---
title: The Uncertainty Principle and Wave-Particle Duality
module: The Wave Nature of Matter
moduleNumber: 2
lessonNumber: 3
order: 203
summary: >
  The packet relations delta-k delta-x about 1 become Heisenberg's principle once
  momentum is hbar times wave number: position and momentum cannot both be sharp,
  nor energy and time. The gamma-ray microscope shows the limit is physical, not
  technical. It fixes the zero-point energy of a confined particle, the size of the
  hydrogen atom, and the natural width of spectral lines, and it frames the
  wave-particle duality of all matter and radiation.
topics: [The Wave Nature of Matter]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 5; §5-5 The Uncertainty Principle, The Gamma-Ray Microscope"
  - book: Tipler & Llewellyn
    ref: "§5-6 Some Consequences of the Uncertainty Principle; §5-7 Wave-Particle Duality"
draft: false
---

The [wave packet](/quantum-mechanics/matter-waves/wave-packets-and-the-probability-interpretation) obeys two
purely mathematical relations between its spatial width and its band of wave
numbers, and between its duration and its band of frequencies:

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

These hold for any wave. What makes them physics is the de Broglie identification of
a particle's momentum and energy with the wave number and frequency of its matter
wave.

## From packet widths to Heisenberg's principle

Multiply each relation by $\hbar$ and use $p = \hbar k$ and $E = \hbar\omega$:

$$
\Delta x\,\Delta p \sim \hbar, \qquad \Delta E\,\Delta t \sim \hbar.
$$

A wave packet narrow in position ($\Delta x$ small) is built from a wide band of
wave numbers, so its momentum $p = \hbar k$ is spread over a wide range: making the
position sharp makes the momentum uncertain, and the reverse. Repeated position
measurements on identically prepared particles scatter with spread $\Delta x$;
repeated momentum measurements scatter with spread $\Delta p$; the two spreads
cannot both be driven to zero.

$$
% caption: A packet sharp in position (left) needs a broad band of wave numbers, so
% its momentum is broad; a packet of definite momentum (right) is spread in position.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% --- left: narrow x, wide p ---
\draw[->, black] (0,0) -- (3.4,0) node[right, font=\scriptsize] {x};
\draw[acc, very thick] plot[domain=0.1:3.3, samples=140] (\x, {2.1*exp(-6.0*(\x-1.6)^2)});
\draw[black, <->] (1.35,-0.5) -- (1.85,-0.5);
\node[font=\scriptsize, anchor=north] at (1.6,-0.5) {small x};
\begin{scope}[yshift=-2.9cm]
\draw[->, black] (0,0) -- (3.4,0) node[right, font=\scriptsize] {p};
\draw[acc, very thick] plot[domain=0.1:3.3, samples=140] (\x, {1.5*exp(-0.9*(\x-1.7)^2)});
\draw[black, <->] (0.75,-0.5) -- (2.65,-0.5);
\node[font=\scriptsize, anchor=north] at (1.7,-0.5) {large p};
\end{scope}
% --- right: wide x, narrow p ---
\begin{scope}[xshift=5.4cm]
\draw[->, black] (0,0) -- (3.4,0) node[right, font=\scriptsize] {x};
\draw[acc, very thick] plot[domain=0.1:3.3, samples=140] (\x, {1.35*exp(-0.8*(\x-1.7)^2)});
\draw[black, <->] (0.75,-0.5) -- (2.65,-0.5);
\node[font=\scriptsize, anchor=north] at (1.7,-0.5) {large x};
\begin{scope}[yshift=-2.9cm]
\draw[->, black] (0,0) -- (3.4,0) node[right, font=\scriptsize] {p};
\draw[acc, very thick] plot[domain=0.1:3.3, samples=140] (\x, {2.1*exp(-6.0*(\x-1.6)^2)});
\draw[black, <->] (1.35,-0.5) -- (1.85,-0.5);
\node[font=\scriptsize, anchor=north] at (1.6,-0.5) {small p};
\end{scope}
\end{scope}
\end{tikzpicture}
$$

The order-of-magnitude relations sharpen into inequalities once the uncertainties
are defined as standard deviations. For Gaussian distributions the product
$\Delta x\,\Delta k$ attains its minimum $1/2$, so with $p = \hbar k$,[^heis]

> **Theorem (Heisenberg uncertainty principle).** For any state, the standard
> deviations of conjugate quantities satisfy
> $$
> \Delta x\,\Delta p \ge \frac{1}{2}\hbar, \qquad \Delta E\,\Delta t \ge \frac{1}{2}\hbar.
> $$
> The lower bound is a property of the wave nature of matter, not a limit of the
> measuring apparatus. Poor measurements make the products larger, never smaller.

