---
title: Wave Packets and the Probabilistic Wave Function
module: The Wave Nature of Matter
moduleNumber: 2
lessonNumber: 2
order: 202
summary: >
  A single de Broglie wave fills all space, but a particle is localized. Adding
  many waves of nearby wavelength builds a wave packet that is confined and moves
  at the group velocity, which equals the particle velocity. Born's rule reads the
  squared amplitude of the wave function as a probability density, the meaning
  confirmed by electron interference building up one detection at a time.
topics: [The Wave Nature of Matter]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 5 — The Wavelike Properties of Particles; §5-3 Wave Packets"
  - book: Tipler & Llewellyn
    ref: "§5-4 The Probabilistic Interpretation of the Wave Function"
draft: false
---

The [de Broglie relation](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction)
$\lambda = h/p$ 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

$$
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2},
$$

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

$$
y(x,t) = y_0 \cos(kx - \omega t),
\qquad
\omega = 2\pi f = \frac{2\pi}{T}, \quad k = \frac{2\pi}{\lambda}.
$$

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**

$$
v_p = f\lambda = \frac{\omega}{k}.
$$

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**.

> **Definition (Wave packet).** A superposition of harmonic waves with a range of
> wave numbers that add constructively over a small region of space and cancel
> elsewhere, producing a localized group. The narrower the region, the wider the
> range of wave numbers required.

## 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

$$
y = y_0\cos(k_1 x - \omega_1 t) + y_0\cos(k_2 x - \omega_2 t)
= 2 y_0 \cos\!\left(\tfrac{\Delta k}{2}x - \tfrac{\Delta\omega}{2}t\right)\cos(\bar{k}x - \bar\omega t),
$$

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
$\cos(\tfrac{1}{2}\Delta k\, x - \tfrac{1}{2}\Delta\omega\, t)$. This is the beat
phenomenon.

$$
% caption: Two waves of nearly equal wavelength beat together. The carrier (solid)
% is confined under a slow envelope (dashed) whose nodes mark destructive overlap.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.6,0) node[right, font=\scriptsize] {x};
% carrier under envelope
\draw[acc, very thick] plot[domain=0:6.2, samples=260]
  (\x, {1.4*cos(45*\x)*cos(340*\x)});
% envelope (dashed)
\draw[black, dashed] plot[domain=0:6.2, samples=120] (\x, {1.4*cos(45*\x)});
\draw[black, dashed] plot[domain=0:6.2, samples=120] (\x, {-1.4*cos(45*\x)});
\node[black, font=\scriptsize, anchor=south] at (5.4,1.05) {envelope};
\node[acc, font=\scriptsize, anchor=north] at (1.0,-0.15) {carrier};
\end{tikzpicture}
$$

The two speeds differ. Each carrier crest moves at the phase velocity
$v_p = \bar\omega/\bar k$. The envelope, written as
$\cos\!\big[\tfrac12\Delta k(x - \tfrac{\Delta\omega}{\Delta k}t)\big]$, moves at the
**group velocity**

$$
v_g = \frac{\Delta\omega}{\Delta k} \;\longrightarrow\; \frac{\d\omega}{\d k}
\qquad (\text{in the limit of many nearby waves}).
$$

Because $\omega = k v_p$, differentiating gives the general relation between the two:

$$
v_g = \frac{\d\omega}{\d k} = v_p + k\frac{\d v_p}{\d k}.
$$

If the phase velocity is the same for every wavelength, $\d v_p/\d k = 0$ 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**, $v_g \neq v_p$, 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.

$$
% caption: A continuous band of wave numbers sums to one localized packet: a
% carrier wave confined under a smooth envelope that vanishes away from the center.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.6,0) node[right, font=\scriptsize] {x};
\draw[acc, very thick] plot[domain=0:6.2, samples=280]
  (\x, {1.7*exp(-0.9*(\x-3.1)^2)*cos(430*\x)});
\draw[black, dashed] plot[domain=0:6.2, samples=120] (\x, {1.7*exp(-0.9*(\x-3.1)^2)});
\draw[black, dashed] plot[domain=0:6.2, samples=120] (\x, {-1.7*exp(-0.9*(\x-3.1)^2)});
\node[black, font=\scriptsize, anchor=south west] at (4.2,0.9) {envelope};
\draw[black, <->] (2.15,-1.05) -- (4.05,-1.05);
\node[font=\scriptsize, anchor=north] at (3.1,-1.05) {packet width};
\end{tikzpicture}
$$

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:

$$
\Delta k \,\Delta x \sim 1, \qquad \Delta\omega\,\Delta t \sim 1.
$$

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
$\Delta\omega \sim 1/\Delta t$.

$$
% caption: A band of wave-number amplitudes (right) of width delta-k produces a
% packet (left) of width delta-x, with the product fixed near one.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left: packet envelope
\draw[->, black] (0,0) -- (3.6,0) node[right, font=\scriptsize] {x};
\draw[acc, very thick] plot[domain=0:3.4, samples=120] (\x, {2.0*exp(-2.2*(\x-1.6)^2)});
\draw[black, <->] (1.1,-0.55) -- (2.1,-0.55);
\node[font=\scriptsize, anchor=north] at (1.6,-0.55) {width x};
% right: spectrum
\begin{scope}[xshift=5.0cm]
\draw[->, black] (0,0) -- (3.6,0) node[right, font=\scriptsize] {k};
\draw[acc, very thick] plot[domain=0:3.4, samples=120] (\x, {2.0*exp(-3.2*(\x-1.6)^2)});
\draw[black, <->] (1.25,-0.55) -- (1.95,-0.55);
\node[font=\scriptsize, anchor=north] at (1.6,-0.55) {width k};
\end{scope}
\end{tikzpicture}
$$

## 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
$\omega = E/\hbar$ and $k = p/\hbar$, with the nonrelativistic free-particle energy
$E = p^2/2m$:

$$
v_p = \frac{\omega}{k} = \frac{E}{p} = \frac{p^2/2m}{p} = \frac{p}{2m} = \frac{v}{2}.
$$

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 $E = \hbar\omega$ and $p = \hbar k$, the group velocity is

$$
v_g = \frac{\d\omega}{\d k} = \frac{\d E}{\d p} = \frac{\d}{\d p}\!\left(\frac{p^2}{2m}\right) = \frac{p}{m} = v.
$$

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.

