Why Is Momentum P Related to Wavelength and Not Frequency?

Before delving into the intriguing question of why momentum p is related to wavelength and not frequency, it's essential to look back at the historical context that shaped our current understanding of light and matter. In the pre-quantum mechanics era, light was widely recognized as a wave, and physicists had already developed methods to calculate both wavelength and frequency for any wave. Indeed, Einstein's equations for the energy and momentum of a photon could be easily stated using either wavelength or frequency.

Then, in the 1920s, physicist Louis de Broglie introduced a groundbreaking idea: matter, just like light, also exhibits wave properties. This concept brought about the idea that particles, like electrons, could also be represented by wavelengths. However, the challenge remained: how to assign a frequency to an electron, which behaves more like a particle than a wave?

The Davisson–Germer experiment was instrumental in addressing this dilemma. By diffraction of electrons through a crystal lattice, scientists could measure the wavelength of electrons using the Bragg equation. It was through this experimental evidence that the wavelength of matter waves became more intuitive and less abstract. Therefore, the de Broglie relation was often stated in terms of wavelength rather than frequency.

Convenience in Expression

One might argue that it's simply a matter of convenience. People are more accustomed to associating wavelength with wave behavior than frequency. This convenience is rooted in the historical development of the equations that govern wave and particle properties. Einstein's famous equation for the energy of a photon, (E hf), combined with the equation (E mc^2) (where c is the speed of light), yields a relationship between mass, speed of light, and frequency and wavelength.

Starting from (E mc^2) and (E hf), one can derive (mc^2 hf). Given that (f frac{c}{lambda}), it follows that (mc^2 hfrac{c}{lambda}), which simplifies to (mc frac{h}{lambda}). Here, (mc) is identified as the momentum p of the photon, leading to the relation (p frac{h}{lambda}).

Vector vs. Scalar Quantities

It's crucial to understand that momentum is a vector quantity, whereas frequency is a scalar quantity. This distinction plays a vital role in the de Broglie relation. Momentum is proportional to wave number, which is the number of cycles per unit of length and also a vector quantity. This vectorial nature of momentum highlights the intrinsic connection between waves and particles, emphasizing how the introduction of wave properties to matter fundamentally altered our perception of physics.

In conclusion, the relationship between momentum and wavelength, rather than frequency, is a natural consequence of the historical development of physics, experimental evidence, and the inherent vectorial nature of wave properties. This relationship is crucial to the understanding of wave-particle duality and has paved the way for numerous advancements in modern physics.