The bound is set by $\hbar = 1.055\times10^{-34}\,\text{J·s}$, so it constrains only
the atomic scale. For a macroscopic object the implied spreads are far below any
detectable value.

## The gamma-ray microscope

Heisenberg's thought experiment shows the limit is unavoidable in principle. To
measure an electron's position, scatter light from it and view it through a
microscope. Diffraction limits the resolution to about the wavelength divided by the
aperture half-angle,

$$
\Delta x \approx \frac{\lambda}{\sin\theta},
$$

so a sharp position calls for short wavelength — gamma rays. But a gamma-ray photon
carries momentum $p_\gamma = h/\lambda$, and scattering it off the electron is a
[Compton collision](/quantum-mechanics/old-quantum-theory/x-rays-and-the-compton-effect)
that recoils the electron. To register on the screen the scattered photon need only
pass somewhere through the lens, so its $x$-momentum is unknown by

$$
\Delta p_x \approx p_\gamma \sin\theta = \frac{h}{\lambda}\sin\theta.
$$

By momentum conservation the electron's recoil momentum is uncertain by at least
this much. The product is independent of $\lambda$ and $\theta$:

$$
\Delta x\,\Delta p_x \approx \frac{\lambda}{\sin\theta}\cdot\frac{h}{\lambda}\sin\theta = h.
$$

Shortening $\lambda$ sharpens the position but recoils the electron harder, spreading
its momentum by the same factor. The two effects trade exactly, and the product
stays of order $h$.

$$
% caption: Heisenberg's gamma-ray microscope. A short-wavelength photon locates the
% electron to within one wavelength but Compton-recoils it, spreading its momentum.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% screen
\draw[black, very thick] (-1.9,3.4) -- (1.9,3.4);
\node[font=\scriptsize, anchor=south] at (0,3.45) {screen};
% lens
\draw[acc, thick] (-1.5,2.2) arc (200:340:1.6 and 0.4);
\node[acc, font=\scriptsize, anchor=south] at (-1.7,2.15) {lens};
% electron
\fill[acc] (0,0) circle (2.4pt);
\node[font=\scriptsize, anchor=north] at (0,-0.1) {electron};
% incident photon
\draw[->, black!70, very thick] (-3.2,-1.0) -- (-0.25,-0.05);
\node[font=\scriptsize, anchor=east] at (-3.2,-1.0) {incident photon};
% scattered photon into lens
\draw[->, black!70, very thick] (0.2,0.1) -- (1.0,2.05);
\node[font=\scriptsize, anchor=west] at (0.7,1.2) {scattered};
% aperture half-angle rays
\draw[black, dashed] (0,0) -- (-1.4,2.15);
\draw[black, dashed] (0,0) -- (1.4,2.15);
\draw[black] (0,0.9) arc (90:57:0.9);
\node[font=\scriptsize] at (0.55,1.05) {angle};
\end{tikzpicture}
$$

## Zero-point energy of a confined particle

A particle held in a region of size $L$ has $\Delta x \lesssim L$, so its momentum
spread is at least $\hbar/L$. Take the standard deviation of momentum as a measure of
$p$. If the box is symmetric the mean momentum is zero, so

$$
(\Delta p)^2 = \langle p^2\rangle - \langle p\rangle^2 = \langle p^2\rangle \gtrsim \left(\frac{\hbar}{L}\right)^2,
$$

and the average kinetic energy cannot vanish:

$$
E = \frac{\langle p^2\rangle}{2m} \gtrsim \frac{\hbar^2}{2mL^2}.
$$

> **Definition (Zero-point energy).** The minimum kinetic energy
> $E_{\min} \sim \hbar^2/2mL^2$ that a particle confined to a region of size $L$
> must have. A confined quantum particle can never be fully at rest; localizing it
> forces a nonzero momentum spread.

$$
% caption: A particle confined between rigid walls of separation L cannot sit at
% the bottom; its lowest state has energy of order hbar-squared over 2 m L-squared.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% walls and floor
\draw[black, very thick] (0,3.0) -- (0,0) -- (4.0,0) -- (4.0,3.0);
\node[font=\scriptsize, anchor=north] at (2.0,-0.05) {width L};
\draw[black, <->] (0,-0.35) -- (4.0,-0.35);
% ground energy level
\draw[acc, dashed] (0,0.8) -- (4.0,0.8);
\node[acc, font=\scriptsize, anchor=west] at (4.1,0.8) {lowest energy};
% ground-state half sine
\draw[acc, very thick] plot[domain=0:4.0, samples=100] (\x, {0.8 + 1.3*sin(45*\x)});
\node[font=\scriptsize, anchor=south, text=black] at (2.0,2.05) {ground state};
\end{tikzpicture}
$$

