Photons (4.5.1)
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In the 1800s, electromagnetic (EM) radiation was solely viewed as continuous waves.
Many phenomena involving electromagnetic radiation can be explained by a wave model:
- EM waves diffract around objects. For example, radio waves diffract around hills.
- EM waves interfere. For example, in Young’s double slit experiment, the waves passing through each slit interfere to produce an interference pattern on a screen. Young’s double slit experiment also demonstrates diffraction of EM waves.
In 1900, our understanding of electromagnetic radiation underwent a revolution. German physicist Max Planck studied the blackbody emission spectrum, which is the spread of wavelengths of EM radiation emitted by an idealised object due to its temperature.
All blackbody emission spectra have the same characteristic shape. The peak wavelength corresponds to the wavelength of light that is emitted most strongly.
The peak wavelength does not necessarily correspond to the colour we see. For example, a blackbody can appear white even if its peak emission is green because its broad spectrum includes significant red and blue light, which, when combined, the brain perceives as white.
The temperature of a blackbody determines its peak wavelength and hence its colour, as shown below for blue, green and red blackbodies, where the intensity is presented in arbitrary units, a.u.
The shape of blackbody spectra could not be explained by the wave model of EM radiation. Instead, Planck postulated the photon model, which states that EM radiation is made up of individual particles rather than continuous waves. We now know these particles as photons.
The wave model could not explain blackbody spectra because it predicted an infinite energy of radiation at low wavelengths (or high frequencies) and became known as the ultraviolet catastrophe or Rayleigh–Jeans catastrophe.
The graph below shows the blackbody spectrum predicted by the wave model (in red) compared to the experimentally observed spectrum shown in black.
Planck’s photon model was able to reproduce the observed spectrum. Planck postulated that the energy emitted at each wavelength was discrete, not continuous.
- The total energy at each wavelength is equal to the energy of the photons emitted, which are small packets of energy.
- At low wavelengths, each of these photons has such a high energy that a blackbody cannot produce them at all, explaining the dip in the spectrum.
In different circumstances, electromagnetic radiation can be considered as either a wave or a photon. Since a photon is a particle, this is known as the wave–particle duality of EM radiation.
When EM radiation propagates through space, it should be modelled as a continuous wave:
- The wave model of EM radiation describes the diffraction and interference of waves.
When EM radiation interacts with matter, it should be modelled as a particle:
- Electrons exist in energy levels in atoms. When these electrons absorb a photon, they can transition to a higher energy level. This cannot be explained by considering EM radiation as continuous waves.
Max Planck demonstrated that photons are the smallest indivisible unit of energy of an electromagnetic wave. A quantum is the smallest indivisible unit. The photon is the quantum of energy of EM radiation.
Planck believed the particulate nature of EM radiation to be a mathematical trick that explained the black body emission spectrum. He argued light was exclusively a wave.
In 1905, Einstein further developed Planck’s ideas by demonstrating that photons are real particles and that several phenomena related to EM radiation could only be explained using the photon model.
Photons are discrete wave-packets of energy.
Photons demonstrate the discrete nature of energy, meaning that energy cannot be divided into smaller portions indefinitely.
For example, an electron in an atom can absorb a photon, thereby gaining energy. The electron cannot absorb ‘part’ of the photon. It can only gain the whole energy of the photon or none at all.
As an analogy, consider a person walking up a set of stairs. His height above the ground increases in discrete steps, and he cannot be at any height in between the steps. On the other hand, if they walk up a slope, then their height increases continuously.
- On a small scale, energy is discrete, as seen in the case of an electron absorbing a photon.
- On a large scale, energy can be seen as continuous. For example, a car gains kinetic energy as it speeds up, and the kinetic energy is observed to increase continuously.
Planck’s constant is a fundamental quantity in quantum physics. Its value is:
In 1900, Max Planck introduced Planck’s constant, a fundamental concept defining the smallest unit of energy, or packet, that constitutes electromagnetic waves. These packets of energy are photons.
