KAHIBARO
Table of Contents
Light as Packets of Energy
In classical physics, light was often treated as a wave. In quantum physics, light also has a particle-like aspect. The particle of light is called a photon. The key idea of this chapter is that each photon carries a definite amount of energy.
This energy is not arbitrary. It depends on the light's frequency. Higher-frequency light has more energetic photons, and lower-frequency light has less energetic photons.
The Photon Energy Formula
The energy of a photon is given by Planck's relation,
$$E = hf$$
where $E$ is the photon energy, $h$ is Planck's constant, and $f$ is the frequency of the light.
Planck's constant is
$$h = 6.626 \times 10^{-34}\ \text{J s}$$
This is a very small number, which is why quantum effects are not obvious in everyday life.
Since the speed of light satisfies $c = f\lambda$, we can also write photon energy in terms of wavelength:
$$E = \frac{hc}{\lambda}$$
where $\lambda$ is the wavelength and $c$ is the speed of light.
Important formulas for photon energy:
$$E = hf$$
$$E = \frac{hc}{\lambda}$$
Photon energy is directly proportional to frequency and inversely proportional to wavelength.
What the Formula Means
These formulas show an important physical rule. Blue light has a higher frequency than red light, so a blue photon has more energy than a red photon. Ultraviolet photons have even more energy, and X ray photons have much more.
This does not mean that brighter light always has higher-energy photons. Brightness can also mean that there are more photons. A beam can be intense because it contains many low-energy photons, or because it contains fewer high-energy photons.
Energy and Frequency
The relationship between energy and frequency is linear. If the frequency doubles, the energy of each photon also doubles.
The table below shows the trend clearly.
| Quantity | Symbol | Relation to photon energy |
|---|---|---|
| Frequency | $f$ | Higher $f$ means higher $E$ |
| Wavelength | $\lambda$ | Higher $\lambda$ means lower $E$ |
| Planck's constant | $h$ | Constant of proportionality |
Units of Photon Energy
If SI units are used, photon energy comes out in joules. In atomic and quantum physics, another unit is often more convenient, the electron volt, abbreviated eV.
One electron volt is
$$1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$$
Because photon energies are often very small in joules, electron volts make the numbers easier to read.
For example, visible light photons typically have energies of a few electron volts.
Useful conversion:
$$1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$$
When converting from joules to electron volts, divide by $1.602 \times 10^{-19}$.
Photon Energy and Color
Visible light covers only a small range of frequencies and wavelengths. Within this range, different colors correspond to different photon energies.
| Color | Wavelength trend | Energy trend |
|---|---|---|
| Red | Longer wavelength | Lower energy |
| Green | Intermediate wavelength | Intermediate energy |
| Blue | Shorter wavelength | Higher energy |
| Violet | Shortest visible wavelength | Highest visible energy |
So color is directly connected to the energy of a single photon.
Example Calculation with Frequency
Suppose light has frequency
$$f = 5.0 \times 10^{14}\ \text{Hz}$$
Its photon energy is
$$E = hf = (6.626 \times 10^{-34})(5.0 \times 10^{14})$$
$$E = 3.31 \times 10^{-19}\ \text{J}$$
Now convert to electron volts:
$$E = \frac{3.31 \times 10^{-19}}{1.602 \times 10^{-19}} \approx 2.07\ \text{eV}$$
So each photon carries about $2.1\ \text{eV}$ of energy.
Example Calculation with Wavelength
Suppose light has wavelength
$$\lambda = 600\ \text{nm} = 6.00 \times 10^{-7}\ \text{m}$$
Then
$$E = \frac{hc}{\lambda}$$
$$E = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{6.00 \times 10^{-7}}$$
$$E \approx 3.31 \times 10^{-19}\ \text{J}$$
which is again about
$$2.07\ \text{eV}$$
Why Photon Energy Matters
Photon energy helps explain many quantum phenomena. When light interacts with matter, it does not always transfer energy continuously. Instead, energy is exchanged in discrete amounts, one photon at a time.
If an atom absorbs light, it absorbs the energy of whole photons. If the photon energy matches an allowed atomic energy change, absorption can occur. If not, the interaction may not happen in the same way. This idea is central to atomic spectra, the photoelectric effect, and many other quantum effects.
A Simple Picture
You can imagine a beam of light as a stream of tiny packets. Each packet carries energy $E = hf$. Changing the frequency changes the energy in each packet. Changing the brightness mainly changes how many packets arrive each second.
Summary
A photon is a quantum of light, and its energy is determined by frequency. The fundamental relations are $E = hf$ and $E = hc/\lambda$. Higher frequency means higher photon energy, and longer wavelength means lower photon energy. This simple idea is one of the foundations of quantum physics.
Core idea:
Light energy comes in discrete packets called photons.
Each photon has energy
$$E = hf = \frac{hc}{\lambda}$$
This is one of the most important formulas in modern physics.