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Physics Lesson 19.5.3 - The Uncertainty Relation for EM Wave Packet (Photon)

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Welcome to our Physics lesson on The Uncertainty Relation for EM Wave Packet (Photon), this is the third lesson of our suite of physics lessons covering the topic of Electromagnetic Wave Packet. The Uncertainty Principle, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

The Uncertainty Relation for EM Wave Packet (Photon)

Let's assume at a certain instant, a photon is at the distance x from the radiating atom as shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Electromagnetic Wave Packet. The Uncertainty Principle

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 = 2π/λ

we have an uncertainty of the wave number, Δk as well. The equation

∆x ∙ ∆λ ≈ λ2/2π

is usually written in terms of Δx and Δk. But first, let's find the formula of Δk. Thus,

∆k = 2π/λ - 2π/λ + ∆λ
= 2π ∙ (λ + ∆λ) - 2π ∙ λ/λ ∙ (λ + ∆λ)
= 2π ∙ ∆λ/λ ∙ (λ + ∆λ)

Since Δλ << λ (as seen in the example solved earlier), we can approximate the above equation as

∆k ≈ 2π ∙ ∆λ/λ2

Thus,

λ2/2π = ∆λ/∆k

Hence, the equation

∆x ∙ ∆λ ≈ λ2/2π

is transformed into

∆x ∙ ∆λ ≈ ∆λ/∆k

Simplifying both sides by Δλ, we obtain

∆x ≈ 1/∆k

or

∆x ∙ ∆k ≈ 1

The above relation is simpler than the other relation that involves Δλ; hence it is known as the uncertainty relation for photon. In fact, this relation is valid for all wave packets, not only for photons.

Example 4

Calculate the uncertainty Δk of the wave number of an electromagnetic wave packet if the radiation time is Δτ = 0.01 s.

Solution 4

Clues:

Δτ = 0.01 s = 10-2 s
(c = 3 × 108 m/s)
Δk = ?

First, we calculate the extension of EM wave packet. We have

∆x = c ∙ ∆τ
= (3 × 108 m/s) ∙ (10-2 s)
= 3 × 106 m

Therefore, we obtain for the uncertainty of wave number Δk

∆x ∙ ∆k ≈ 1
∆k ≈ 1/∆x
= 1/3 × 106 m
= 3.33 × 10-7 m-1

You have reached the end of Physics lesson 19.5.3 The Uncertainty Relation for EM Wave Packet (Photon). There are 5 lessons in this physics tutorial covering Electromagnetic Wave Packet. The Uncertainty Principle, you can access all the lessons from this tutorial below.

More Electromagnetic Wave Packet. The Uncertainty Principle Lessons and Learning Resources

Modern Physics Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
19.5Electromagnetic Wave Packet. The Uncertainty Principle
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
19.5.1Recalling the Simple Harmonic Wave (Oscillation)
19.5.2Definition and Features of Wave Packet
19.5.3The Uncertainty Relation for EM Wave Packet (Photon)
19.5.4De Broglie's Wave Packet. Wave Width and Uncertainty of Momentum
19.5.5Heisenberg's Uncertainty Principle

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