Wave Packets and the Probabilistic Wave Function

The de Broglie relation$\lambda =h/p$ assigns a wave to a particle, but a wave of one definite wavelength is a problem. A pure harmonic wave $\cos (kx-\omega t)$ extends from $-\infty$ to $+\infty$ with the same amplitude everywhere, so it locates the particle nowhere. A particle is found at one place. Reconciling the two requires superposing many harmonic waves into a localized group, and then deciding what the amplitude of that group physically means.

Harmonic waves and phase velocity

Classical waves obey the wave equation

whose basic solution is the harmonic wave of amplitude $y_0$ traveling in the $+x$ direction,

Here $\omega$ is the angular frequency and $k$ the wave number. A point of fixed phase $kx-\omega t=\text{const}$ moves at the phase velocity

This single wave carries no information about where anything is. A localized disturbance — the flip of a rope, a brief pulse of light through a shutter — cannot be a single harmonic; it is a superposition of harmonic waves of different wavelengths, and the superposition is called a wave packet.

Two waves and the group velocity

The simplest packet superposes two waves of equal amplitude and nearly equal wave numbers $k_1,k_2$ and frequencies ${\omega }_1,{\omega }_2$. Their sum, by the sum-to-product identity, is

with $\Delta k=k_2-k_1$, $\Delta \omega ={\omega }_2-{\omega }_1$, and mean values $\bar{k}=(k_1+k_2)/2$, $\bar{\omega }=({\omega }_1+{\omega }_2)/2$. The result is a fast carrier wave $\cos (\bar{k}x-\bar{\omega }t)$ modulated by a slow envelope $\cos (\frac{1}{2}\Delta kx-\frac{1}{2}\Delta \omega t)$. This is the beat phenomenon.

The two speeds differ. Each carrier crest moves at the phase velocity $v_p=\bar{\omega }/\bar{k}$. The envelope, written as $\cos [\frac{1}{2}\Delta k(x-\frac{\Delta \omega }{\Delta k}t)]$, moves at the group velocity

Because $\omega =k{v}_p$, differentiating gives the general relation between the two:

If the phase velocity is the same for every wavelength, $\mathbf{d}{v}_p/\mathbf{d}k=0$ and $v_g=v_p$: the packet holds its shape as it moves. Such a medium is nondispersive (waves on an ideal string, sound in air, light in vacuum). If $v_p$ depends on wavelength the medium is dispersive, $v_g\ne v_p$, and the packet spreads as it travels. Matter waves in free space are dispersive.

Building a localized packet

Superposing a handful of waves with wave numbers in a band $\Delta k$ produces a central group plus repeats where all the components happen to realign. Superposing a continuous range instead removes the repeats — no finite length can hold a whole number of every wavelength at once — leaving a single localized packet.

The tradeoff between the packet's spatial width $\Delta x$ and the band of wave numbers $\Delta k$ needed to build it is a general property of Fourier superposition. A narrow packet demands a broad band, and a narrow band produces a broad packet:

These are the classical uncertainty relations, order-of-magnitude statements whose exact constants depend on how the widths are defined and on the shape of the packet. The second is the response-time–bandwidth relation of signal processing: an amplifier that must respond to a pulse of duration $\Delta t$ needs bandwidth $\Delta \omega \sim 1/\Delta t$.

The matter wave packet moves with the particle

For matter the quantity that plays the role of displacement is the wave function$\Psi (x,t)$, and a free particle of definite momentum can be written as $\Psi =A\cos (kx-\omega t)$, $A\sin (kx-\omega t)$, or $A{e}^{i(kx-\omega t)}$. Take a single such wave and compute its phase velocity using the de Broglie relations $\omega =E/\hslash$ and $k=p/\hslash$, with the nonrelativistic free-particle energy $E=p^2/2m$:

The phase velocity of a single matter wave is half the particle's velocity — it does not equal the particle speed, and a single wave is not localized anyway. Both defects vanish for a packet. Using $E=\hslash \omega$ and $p=\hslash k$, the group velocity is

The packet travels at exactly the particle's velocity. De Broglie chose the relations $E=hf$ and $p=h/\lambda$ in part because they force this agreement; the relativistic energy-momentum relation gives the same result.

The probabilistic interpretation

What waves in a matter wave is neither a medium nor a field but a probability. The reading comes from light. The energy density of a light wave is proportional to ${\mathcal{E}}^2$, yet light energy arrives in photons of energy $hf$; the number of photons per unit volume is therefore proportional to ${\mathcal{E}}^2$. At low intensity ${\mathcal{E}}^2$ is the probability of detecting a photon in unit volume. Born carried the same reading to matter: the squared magnitude of $\Psi$ is a probability density.¹

The wave function is in general complex, with a real and an imaginary part, so $\Psi$ has no direct physical meaning on its own. Only $|\Psi {|}^2$, which is always real, is measured. The particle is most likely to be found where $|\Psi {|}^2$ is large and never where $|\Psi {|}^2$ vanishes.

The interpretation is not a statement about our ignorance of a definite but unknown position. It is confirmed directly by interference. Send light — or electrons — through a double slit at intensity so low that one particle crosses the apparatus at a time. Each particle registers as a single dot on the detector, a particle-like event at a definite point. But the positions of successive dots are not reproducible; they scatter. As dots accumulate, they pile up into the interference fringes predicted by the wave theory, with no dots where the waves from the two slits cancel. The pattern is built from individual particle detections whose probabilities are set by $|\Psi {|}^2$.

The double slit fixes the vocabulary for everything that follows. Between preparation and detection the electron is a wave that interferes with itself; at detection it is a particle at a point, drawn from the distribution $|\Psi {|}^2$. The next lesson turns the packet width relations $\Delta k\Delta x\sim 1$ and $\Delta \omega \Delta t\sim 1$ into the Heisenberg uncertainty principle and traces its physical consequences.

The Schrödinger equation supplies the wave equation whose solutions are these functions $\Psi$; there the packet, the probability density, and the boundary conditions become a computational system rather than a qualitative picture.