de Broglie Wavelength

Equivalent forms

\lambda = h/(mv)

p = h/\lambda

One of physics' shortest equations launches all of quantum mechanics.

Symbol	Name	SI unit	Dimension	Range
$lambda$	Wavelengthoutput de Broglie wavelength of the particle	m	L	1e-35 – 0.000001
$h$	Planck's constant Fundamental quantum of action	J*s	ML^2T^-1	6.62607015e-34 – 6.62607015e-34
$p$	Momentum Linear momentum of the particle	kg*m/s	MLT^-1	1e-30 – 10000000000

Unit systems

Symbol	SI	CGS	Imperial
$lambda$	m	cm	ft
$h$	J*s	erg*s	ftlbfs
$p$	kg*m/s	g*cm/s	lb*ft/s

Where it holds

Valid for non-interacting free particles. Requires p > 0. Relativistic correction

p = \gamma *m*v

for v comparable to c.

Dimensional analysis

\lambda \to [L]

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

Both sides are length.

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Discovery

Louis de Broglie · 1924

In his PhD thesis, de Broglie proposed that if light could be particle-like, particles could be wave-like — confirmed by Davisson–Germer in 1927.

Try this

Does a thrown baseball have a wavelength?

Every moving particle behaves like a wave. What is the wavelength of a 0.145 kg baseball thrown at 40 m/s?

Research status: stable

Real-world applications

Electron microscopes $(\lambda \approx 0.004\,\mathrm{nm}$ at 100 kV)
Neutron scattering for material science
Atom interferometry for precision gravimetry
Low-energy electron diffraction (LEED) for surface studies

Common misconceptions

lambda is not a 'size' of the particle — it is a length scale of interference
Macroscopic objects do have wavelengths, but they are unmeasurably small
The wave is not a physical oscillation of the particle, but of the probability amplitude

Experimental verification

Davisson–Germer (1927) observed electron diffraction from nickel crystals matching

\lambda = h/p

. Thomson's electron diffraction (1927), neutron diffraction (1936), and C60 fullerene interferometry (1999) all confirm the relation.

Derivation

Starting from Einstein's photon relation

E = hf

and the photon momentum

p = E/c = hf/c = h/\lambda

.

De Broglie postulated the same relation

p = h/\lambda

holds for matter, giving

\lambda = h/p

.

For a non-relativistic particle of mass m and speed v,

p = mv

so

\lambda = h/(mv)

.

Limiting cases

p \to 0

⟶

\lambda \to \infty

Slow, low-momentum particles become highly delocalized waves.

p \to

large (macroscopic)⟶

\lambda \to 0

Baseball at 40 m/s has

\lambda \approx 1.14e-34\,\mathrm{m}

— 19 orders of magnitude below a proton.

p = m_e * 1e_{6} m/s

(fast electron)⟶

\lambda \approx 0.73\,\mathrm{nm}

Comparable to atomic spacings — basis of electron diffraction.

What if…

What if h were zero?

All matter would be purely classical — no wave behavior, no quantum mechanics, no stable atoms.

What if you slowed an electron to 1 mm/s?

$\lambda \approx 0.73\,\mathrm{m}$ — a macroscopic matter wave, achievable in ultracold atom experiments.

What if the particle is a photon?

$p = E/c = h/\lambda$ recovers the original Einstein relation — de Broglie's law reduces to it.

1

Baseball wavelength

Given ·

m:: 0.145
v:: 40

Find · lambda

Steps

$p = m*v = 0.145 * 40 = 5.8\,\mathrm{kg}*m/s$
$\lambda = h/p = 6.62607015e-34 / 5.8$
$\lambda \approx 1.14e-34\,\mathrm{m}$ — far below any detectable scale

Result ·

\lambda = h/(mv) = 6.62607015e-34 / (0.145 * 40) \approx 1.14e-34\,\mathrm{m}

2

Electron at 1% c

Given ·

m:: 9.1093837015e-31
v:: 2997924.58

Find · lambda

Steps

$p = m_e * v = 9.1093837015e-31 * 2997924.58 \approx 2.73e-24\,\mathrm{kg}*m/s$
$\lambda = 6.62607015e-34 / 2.73e-24$
$\lambda \approx 2.43e-10\,\mathrm{m}$

Result ·

\lambda \approx 2.43e-10\,\mathrm{m} \approx 0.24\,\mathrm{nm}

Photon Energy (Planck Relation)→Heisenberg Uncertainty Principle→Time-Dependent Schrödinger Equation→