de Broglie Wavelength Calculator

The de Broglie Hypothesis: Wave–Particle Duality

In 1924, French physicist Louis de Broglie proposed one of the most revolutionary ideas in physics: that all matter exhibits both particle-like and wave-like properties. This dual nature was previously known for light, where experiments like the photoelectric effect demonstrated particle behavior and Young’s double-slit experiment showed wave interference. De Broglie extended this concept to matter, suggesting that electrons, protons, neutrons, and even macroscopic objects possess wavelengths, though the effect becomes noticeable only at very small scales.

The Formula for de Broglie Wavelength

The de Broglie wavelength λ is given by:

λ = h / p

where:

• λ: wavelength associated with the particle
• h: Planck’s constant (6.626 × 10⁻³⁴ J·s)
• p: momentum of the particle (p = mv for non-relativistic cases)

Relativistic Considerations

For particles moving close to the speed of light, the momentum must include the Lorentz factor:

p = γmv

In this case, the de Broglie wavelength becomes:

λ = h / (γmv)

This shows that as velocity increases, momentum increases dramatically, and the corresponding wavelength becomes shorter.

Experimental Verification

De Broglie’s hypothesis was experimentally verified in 1927 by Davisson and Germer, who observed electron diffraction patterns when a beam of electrons scattered off a nickel crystal. The diffraction confirmed that electrons indeed behaved like waves with wavelengths consistent with de Broglie’s formula. Similar experiments with neutrons and atoms further validated the hypothesis.

Applications of de Broglie Wavelength

• Electron Microscopy: The resolving power of electron microscopes is due to the small de Broglie wavelength of fast-moving electrons, which allows imaging at atomic scales.
• Quantum Mechanics: The Schrödinger wave equation was directly inspired by de Broglie’s wave hypothesis.
• Crystallography: Neutron diffraction relies on de Broglie wavelengths of neutrons to probe atomic arrangements.
• Semiconductor Physics: Quantum tunneling and confinement effects are governed by matter-wave properties.
• Astrophysics: Used in modeling dense quantum systems like neutron stars and white dwarfs.

Worked Example

Suppose an electron (mass = 9.11 × 10⁻³¹ kg) moves at 1 × 10⁶ m/s. Its momentum is:

p = mv = (9.11 × 10⁻³¹)(1 × 10⁶) = 9.11 × 10⁻²⁵ kg·m/s

The de Broglie wavelength is:

λ = h / p = 6.626 × 10⁻³⁴ / 9.11 × 10⁻²⁵ ≈ 7.27 × 10⁻¹⁰ m

This wavelength is on the order of an angstrom (10⁻¹⁰ m), which is comparable to interatomic distances, explaining why electron diffraction is so effective in studying crystals.

Macroscopic Limits

While the de Broglie wavelength applies to all matter, for macroscopic objects it becomes negligible. For example, a baseball of mass 0.145 kg moving at 40 m/s has a de Broglie wavelength of about 10⁻³⁴ m, far too small to be observed. Thus, wave behavior is only observable for microscopic particles.

Philosophical Implications

The de Broglie hypothesis forced physicists to rethink the nature of reality. Objects are not exclusively particles or waves but exhibit both properties depending on how they are observed. This principle underlies quantum mechanics and challenges classical intuitions about the world.

Conclusion

The de Broglie wavelength remains one of the most important concepts in modern physics. It bridges classical mechanics and quantum theory, explains wave–particle duality, and forms the basis for technologies like electron microscopes and semiconductor devices. Understanding de Broglie’s idea is fundamental for anyone exploring the quantum world.