along z, x or y.
Worked example. At θ = 90°, φ = 0 the state is |+⟩ = (|0⟩ + |1⟩)/√2: a z measurement gives 0 or 1 with probability ½ each, while an x measurement gives + with certainty.
Watch out. The sphere is a picture of state space, not of a particle moving in real space. Opposite points on it are orthogonal states, so the angle between states on the sphere is twice the angle between their vectors.
What does azimuth encode?
The relative phase between the basis amplitudes.