> **Example (electron in an atomic box).** For $L = 0.1\,\text{nm}$, using
> $\hbar c = 197.3\,\text{eV·nm}$ and $mc^2 = 0.511\times10^6\,\text{eV}$,
> $$
> E \approx \frac{(\hbar c)^2}{2(mc^2)L^2} = \frac{(197.3\,\text{eV·nm})^2}{2(0.511\times10^6\,\text{eV})(0.1\,\text{nm})^2} = 3.81\,\text{eV}.
> $$
> This is the right order of magnitude for the kinetic energy of an electron bound
> in an atom. The same estimate for a macroscopic bead ($m = 10^{-9}\,\text{kg}$,
> $L = 10^{-6}\,\text{m}$) gives $E \sim 10^{-29}\,\text{eV}$, unobservable.

## The size of the hydrogen atom

The uncertainty principle predicts the size and binding energy of hydrogen without
solving any wave equation. An electron a distance $r$ from the proton has energy

$$
E = \frac{p^2}{2m} - \frac{ke^2}{r}.
$$

Confinement to within $r$ forces $\Delta x \sim r$, hence $p \sim \hbar/r$, so

$$
E(r) \approx \frac{\hbar^2}{2mr^2} - \frac{ke^2}{r}.
$$

Squeezing the electron inward raises the kinetic term as $1/r^2$ while lowering the
potential term as $1/r$; the balance sets a preferred radius. Setting $\d E/\d r = 0$,

$$
-\frac{\hbar^2}{mr^3} + \frac{ke^2}{r^2} = 0
\quad\Longrightarrow\quad
r_m = \frac{\hbar^2}{ke^2 m} = a_0 = 0.0529\,\text{nm},
$$

the Bohr radius. The corresponding energy is

$$
E_m = -\frac{k^2 e^4 m}{2\hbar^2} = -13.6\,\text{eV},
$$

the [ground-state energy of hydrogen](/quantum-mechanics/central-potentials/the-hydrogen-atom).
The atom is as small as it can be before the kinetic penalty of confinement
outweighs the electrostatic attraction. The exact numerical agreement is a
coincidence of the choice $\Delta x \sim r$, but any reasonable choice gives the
correct order of magnitude.

$$
% caption: Total energy versus radius for the electron in hydrogen. The rising
% confinement term and the falling attraction term meet at a minimum near the Bohr radius.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (0,-2.2) -- (0,2.6) node[above, font=\scriptsize] {energy};
\draw[->, black] (0,0) -- (5.8,0) node[right, font=\scriptsize] {radius};
% kinetic term ~ +a/r^2
\draw[black, thick, densely dashed] plot[domain=0.55:5.4, samples=120] (\x, {1.4/(\x*\x)});
\node[black, font=\scriptsize, anchor=west] at (1.05,1.6) {kinetic term};
% potential term ~ -b/r
\draw[black, thick] plot[domain=0.55:5.4, samples=120] (\x, {-3.0/\x});
\node[black, font=\scriptsize, anchor=west] at (3.6,-0.95) {attraction};
% total E(r) = 1.4/r^2 - 3.0/r
\draw[acc, very thick] plot[domain=0.6:5.4, samples=160] (\x, {1.4/(\x*\x) - 3.0/\x});
% minimum marker near r=0.93
\fill[acc] (0.93,-1.6) circle (2.2pt);
\draw[black, dashed] (0.93,0) -- (0.93,-1.6);
\node[acc, font=\scriptsize, anchor=west] at (1.05,-1.75) {minimum at Bohr radius};
\end{tikzpicture}
$$

## Natural width of spectral lines

The energy-time relation $\Delta E\,\Delta t \ge \tfrac12\hbar$ limits how precisely
an energy can be defined in a finite time. An atom in an excited state does not stay
there; it decays after a mean lifetime $\tau$, so the state's energy is available for
measurement only for a time $\sim\tau$. Its energy is therefore uncertain by

$$
\Delta E \gtrsim \frac{\hbar}{\tau}.
$$

> **Definition (Natural line width).** The spread $\Gamma_0 = \hbar/\tau$ in the
> energy of an excited state of lifetime $\tau$. It sets a minimum width to the
> spectral line emitted in the decay, present even when Doppler and collisional
> broadening are removed.

For a typical atomic lifetime $\tau \approx 10^{-8}\,\text{s}$,

$$
\Delta E \gtrsim \frac{6.58\times10^{-16}\,\text{eV·s}}{10^{-8}\,\text{s}} \approx 10^{-7}\,\text{eV},
$$

which spreads the emitted wavelength by $\Delta\lambda/\lambda \approx \Delta E/(E -
E_0)$. Only the ground state, with infinite lifetime, has a perfectly sharp energy.
Doppler and recoil effects usually broaden lines beyond the natural width, but in
special cases — the Mössbauer effect in solids at low temperature — the observed
width reduces to $\Gamma_0$ alone, giving photons of extraordinarily well-defined
energy.

