Book Notes / Quantum Mechanics: Concepts and Applications

Nouredine Zettili / Reading notes

A probability distribution leaves out phase

The probability density is the squared magnitude of the wavefunction. Taking that magnitude discards phase information that can affect a later prediction. Multiplying a packet by a constant phase leaves its predictions unchanged; giving it a position-dependent phase can change its momentum distribution while leaving its initial position density unchanged.
A Gaussian envelope makes the distinction concrete. Multiplying it by the carrier wave exp(ik₀x) shifts its mean momentum to ℏk₀. Two such packets can begin with identical position distributions but move in different directions. The initial density alone does not specify how either one will evolve.
Interference is another test of what was discarded. For two coherently prepared alternatives, the probability density comes from |ψ₁ + ψ₂|², which includes 2 Re(ψ₁ψ₂*). Add the separate densities instead and that term disappears. Changing relative phase can change the observed pattern without changing either alternative's individual intensity.

An ideal plane wave has one wavenumber and a spatially uniform magnitude. Localizing a particle requires superposing wavenumbers so that their contributions reinforce in a region and cancel elsewhere. The Fourier transform connects that spatial envelope to its momentum amplitudes through p = ℏk.
For a minimum-uncertainty Gaussian whose position standard deviation is σ₀, the momentum standard deviation is ℏ/(2σ₀). Halving the position spread doubles the momentum spread. These spreads describe the same preparation; they are not corrections for an imprecise instrument.
For a free particle with zero mean momentum, the spread contributes a mean kinetic energy of ℏ²/(8mσ₀²). Halving σ₀ therefore multiplies that energy by four. A narrow spatial envelope imposes a specific momentum and energy requirement before any measurement is performed.

Each momentum component of a free nonrelativistic particle acquires the phase set by its energy, p²/(2m). Because this dependence is quadratic, the relative phases change in a way that reshapes the packet. Its center moves with velocity p₀/m; the envelope can spread while that center follows uniform motion.
For the initially minimum-uncertainty Gaussian above, with no initial position–momentum covariance, the exact width is σₓ(t) = σ₀ √[1 + (t/τ)²], where τ = 2mσ₀²/ℏ. At t = τ, its width is √2 times its initial width. The growth is not linear in time; linear growth is the long-time limit.
Preparing the packet twice as narrowly makes τ four times shorter. At late times, its width is approximately ℏ|t|/(2mσ₀), so the initially narrower packet becomes twice as broad. Better localization at the starting time does not imply better localization later. The momentum spread needed to produce it drives the subsequent dispersion.
The assumptions do real work. An initial position-dependent quadratic phase introduces position–momentum correlations and changes the width evolution; such a Gaussian can initially contract. A confining potential changes the dynamics again, and a bound energy eigenstate can have a stationary density. The free, initially uncorrelated Gaussian calculation does not establish that every localized state continually spreads.