> **Theorem (Group velocity equals particle velocity).** For a free particle whose
> de Broglie waves satisfy $\omega = E/\hbar$ and $k = p/\hbar$, the group velocity
> $v_g = \d\omega/\d k = \d E/\d p$ equals the classical particle velocity $v$, while the
> phase velocity $v_p = E/p$ does not. A localized packet therefore tracks the
> particle.

$$
% caption: Between two instants the envelope advances at the group velocity v_g
% (equal to the particle speed) while an individual carrier crest advances less.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% earlier packet
\draw[acc, very thick] plot[domain=0:3.6, samples=200]
  (\x, {1.5*exp(-1.4*(\x-1.5)^2)*cos(430*\x)});
\draw[black, dashed] plot[domain=0:3.6, samples=100] (\x, {1.5*exp(-1.4*(\x-1.5)^2)});
\node[acc, font=\scriptsize, anchor=south] at (1.5,1.55) {time t};
% later packet (shifted right by group velocity)
\begin{scope}[yshift=-3.3cm]
\draw[acc, very thick] plot[domain=1.5:5.2, samples=200]
  (\x, {1.5*exp(-1.4*(\x-3.3)^2)*cos(430*\x)});
\draw[black, dashed] plot[domain=1.5:5.2, samples=100] (\x, {1.5*exp(-1.4*(\x-3.3)^2)});
\node[acc, font=\scriptsize, anchor=south] at (3.3,1.55) {later};
\end{scope}
% displacement arrows
\draw[->, black!70] (1.5,-1.7) -- (3.3,-1.7);
\node[font=\scriptsize, anchor=south] at (2.4,-1.7) {envelope moves at group speed};
\end{tikzpicture}
$$

## 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.[^born]

> **Definition (Born rule).** For a particle described by wave function
> $\Psi(x,t)$, the probability of finding it in the interval $\d x$ about $x$ is
> $$
> P(x)\,\d x = |\Psi(x,t)|^2\,\d x, \qquad |\Psi|^2 \equiv \Psi^\ast\Psi,
> $$
> where $\Psi^\ast$ is the complex conjugate. The density $|\Psi|^2$ is real and
> non-negative even though $\Psi$ itself is generally complex.

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.

$$
% caption: The probability density is the squared envelope of the packet. The
% classical particle sits at the peak; the packet spreads over a range of x.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.6,0) node[right, font=\scriptsize] {x};
\draw[->, black] (0,0) -- (0,2.9) node[above, font=\scriptsize] {probability density};
\draw[acc, very thick] plot[domain=0.2:6.2, samples=160]
  (\x, {2.4*exp(-1.1*(\x-3.2)^2)});
\fill[acc] (3.2,0) circle (2.2pt);
\node[acc, font=\scriptsize, anchor=north] at (3.2,-0.05) {likely position};
\node[acc, font=\scriptsize, anchor=west] at (3.4,2.1) {squared amplitude};
\end{tikzpicture}
$$

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 $|\Psi|^2$.

$$
% caption: Electron two-slit detections accumulate. Each hit is a point; after
% many, the count follows the wave interference pattern set by the squared amplitude.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% few dots
\draw[black] (0,0) rectangle (2.0,3.0);
\node[font=\scriptsize, anchor=south] at (1.0,3.05) {few};
\foreach \p in {(0.5,0.7),(1.4,2.1),(0.8,1.4),(1.6,0.5),(0.4,2.4),(1.1,1.0)}
  \fill[black!70] \p circle (1.3pt);
% many dots forming bands
\begin{scope}[xshift=4.0cm]
\draw[black] (0,0) rectangle (2.0,3.0);
\node[font=\scriptsize, anchor=south] at (1.0,3.05) {many};
\foreach \yb in {0.35,1.05,1.75,2.45}{
  \foreach \x in {0.25,0.5,0.75,1.0,1.25,1.5,1.75}{
    \fill[acc] (\x, {\yb + 0.09*sin(1400*\x)}) circle (1.1pt);
    \fill[acc] (\x, {\yb - 0.05 + 0.07*sin(900*\x+40)}) circle (1.1pt);
  }
}
\end{scope}
% intensity curve at right
\begin{scope}[xshift=7.2cm]
\draw[->, black] (0,0) -- (0,3.0);
\draw[acc, very thick] plot[domain=0:3.0, samples=120]
  ({0.9*(0.5+0.5*cos(240*\x))*exp(-0.15*(\x-1.5)^2)}, \x);
\node[acc, font=\scriptsize, anchor=west] at (0.2,2.7) {count};
\end{scope}
\end{tikzpicture}
$$

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](/quantum-mechanics/matter-waves/the-uncertainty-principle) 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](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
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.

[^born]: Tipler & Llewellyn, §5-4 — the probabilistic interpretation: $|\Psi|^2 = \Psi^\ast\Psi$ as the probability density $P(x)\,\d x = |\Psi|^2\,\d x$, argued by analogy with the photon number density proportional to $\mathcal{E}^2$ and confirmed by low-intensity two-slit interference building up from individual detections.