The energy of a photon of electromagnetic radiation is equal to:
Where:
- is energy, measured in joules ,
- is the Planck constant,
- is the frequency of the EM radiation in
The energy of a photon is directly proportional to its frequency, and Planck’s constant is the proportionality constant.
The wave equation for EM waves states that:
Where:
- is the speed of light in a vacuum,
- is the frequency of the wave in ,
- is the wavelength of the wave in
The wave equation can be rearranged to:
If we substitute this expression for frequency into the equation for the energy of a photon we get:
This equation allows the energy of EM radiation to be calculated if its wavelength is known.
The energy of a photon is inversely proportional to its wavelength:
- Short-wavelength EM radiation, such as X-rays, consists of high-energy photons.
- Long-wavelength EM radiation, such as radio waves, consists of low-energy photons.
Question walkthrough
Finding Photon Frequency from Energy in Terahertz
Calculate the frequency of a photon of red light in terahertz, given its energy, using the Planck relation.
Question walkthrough
Calculating Photon Energy from Wavelength
Calculate the energy of a photon of green light given its wavelength, using the Planck-wavelength relation.
An electronvolt is defined as the kinetic energy gained by an electron after being accelerated through a potential difference of The kinetic energy gained by a charged particle accelerated through a potential difference is given by:
Where:
- is the magnitude of the charge of the particle in
- is the potential difference in
The electron charge is so one electronvolt is equal to:
Planck’s constant, is an extremely small quantity. The energy of a photon is equal to:
Therefore, the energies in of individual photons are extremely small. The electronvolt is a more convenient unit of energy for photons. An example of this is the energy in electronvolts of a red visible light photon of wavelength is , which is easier to read and use than .
It is important to note that the following units are commonly used as units of energy:
Question walkthrough
Converting a Proton's Kinetic Energy from eV to Joules
Calculate the final kinetic energy of a proton accelerated from rest through a given potential difference, in electron-volts and joules.
Light-emitting diodes (LEDs) emit visible light when the potential difference across them exceeds the threshold voltage.
An LED emits photons in a narrow band of wavelengths. The circuit below shows how a potential difference can be applied across an LED.
Whether an LED emits light depends on the bias of the battery, which refers to the orientation of the positive and negative terminals:
- In reverse bias, no current flows through the LED, and it does not emit visible light.
- In forward bias, a current flows when the potential difference exceeds the threshold voltage. The LED in the circuit above is connected in forward bias.
For an LED in forward bias, a current flows at voltages above the threshold voltage.
The graph below shows the current–voltage curves for different colour LEDs.
- The threshold voltage is the voltage at which the curve begins to slope upwards.
- Higher frequency LEDs typically have a higher threshold voltage. Blue light has a higher frequency than green and red; therefore, a blue LED has a higher threshold voltage.
The voltage across an LED corresponds to the potential difference through which electrons in the LED are accelerated.
At the threshold voltage the energy, transferred to an electron moving through the LED is:
Where:
- is the electron charge,
- is the threshold voltage in
This is approximately equal to the energy of each photon emitted by the LED, which is also given by:
Where:
- is Planck’s constant,
- is the speed of light,
- is the wavelength of the photons in
Equating these two energies gives:
Therefore, Planck’s constant can be calculated by measuring the threshold voltage for an LED which emits a known wavelength of light.
Although Planck’s constant can be estimated using a single LED, a more accurate value can be obtained by measuring the threshold voltage for multiple LEDs with different emission wavelengths.
A circuit to measure the threshold voltage across an LED is shown below:
- An ammeter is connected in series with the LED to measure the current flowing through it.
- A safety resistor is connected in series with the LED to limit the current.
- A potentiometer is connected in parallel to the LED to allow for continuous voltage variation.
- A voltmeter is connected in parallel to the LED to measure the voltage across it.
- A battery produces a potential difference to drive current through the circuit.
The graph below shows an example of some LED threshold voltages plotted against their inverse wavelengths:
The error bars are due to the uncertainty in the exact voltage at which the current begins to increase.
The threshold voltages are related to the emission wavelengths by:
Therefore, the gradient is:
This can be used to calculate the planck constant