## Wave-particle duality

Electrons diffract; light photoemits. Every entity — electrons, atoms, light, sound
— carries both particle and wave characteristics, in symmetry between matter and
radiation. In classical physics the two pictures are exclusive: a particle is
localized, exchanges energy in a lump, and follows conservation laws in collisions
but does not interfere; a wave spreads its energy continuously and does interfere.
Neither classical picture describes matter or radiation completely.

The two aspects appear in different circumstances:

- **Emission and absorption** — events at a definite place and time, exchanging
  energy and momentum in discrete amounts — are described by the particle picture.
  A photon strikes one rod of the retina; an electron registers as one dot.
- **Propagation through space**, including diffraction and interference, is described
  by the wave picture. The wave function $\Psi$ solves a wave equation, spreads
  through the apparatus, and interferes with itself.

The division sorts observation from prediction. An observation of a particle or
photon is a localized interaction, read in particle language. A prediction of where
the next particle will probably be found is a statement about the amplitude
$|\Psi|^2$ of a wave that has propagated and diffracted, read in wave language.

> **Definition (Wave-particle duality).** The principle that matter and radiation
> each exhibit both wave and particle properties. Propagation and interference are
> governed by a wave function; emission, absorption, and detection are discrete,
> particle-like events whose probabilities are set by the squared wave amplitude.

The [two-slit experiment](/quantum-mechanics/matter-waves/wave-packets-and-the-probability-interpretation)
holds both aspects at once. One electron at a time crosses the apparatus and lands
as a single dot — particle-like detection. Yet the dots accumulate into
interference fringes with none where the two paths cancel — wave-like propagation.
When the wavelength is far smaller than every aperture, interference is
unobservable and the particle picture alone suffices; this is the classical limit,
in which geometric optics and Newtonian trajectories are recovered.

$$
% caption: A plane wave through a slit of width a spreads into a diffraction angle,
% giving the transmitted particle a transverse momentum spread set by the slit width.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% incoming plane-wave fronts
\foreach \x in {-3.4,-3.0,-2.6,-2.2}{ \draw[acc] (\x,-1.3) -- (\x,1.3); }
\draw[->, acc, very thick] (-3.6,0) -- (-2.0,0);
\node[acc, font=\scriptsize, anchor=south] at (-2.8,1.35) {plane wave};
% slit
\draw[black, very thick] (-1.6,1.3) -- (-1.6,0.35);
\draw[black, very thick] (-1.6,-0.35) -- (-1.6,-1.3);
\node[font=\scriptsize, anchor=east] at (-1.65,0) {width a};
% diffraction spread
\draw[->, acc, thick] (-1.6,0) -- (2.4,1.1);
\draw[->, acc, thick] (-1.6,0) -- (2.4,0);
\draw[->, acc, thick] (-1.6,0) -- (2.4,-1.1);
\draw[black] (-1.6,0.7) arc (90:56:0.7);
\node[font=\scriptsize] at (-1.0,0.65) {spread};
% screen
\draw[black, very thick] (3.0,-1.6) -- (3.0,1.6);
\node[font=\scriptsize, anchor=west] at (3.05,0) {screen};
\end{tikzpicture}
$$

A slit of width $a$ localizes the transverse position of a passing particle to
$\Delta x \sim a$. Its transverse momentum then spreads by $\Delta p_x \sim \hbar/a$,
which is the diffraction that broadens the beam — the same statement as
$\Delta x\,\Delta p_x \sim \hbar$, now read as a single experiment. Narrowing the
slit sharpens the position and widens the diffraction pattern in lockstep.

The [Schrödinger equation](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
replaces these order-of-magnitude estimates with an exact wave equation for
$\Psi(x,t)$. The zero-point energy, the confined states, and the tunneling that the
uncertainty principle only sketches become quantitative predictions in the
[square-well](/quantum-mechanics/wave-mechanics-1d/particle-in-infinite-and-finite-square-wells)
and barrier problems of the next module.

[^heis]: Tipler & Llewellyn, §5-5 — the uncertainty principle $\Delta x\,\Delta p \ge \tfrac12\hbar$ and $\Delta E\,\Delta t \ge \tfrac12\hbar$ obtained from the classical packet relations times $\hbar$, with the standard-deviation definition attaining the bound for Gaussian distributions; the gamma-ray-microscope argument that the product cannot fall below order $h$ even in an ideal measurement.