Wave-Particle Duality (de Broglie)

Equivalent forms

\lambda=\frac{h}{\sqrt{2mK}}

p=\hbar k

The same h that quantizes light also gives every electron, atom, and buckyball a wavelength — duality is universal, just hidden by scale.

Symbol	Name	SI unit	Dimension	Range
$\lambda$	de Broglie wavelengthoutput Wavelength associated with a moving particle	m	L	—
$p$	Momentum Particle momentum mv	kg*m/s	MLT^-1	1e-25 – 1e-22
$h$	Planck constant Quantum of action	J*s	ML^2T^-1	—
$m$	Mass Particle rest mass	kg	M	—
$v$	Speed Particle speed (non-relativistic)	m/s	L*T^-1	—

Where it holds

Non-relativistic form

\lambda = h/mv

; for fast particles use the relativistic momentum

p = \gamma mv

. Wave effects are observable only when

\lambda is

comparable to apparatus dimensions.

Dimensional analysis

\lambda \to [L]

h/p \to [M*L^2*T^-1]/[M*L*T^-1] = [L]

Action over momentum yields length.

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Discovery

Louis de Broglie · 1924

In his PhD thesis de Broglie proposed matter waves almost as a symmetry argument. His examiners were unsure until Einstein endorsed it. Three years later Davisson and Germer, investigating a lab accident that had recrystallized a nickel target, saw electrons diffract exactly as λ = h/p predicted. De Broglie won the 1929 Nobel Prize.

Research status: stable

Real-world applications

Electron microscopes (short $\lambda$ → atomic resolution)
Neutron diffraction for crystal and magnetic structure
Matter-wave interferometry and atom optics

Common misconceptions

The particle is not literally spread into a wave in space — $\lambda$ sets the scale of quantum interference, not a physical smear
Macroscopic objects have wavelengths $(\sim 10^{-34} m) far too$ small to ever detect
Higher speed means shorter wavelength, the opposite of everyday waves on water

Experimental verification

Davisson–Germer (1927) and G.P. Thomson's electron-diffraction rings; later neutron, atom, and

C_{60}

-molecule interference (Zeilinger, 1999) confirmed matter waves for ever-larger objects.

Derivation

Start from the photon relations

E = hf

and

p = E/c = h/\lambda

, valid for massless light.

De Broglie postulated the momentum relation

p = h/\lambda

holds for matter too,

so \lambda = h/p

.

For a non-relativistic particle

p = mv

, giving

\lambda = h/mv

.

Expressed through kinetic energy

K = p^{2}/2m

,

\lambda = h/\sqrt{2mK}

, the form used for electrons accelerated through a voltage

V (K = eV)

.

What if…

What if a 145 g baseball is thrown at 40 m/s?

$\lambda = h/mv \approx 1.1e-34\,\mathrm{m}$ — twenty orders of magnitude smaller than a nucleus, utterly unobservable.

What if the particle slows down?

Its momentum drops $so \lambda$ grows — ultracold atoms have micron-scale wavelengths, enabling atom interferometers.

1

Wavelength of a 100 eV electron

Given ·

K:: 100 eV
m:: 9.11e-31 kg

Find · de Broglie wavelength

Steps

$K = 100\,\mathrm{eV} = 1.60e-17\,\mathrm{J}$
$p = \sqrt{2mK} = \sqrt{2\times 9.11e-31\times 1.60e-17} = 5.4e-24 kg\cdot m/s$
$\lambda = 6.626e-34 / 5.4e-24 = 1.23e-10\,\mathrm{m}$

Result ·

\lambda = h/\sqrt{2mK} \approx 1.23e-10\,\mathrm{m} (0.12\,\mathrm{nm})

Photon Energy (Planck-Einstein Relation)→The Schrödinger Equation→Heisenberg Uncertainty Principle→