Photons

### A photon is a "particle" of light

Photons are funny things. Not funny "ha ha," but funny-sometimes-confusing.

A photon is an elementary particle of light. But as you may know, light is an electromagnetic wave, a dual wave consisting of periodic disturbances in both the electric and magnetic fields.

But viewed in the right way, those waves "collapse" into photons, small "packets" of wave energy can be measured as discrete particles in the right circumstances.

Another strange thing about photons is that while they have no mass (no resting mass), they do carry momentum, and that momentum can be measured. Any way you look at it, the photon is a paradox. Still, photons are what they are, and our job is to understand them as they exist and behave, and to model them mathematically so that we can

• better understand them,
• & use them to accomplish other goals.

Albert Einstein first proposed that light is "quantized," or that it comes in discrete "packets" that later came to be called photons, from the Greek word (phôs) for light.

Photons carry no fundamental charge, and are sometimes referred to as the force carriers of the electromagnetic field.

### Energy of a photon

The energy of a photon is proportional to the wavelength or frequency of light

$$E \propto \nu$$

where ν (the Greek letter "nu") is the frequency in Hz (1 Hz = 1 s-1). And because λ · ν = c, where c is the speed of light (c = 2.99 × 108 m/s), we also have

$$E \propto \frac{c}{\lambda}$$

The constant of proportionality is called Planck's constant, h = 6.626 × 10-34 J·s, and has units of energy × time. So we have two equations for the energy of a photon, in Joules:

$$E = \frac{hc}{\lambda} \phantom{00} \color{magenta}{\text{or}} \phantom{00} E = h\nu$$

Sometimes the frequency of a wave is given in terms of angular frequency, ω (Greek "omega"),

which references the unit circle. Angular frequency has units of radians per second (rad·s-1), so

$$\omega = 2 \pi \cdot \nu$$

When angular frequency is used, we generally use a modified version of Planck's constant, which we call "h-bar." It is

$$\hslash = \frac{h}{2 \pi}$$

Thus, the energy of a photon can also be written in terms of ℏ:

$$\bf{E = \hslash \cdot \omega}$$

So high frequency and short wavelength means higher energy. You can remember this by thinking about the number of wiggles per unit time of a wave. More wiggles means more energy.

#### Energy of a photon

The energy of a photon of light can be calculated using either the frequency (ν) or the wavelength (λ) of the light.

$$E = \frac{hc}{\lambda} \phantom{000} \color{magenta}{\text{or}} \phantom{000} E = h \nu$$

where h is Planck's constant, $h = 6.626 \times 10^{-34} \, J·s.$

### Example 1

A specific band of UV light has a frequency of   6.18 × 1015 Hz. Calculate the energy of a photon of this light.

Solution: The energy of a photon, given its frequency, is

$$E = h\nu$$

By the way, a narrow range of frequency or wavelength is often called a "band" of the EM spectrum.

So the energy delivered by one photon is

\begin{align} &= (6.626 \times 10^{-34} \, Js)(6.18 \times 10^{15} \, s^{-1}) \\[5pt] &= 4.09 \times 10^{-18} \; J \end{align}

A mole of such photons would have an energy of the order of 105 Joules, or about 100 KJ/mol, roughly ¼ of a typical single chemical bond.

### Example 2

X-ray radiation at a wavelength of 1 nm is capable of breaking bonds in the DNA of cells. Calculate the energy of a photon of λ = 1.0 nm x-ray radiation.

Solution: The energy of this x-ray photon is

$$E = \frac{hc}{\lambda}$$

Recalling that 1 nanometer (nm) is 1.0 × 10-9 Hz, we have

\begin{align} &= \frac{(6.626 \times 10^{-34} \, Js)(2.99 \times 10^8 \, \frac{m}{s})}{1.0 \times 10^{-9} \, m} \\[5pt] &= 1.98 \times 10^{-16} \; J \end{align}

That's about 1 MJ/mol, enough to break any chemical bond. That's why x-rays are called ionizing radiation.

### Wave packets

Here's one way to imagine the transition of light from wave to photon. One thing we have to remember is that light, like electrons bound to atoms, is actually neither light nor particle, but something else that defies the expectations we get from our life in the macroscopic world. Einstein is said to have imagined riding on the "front" of a light wave in order to visualize some parts of his relativity theory.

In this figure, nine waves of slightly different frequency (from 4.0 to 4.8 cycles per 2π interval) are added to form the magenta wave at the bottom. There is good overlap between peaks and troughs in the middle of the wave, and poor overlap elsewhere, resulting in a wave "packet" superimposed on a somewhat noisy background.

We can think of photons in this way, as packets of wave energy that carry their own momentum and energy, just like particles with mass, such as neutrons.

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#### The Greek alphabet

 alpha Α α beta Β β gamma Γ γ delta Δ δ epsilon Ε ε zeta Ζ ζ eta Η η theta Θ θ iota Ι ι kappa Κ κ lambda Λ λ mu Μ μ nu Ν ν xi Ξ ξ omicron Ο ο pi Π π rho Ρ ρ sigma Σ σ tau Τ τ upsilon Υ υ phi Φ φ chi Χ χ psi Ψ ψ omega Ω ω

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