Energy and Momentum of a Photon

Section: Quantum Physics  |  Syllabus: Cambridge A Level Physics (9702)

Particulate Nature of Electromagnetic Radiation

Electromagnetic radiation consists of self-propagating oscillating electric and magnetic fields. While interference and diffraction provide evidence for a wave model, the photoelectric effect provides evidence for a particulate nature.

Photon A quantum of energy of electromagnetic radiation. Photons are massless but carry energy and momentum.

Photon Energy

The energy of a photon is directly proportional to the frequency of the radiation.

Photon Energy:

E = hf

where:

Substituting the wave equation c = fλ, we can write:

E = (hc/λ)

This shows that higher frequency (or shorter wavelength) radiation (like gamma rays) has higher energy photons than lower frequency radiation (like radio waves).

Ionising Radiation

High-energy photons (UV, X-rays, Gamma) can remove electrons from atoms, creating ions. This is why they are dangerous to living tissue.

Worked Examples

Worked Example 1: Energy of Blue Light

Question: Calculate the energy of a photon of blue light with a wavelength of 450 nm.

Solution

1. Convert wavelength to metres: λ = 450 × 10^-9 m

2. Use E = (hc/λ):

E = (6.63 × 10^-34)(3.00 × 10^8)450 × 10^-9

E = 4.42 × 10^-19 J

Worked Example 2: Photons from a Laser

Question: A laser pointer emits red light (λ = 650 nm) with a power of 3.0 mW. Calculate the number of photons emitted per second.

Solution

1. Energy of one photon: E = (hc/λ) = (6.63 × 10^-34)(3.00 × 10^8)650 × 10^-9 = 3.06 × 10^-19 J

2. Power P = Total Energy / time = N × E (where N is number of photons/sec).

3. N = (P/E) = 3.0 × 10^-33.06 × 10^-19

Answer: 9.8 × 10^15 s^-1

Worked Example: Conversions

Question: (a) Convert 5.0 MeV to Joules. (b) Convert 8.0 × 10^-13 J to eV.

Solution

(a) 5.0 MeV = 5.0 × 10^6 eV

E = (5.0 × 10^6) × (1.60 × 10^-19) = 8.0 × 10^-13 J

(b) E = 8.0 × 10^-131.60 × 10^-19 = 5.0 × 10^6 eV = 5.0 MeV

Worked Example: Momentum of a Gamma Ray

Question: A gamma photon has energy 1.0 MeV. Calculate its momentum.

Solution

1. Convert E to Joules: 1.0 × 10^6 × 1.60 × 10^-19 = 1.60 × 10^-13 J

2. Use p = E/c: p = 1.60 × 10^-133.00 × 10^8 = 5.3 × 10^-22 kg m s^-1

Worked Example: Photon Energy from Frequency

Question: Calculate the energy of a photon of red light with frequency 4.30 × 10^14 Hz.

Solution

Using the photon energy equation E = hf:

E = (6.63 × 10^-34)(4.30 × 10^14)

E = 2.85 × 10^-19 J

Worked Example: Energy from Photon Momentum

Question: Calculate the energy of a photon with momentum 8.65 × 10^-26 kg m s^-1. Give your answer in electronvolts.

Solution

1. Using E = pc:

E = (8.65 × 10^-26)(3.00 × 10^8) = 2.60 × 10^-17 J

2. Convert to electronvolts:

E = 2.60 × 10^-171.60 × 10^-19 = 163 eV

Electronvolt (eV)

The joule is a very large unit for atomic-scale energies. A more convenient unit is the electronvolt.

Electronvolt (eV) The energy transferred to or from an electron when it moves through a potential difference of 1 volt. 1 eV = 1.60 × 10^-19 J

Common multiples:

Photon Momentum

Although photons have no mass, they have momentum. This was a surprising result of quantum theory.

p = (E/c)

Substituting E=hf:

p = (hf/c) = (h/λ) Units of Momentum

SI unit: kg m s^-1 or N s.

Common Misconception

It is often thought that because p=mv, a massless particle must have zero momentum. This is only true in classical mechanics. In relativity/quantum physics, energy and momentum are linked even for massless particles.

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