Library / Foundations: Physics / FPH-38

Foundations: Physics

de Broglie wavelength

Wavelength associated with a moving particle.

Ref: FPH-38
Rev: v1
Field: Foundations: Physics
Result: λ [nm]

λ = \frac{h}{p} = \frac{h}{m v}

p is particle momentum. h = 6.626×10⁻³⁴ J·s.

Moving particles behave like waves, with a shorter wavelength the more momentum they have.

Calculator

Solve for

λde Broglie wavelengthComputed

mMass

10⁻³¹ kg

520,000

vSpeed

10⁶ m/s

0.1100

Momentum0.9109 10⁻²⁴ kg·m/s
Kinetic energy2.843 eV

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

de Broglie wavelength (nm) current value 0.7274 nm

vSpeedKey driver

0.007277.27

mMass

3.31e-41.33

Electrons accelerated through about 100 V have a wavelength near 0.12 nm, comparable to atom spacings, which is why electron diffraction reveals crystal structure.
Everyday objects have wavelengths so small that wave behaviour is unobservable.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (nm)	Local step	Change in result
v Speed	0.1 to 100 10⁶ m/s	0.007274	7.274	7.267	+10%	−9.09 %
m Mass	5 to 20,000 10⁻³¹ kg	3.313e-4	1.325	1.325	+10%	−9.09 %

Sweep of Speed

Speed (10⁶ m/s)	de Broglie wavelength (nm)
0.1	7.274
0.2371	3.068
0.5623	1.294
1.334	0.5455
3.162	0.23
7.499	0.097
17.78	0.04091
42.17	0.01725
100	0.007274

Sweep one variable

Vary

de Broglie wavelength (nm)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

de Broglie wavelength

Result · nm

What the equation gives you. Everything else feeds it.

Mass

10⁻³¹ kg · typical 5 to 20,000

Particle mass (electron 9.109). Inversely proportional.

Speed

10⁶ m/s · typical 0.1 to 100

Particle speed. Inversely proportional.

Worked example with the values above

Momentum: p = mv = 9.109×10⁻³¹ × 1,000,000 = 9.109×10⁻²⁵ kg·m/s.
λ = h/p = 6.626×10⁻³⁴ / 9.109×10⁻²⁵.
λ = 0.7274 nm.

Assumptions and limits

Non-relativistic speeds.
Free particle.

Common mistakes

Using energy instead of momentum.
Mixing powers of ten.

Statistics and uncertainty

Relative errors in mass and speed add in quadrature; electron masses are very precisely known.

Where it is used

Electron microscopes.
Neutron and electron diffraction.

Maths you need

Momentum: p = mv
Planck constant: h

Related formulas

Practise this formula