The Uncertainty Principle and Wave-Particle Duality

The wave packet obeys two purely mathematical relations between its spatial width and its band of wave numbers, and between its duration and its band of frequencies:

Δ k Δ x \sim 1, Δ ω Δ 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 $ℏ$ and use $p = ℏ k$ and $E = ℏ ω$ :

Δ x Δ p \sim ℏ, Δ E Δ t \sim ℏ.

A wave packet narrow in position ( $Δ x$ small) is built from a wide band of wave numbers, so its momentum $p = ℏ k$ 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 $Δ x$ ; repeated momentum measurements scatter with spread $Δ p$ ; the two spreads cannot both be driven to zero.

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.

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

The bound is set by $ℏ = 1.055 \times 1 0^{- 34} J\cdots$ , 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,

Δ x \approx \frac{λ}{sin θ},

so a sharp position calls for short wavelength — gamma rays. But a gamma-ray photon carries momentum $p_{γ} = h / λ$ , and scattering it off the electron is a Compton collision 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

Δ p_{x} \approx p_{γ} sin θ = \frac{h}{λ} sin θ .

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

Δ x Δ p_{x} \approx \frac{λ}{sin θ} \cdot \frac{h}{λ} sin θ = h .

Shortening $λ$ 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 $h$ .

Heisenberg's gamma-ray microscope. A short-wavelength photon locates the electron to within one wavelength but Compton-recoils it, spreading its momentum.

Zero-point energy of a confined particle

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

(Δ p)^{2} = ⟨ p^{2} ⟩ - ⟨ p ⟩^{2} = ⟨ p^{2} ⟩ ≳ (\frac{ℏ}{L})^{2},

and the average kinetic energy cannot vanish:

E = \frac{⟨ p ^{2} ⟩}{2 m} ≳ \frac{ℏ ^{2}}{2 m L ^{2}} .

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.

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}}{2 m} - \frac{k e ^{2}}{r} .

Confinement to within $r$ forces $Δ x \sim r$ , hence $p \sim ℏ/ r$ , so

E (r) \approx \frac{ℏ ^{2}}{2 m r ^{2}} - \frac{k e ^{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{ℏ ^{2}}{m r ^{3}} + \frac{k e ^{2}}{r ^{2}} = 0 ⟹ r_{m} = \frac{ℏ ^{2}}{k e ^{2} m} = a_{0} = 0.0529 nm,

the Bohr radius. The corresponding energy is

E_{m} = - \frac{k ^{2} e ^{4} m}{2 ℏ ^{2}} = - 13.6 eV,

the ground-state energy of hydrogen. 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 $Δ x \sim r$ , but any reasonable choice gives the correct order of magnitude.

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.

Natural width of spectral lines

The energy-time relation $Δ E Δ t \geq \frac{1}{2} ℏ$ 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 $τ$ , so the state's energy is available for measurement only for a time $\sim τ$ . Its energy is therefore uncertain by

Δ E ≳ \frac{ℏ}{τ} .

For a typical atomic lifetime $τ \approx 1 0^{- 8} s$ ,

Δ E ≳ \frac{6.58 \times 1 0 ^{- 16} eV\cdots}{1 0 ^{- 8} s} \approx 1 0^{- 7} eV,

which spreads the emitted wavelength by $Δ λ / λ \approx Δ E / (E - E_{0})$ . 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 $Γ_{0}$ 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 $Ψ$ 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 $∣Ψ ∣^{2}$ of a wave that has propagated and diffracted, read in wave language.

The two-slit experiment 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.

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.

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

The Schrödinger equation replaces these order-of-magnitude estimates with an exact wave equation for $Ψ (x, t)$ . The zero-point energy, the confined states, and the tunneling that the uncertainty principle only sketches become quantitative predictions in the square-well and barrier problems of the next module.

Tipler & Llewellyn, §5-5 — the uncertainty principle $Δ x Δ p \geq \frac{1}{2} ℏ$ and $Δ E Δ t \geq \frac{1}{2} ℏ$ obtained from the classical packet relations times $ℏ$ , 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. ↩