Discover one of the cornerstones of quantum mechanics with our de Broglie wavelength calculator. Enter the realm of probability and superimposed cats, and learn about the fundamental nature of matter.

Our tool is straightforward to use; however, the physics behind it is of exceptional conceptual depth; keep reading to learn:

  • What is the de Broglie wavelength;
  • The de Broglie wavelength equation;
  • How to calculate the de Broglie wavelength of electrons; and
  • If we can apply the de Broglie wavelength formula to cats.

What is the de Broglie wavelength? A few words on the mysterious nature of matter

Our story begins in the late XIXth century. Physicists were laying down the foundations of quantum theory, preparing the world for the frenzy of the early XXth century.

At the time, light and matter were considered well distinct concepts. For scientists, light was nothing but waves: this was easy to prove: we can set up a quick diffraction experiment and appreciate interference patterns typical of undulatory motions.

On the other hand, physicists considered matter made of... matter: particles of varying size and mass make up everything we can touch. The corpuscular nature was not difficult to prove with many experiments.

Scientists, in the past, already challenged the wave-like nature of light. For centuries, the most widely accepted theory for light was corpuscular. Only with better experimental techniques, we started appreciating the wave behavior.

So what's the catch? Max Planck, who first postulated the quantized nature of light, undermined the accepted hypothesis and proposed a model for light where particles — photons — are at the base of electromagnetic radiation. Photons share properties of both waves and particles: this was needed to justify, at the same time, interference experiments (for the wave-like nature) and the photoelectric phenomenon (a corpuscular approach best explains that).

It's 1924, and Louis de Broglie, an aristocratic French physicist, changed physicists' perspective on Nature with his doctoral thesis. In his work, he expanded the dual nature of light to electrons.

This concept expanded to all matter: the de Broglie wavelength is now used by physicists to explain the double nature of matter.

The formula of de Broglie wavelength: calculate the matter wave

To calculate the de Broglie wavelength, we start with the Planck's equation:

E=hνE = h\cdot\nu

Where we can identify:

  • EE — The photon energy;
  • hhPlanck's constant;
  • ν\nu — The photon's frequency.

De Broglie translated this equation, after a bit of rearrangement, to isolate the wavelength. Writing the momentum of the photon with:

p=Ec=hννλ=hλp = \frac{E}{c} = \frac{h\cdot \nu}{\nu\cdot\lambda} = \frac{h}{\lambda}

🙋 Here we applied the relationship between wavelength and frequency. You can learn more at our dedicated tools, the wavelength calculator and the frequency calculator.

Allows us to write:

λ=hp\lambda = \frac{h}{p}

pp is the momentum of a massive object (photons don't have mass): p=mvp = m\cdot v, where mm is the mass and vv the speed of the body.

This was a theory at the beginning, but soon after its publication, physicists started searching for this duality and found traces of it.

The de Broglie wavelength of the electron: the most beautiful experiment in physics

The de Broglie wavelength formula allows us to calculate the de Broglie wavelength of the electron if we know its momentum.

🙋 Many scientists consider the experiment that proves the double nature of electrons (wave and particle) as the most beautiful experiment in physics: we are talking of the double slit experiment applied to single electrons. You can learn more about double slits with our diffraction grating calculator.

We can compute the momentum of an electron if we know its mass and speed. The mass of the electron is a known quantity:

me=9.10938359×1031 kgm_{\text{e}} = 9.10938359\times10^{-31}\ \text{kg}

The speed of an electron varies greatly. In this example, we will assume our particle is traveling at 1010% of the speed of light:

ve=29, ⁣979, ⁣245.8 msv_{\text{e}} = 29,\!979,\!245.8\ \frac{\text{m}}{\text{s}}

The de Broglie wavelength of this electron is:

λ=hp=hmeve=6.6261×1034 J ⁣ ⁣s9.109 ⁣× ⁣1031 kg ⁣ ⁣2.998 ⁣× ⁣108ms=0.024 nm\begin{align*} \lambda&=\frac{h}{p}= \frac{h}{m_{\text{e}} \cdot v_{\text{e}}} \\ & = \frac{6.6261\times 10^{34}\ \text{J}\!\cdot\!\text{s}}{ 9.109\!\times\!10^{-31}\ \text{kg} \!\cdot\! 2.998\!\times\!10^8 \frac{\text{m}}{\text{s}}} &=0.024\ \text{nm} \end{align*}

The wavelength of an electron is a fifth of an angstrom: we are talking of sub-subatomic length. As you can see from the de Broglie wavelength equation, the wave-like behavior of a massive object decreases in magnitude with an increase in both mass and velocity.

Can we calculate the de Broglie wavelength of a cat? Does it even make sense?
Yes, to both questions: every object has a de Broglie wavelength, and it somehow possesses a quantum nature; the consequences, however, are too small to notice.

Let's take a rather chonky cat with mass m=7 kgm=7\ \text{kg}. Our feline friend is running at max speed trying to avoid a mad quantum physicist trying to prove overquoted experiments: a cat can run at v=48 km/h=13.33 m/sv = 48\ \text{km}/\text{h} = 13.33\ \text{m}/\text{s}.

Calculating the wavelength is a game of orders of magnitude. The result is:

λcat=0,71×1035 m\lambda_{\text{cat}} = 0,71\times10^{-35}\ \text{m}

Roughly half the Planck's length, that is, according to the most recent physical theories, the smallest length with a physical meaningfulness.

Davide Borchia
Rest mass (m)
x10⁻³¹
kg
Velocity (v)
m/s
Momentum (p)
x10⁻²⁴
kg·m/s
Wavelength (λ)
nm
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