The de Broglie hypothesis says that every moving particle has an associated wavelength. This idea was one of the major
conceptual steps in early quantum physics because it extended the wave-particle duality of light to matter. If photons
can behave like particles, then electrons, protons, neutrons, and other particles can also behave like waves under the
right conditions.
de Broglie relation.
\[
\begin{aligned}
\lambda &= \frac{h}{p}
\end{aligned}
\]
Here, \(\lambda\) is the de Broglie wavelength, \(h\) is Planck’s constant, and \(p\) is the particle momentum. The
meaning of the formula is simple: the larger the momentum, the smaller the wavelength. Fast or massive particles
therefore tend to have very short wavelengths, while light particles moving more slowly can have wavelengths large enough
to produce measurable diffraction and interference.
Momentum from velocity
In the non-relativistic regime, momentum is
Momentum from mass and speed.
\[
\begin{aligned}
p &= m v
\end{aligned}
\]
Substituting this into the de Broglie formula gives
Wavelength from mass and speed.
\[
\begin{aligned}
\lambda &= \frac{h}{m v}
\end{aligned}
\]
This form is especially useful when the mass and speed are known directly. It makes the inverse dependence on velocity
very clear: doubling the speed halves the wavelength.
Momentum from kinetic energy
In many physics problems the particle is described by its kinetic energy instead of its speed. For a non-relativistic
particle,
Kinetic-energy relation.
\[
\begin{aligned}
K &= \frac{p^2}{2m}
\end{aligned}
\]
Solving for momentum gives
Momentum from kinetic energy.
\[
\begin{aligned}
p &= \sqrt{2mK}
\end{aligned}
\]
Therefore the de Broglie wavelength can also be written as
Wavelength from kinetic energy.
\[
\begin{aligned}
\lambda &= \frac{h}{\sqrt{2mK}}
\end{aligned}
\]
This form is extremely common for electrons accelerated through a potential difference. If a singly charged particle is
accelerated through a voltage \(V\), then its kinetic energy is numerically \(K = V\,\mathrm{eV}\). For example, an
electron accelerated through \(100\ \mathrm{V}\) has kinetic energy \(100\ \mathrm{eV}\).
Sample calculation: electron accelerated through 100 V
Take an electron with mass
Electron mass and kinetic energy.
\[
\begin{aligned}
m_e &= 9.109\times 10^{-31}\ \mathrm{kg}, \\
K &= 100\ \mathrm{eV}
\end{aligned}
\]
First convert the kinetic energy into joules:
Convert eV to J.
\[
\begin{aligned}
K &= 100 \cdot 1.602176634\times 10^{-19}\ \mathrm{J} \\
&\approx 1.602\times 10^{-17}\ \mathrm{J}
\end{aligned}
\]
Next compute the momentum:
Momentum of the electron.
\[
\begin{aligned}
p &= \sqrt{2m_eK} \\
&= \sqrt{2 \cdot 9.109\times 10^{-31} \cdot 1.602\times 10^{-17}} \\
&\approx 5.40\times 10^{-24}\ \mathrm{kg\cdot m/s}
\end{aligned}
\]
Finally, apply the de Broglie relation:
de Broglie wavelength of the electron.
\[
\begin{aligned}
\lambda &= \frac{h}{p} \\
&= \frac{6.626\times 10^{-34}}{5.40\times 10^{-24}} \\
&\approx 1.23\times 10^{-10}\ \mathrm{m} \\
&\approx 0.1227\ \mathrm{nm}
\end{aligned}
\]
This wavelength is on the order of atomic spacing in crystals. That is why electrons can produce diffraction patterns in
crystal lattices, just as X-rays do. This was confirmed experimentally in the Davisson–Germer experiment, which became a
major validation of wave mechanics.
Physical interpretation
The de Broglie wavelength becomes significant when it is comparable to characteristic dimensions in a physical system.
If the wavelength is comparable to the spacing between atoms, then diffraction and interference can be observed. If the
wavelength is much smaller than the system size, the particle behaves more classically and wave effects are harder to
detect.
This explains why wave behavior is easy to observe for electrons in microscopy and diffraction experiments but not for
macroscopic objects. A baseball has such a huge momentum that its de Broglie wavelength is unimaginably small, far below
any measurable scale in ordinary life.
Limitation of the calculator
The formulas used here are non-relativistic. They work very well when the particle speed is much smaller than
the speed of light. At higher speeds, the classical expressions \(p=mv\) and \(K=p^2/(2m)\) are no longer exact, and
special relativity must be used. A relativistic treatment replaces the simple momentum-energy relations with more
accurate expressions involving the Lorentz factor.
Even so, the non-relativistic model is the standard starting point in introductory and intermediate quantum physics,
especially for electron diffraction, matter-wave ideas, and the early interpretation of quantum mechanics.
| Concept |
Main relation |
Meaning |
| de Broglie wavelength |
\(\lambda = h/p\) |
Matter-wave wavelength associated with momentum |
| Momentum from speed |
\(p = mv\) |
Non-relativistic momentum |
| Momentum from energy |
\(p = \sqrt{2mK}\) |
Useful when kinetic energy is known |
| Wavelength from energy |
\(\lambda = h/\sqrt{2mK}\) |
Direct way to compute matter wavelength from kinetic energy |
| Accelerating voltage |
\(K = V\,\mathrm{eV}\) |
For a singly charged particle accelerated through \(V\) volts |
| Relativistic caution |
Needed when \(v\) is not small compared with \(c\) |
Non-relativistic formulas lose accuracy at high speed |