# Electromagnetic Wave Packet. The Uncertainty Principle Revision Notes

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19.5Electromagnetic Wave Packet. The Uncertainty Principle

In these revision notes for Electromagnetic Wave Packet. The Uncertainty Principle, we cover the following key points:

• What is a wave packet? What are its properties? Why is it used for?
• What does a light wave packet represent?
• What is the spatial extension of wave packet?
• What is the uncertainty relation for photons?
• What does the uncertainty of wave number represent?
• What is the uncertainty principle of Heisenberg?
• What is De Broglie's Wave Packet?
• What are the two mathematical expressions of Heisenberg's uncertainty principle?

## Electromagnetic Wave Packet. The Uncertainty Principle Revision Notes

A single radiating action produced by atoms is very short; it lasts a few nanoseconds (1 ns = 10-9 s). Therefore, when studying the radiation of atoms, we always refer to "wave packets" instead of individual waves. Likewise, photons are also emitted in wave packets.

A "wave packet" is a fragmented or a discrete wave. It represents a short or a burst wave All sources that operate periodically in short intervals emit wave packets. Therefore, a wave packet has a limited width in space. The spatial extension Δx of a wave packet is given by the equation

∆x ≈ v ∙ ∆τ

where v is the packet's speed and Δτ is the time interval between two consecutive emissions.

EM wave packets emitted by atoms are very concentrated. This allows us visualize the photon as a particle.

Advanced theoretical calculations show that any wave packet does not contain a single a single wavelength but an infinite number of wavelengths instead. These values range from λ - λ/2 to λ + λ/2, where λ represents the mean wavelength of wave packet. The width Δλ of such wave packets relates to its spatial extension through the relation:

∆x ∙ ∆λ ≈ λ2/

This relation is characteristic for any wave group or packet.

The exact position of photon is not known; all we know is that the photon lies somewhere between x and x + Δx from the parent atom. Here, the uncertainty Δx corresponds to the spatial extension of photon's wave packet. It is known as the "uncertainty of coordinate".

Likewise, it is clear that the wavelength of photon is uncertain too, as its range of wavelength varies is Δλ. This is another uncertainty, which we call as "uncertainty of wavelength".

The existence of these two uncertainties is a fundamental feature of photon. Given that the wave number k is given by

k = /λ

we have an uncertainty of the wave number, Δk as well. It is given by

∆k ≈ 2π ∙ ∆λ/λ2

The uncertainty relation for photon in terms of Δx and Δk is

∆x ∙ ∆k ≈ 1

The relationship between the width of De Broglie packet Δk and the uncertainty of electron's momentum Δp is given by

∆k = ∆p/

where is the reduced Planck's constant.

Heisenberg Uncertainty principle states that:

"It is impossible to determine with absolute precision both the position and impulse of a particle at the same instant."

Mathematically, this principle is written as:

∆x ∙ ∆p ≈ ℏ

When this principle is written in terms of energy and time, it becomes

∆E ∙ ∆τ ≈ ℏ

The meaning of the above relation is as follows:

"It is impossible to determine with absolute precision both the energy of a particle and the instant at which this particle possesses this energy."

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