# De Broglie relation using four momentum in particle physics

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• crick
In summary, the wavelength of the probing photon in particle physics is an important factor in understanding the proton substructure. This is related to the transferred momentum by the De Broglie relation, which can be expressed using four-vectors as a frame-independent equation. The four-momentum and four-wavevector are used to reduce the De Broglie relations to a single equation. This concept is also applied to the relation between group/particle velocity and phase velocity.
crick
In particle phyisics four-momentum is used and De Broglie relation is used to understand what lenghts can be "seen" in an experiment.
Here (page 6) https://people.phys.ethz.ch/~pheno/PPP/PPP2.pdf it is claimed

The key factor for investigating the proton substructure is the wavelength of the probing
photon, which is related to the transferred momentum by
$$\lambda\sim \frac{1}{\sqrt{Q^2}}$$

Where ##Q^2## is not actually "momentum" but its the square of the four momentum transferred by the photon.

I can't understand why this is used, since the square root of square of four momentum is

$$\sqrt{p\cdot p}=\sqrt{|\textbf{p}|^2-E^2/c^2}$$

While De Broglie relation usually involves the three momentum ##\textbf{p}##
$$\lambda\sim \frac{1}{|\textbf{p}|}$$

So is in this case ##\sqrt{Q^2}## approximately the three momentum? Or is the De Broglie relation in the relativistic case to be written using ##\sqrt{Q^2}## instead of three momentum?

Not sure if this will help but wikipedia has an article on matter-waves:

https://en.wikipedia.org/wiki/Matter_wave

and toward the end they describe how they use four-vectors to reduce the deBroglie relations to a single equation:

https://en.wikipedia.org/wiki/Matter_wave

Four-vectors
Main article: Four-vector
Using four-vectors, the De Broglie relations form a single equation:

which is frame-independent.

Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:

where

Four-momentum

Four-wavevector

Four-velocity

crick

## What is the De Broglie relation using four momentum in particle physics?

The De Broglie relation using four momentum is a fundamental concept in particle physics that describes the wave-like properties of particles. It states that the wavelength of a particle is inversely proportional to its momentum, with the proportionality constant being Planck's constant divided by the particle's mass.

## How is the De Broglie relation used in particle physics experiments?

The De Broglie relation is used in particle physics experiments to understand and predict the behavior of particles. It is particularly useful in experiments that involve high-speed particles, such as those in particle accelerators, where the wave-like nature of particles becomes more apparent.

## What is the significance of the De Broglie relation in quantum mechanics?

The De Broglie relation was a groundbreaking concept in the development of quantum mechanics. It helped bridge the gap between classical mechanics and quantum mechanics by showing that particles can exhibit both wave-like and particle-like behavior.

## Can the De Broglie relation be applied to macroscopic objects?

No, the De Broglie relation is only applicable to microscopic particles, such as electrons and photons. This is because the wavelength of macroscopic objects would be so small that it becomes practically impossible to measure.

## How does the De Broglie relation relate to Heisenberg's uncertainty principle?

The De Broglie relation is closely related to Heisenberg's uncertainty principle, which states that the more precisely we know the position of a particle, the less accurately we can know its momentum, and vice versa. This is because the De Broglie relation implies that the wavelength of a particle is inversely proportional to its momentum, making it impossible to know both values simultaneously with absolute certainty.

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