Energy and Momentum of a Photon: Key Concepts Simplified

Photons were predicted to form between 10 seconds and 370,000 years after the Big Bang, which was considered the epoch of light formation. As the bearers of electromagnetic waves, photons are massless particles that are generally thought of as unit of light. Being a light package, it travels at the speed of light, having zero rest mass but discrete energy. As postulated by Einstein, the corpuscular behavior of radiation gives the idea of a photon with no successive classical theory to describe this nature of light.

Planck’s theory of radiation and Einstein’s photoelectric effect independently pointed towards the discrete energy bags, which were later demonstrated by the Compton scattering experiment. The principles of quantum mechanics emerged from the photon hypothesis, which posits that a particle cannot concurrently exhibit both mass and wave properties while still attributing proportional momentum to the light particle. Subsequently, photons move periodically throughout space with varying energy, producing high-frequency gamma-rays to X-rays and low-frequency infrared to radio waves with the same speed as that of light.

Wave-Particle Duality of Light

The wave and particle parameters (energy E and momentum P) of a photon are related as:

E = հν

P = հ/λ (equation 1)

Where, հ is called the Planck constant having value 6.626 Χ 10^-34 J s, ν is the frequency and λ being the wavelength of the radiation.

The discovery of thin lines in both the emission and absorption of atomic spectra was resolved by quantum mechanics, which established discrete energy values:

E_n= հν_n…, n = 1, 2, ,…     (equation 2)

And by energy conservation, the excited photon could have a frequency such that

hν_mn= І E_m –  E_n І (equation 3)

Equations (2) and (3) concluded with the result of discrete energy.

De Broglie in 1923 hypothesized that every moving particle, regardless of its nature, shows wavelike fashion as (1). Thus, the matter of mass ‘m’ acquires momentum p with an associated wavelength given as:

λ = հ/mν (equation 4)

The de Broglie wave approach indicates that matter in motion involves two different velocities: one for mechanical motion (v) and the other for wave propagation (u).  The relationship between two velocities is expressed as,

u = c²/v (c is the speed of light, 3 Χ 10 ⁸ ms^-1) (equation 5)

For the non-relativistic case, energy E of a particle is associated with momentum by 

E = p²/2m, OR  p = (2mE) ^1/2 (equation 6)

The particle parameter p and wave parameter λ are correlated to each other by the de Broglie relation, which applies to matter and waves alike. This wave theory led to the development of wave mechanics.

Energy and Momentum of a Photon

Max Planck developed the quantum theory of radiation in 1900, which was innovative at the time. As per the theory, a photon’s energy is proportional to its frequency with հ constant known as Planck’s constant. 

E = հf (equation 7)

Here f is the frequency of radiation (photon)

Similarly, the frequency of a photon traveling with the velocity c and wavelength λ is given by f = c/λ (equation 8)

Hence equation (1) can be written as

E = հc/λ

λ = հc/E (equation 8)

Substituting this λ in equation (1) we get P = E/c (equation 9)

This relation of energy and momentum concluded that even if the mass of a particle is zero, it still obtains a proportional momentum.

Relation (1) can be expressed as,

E = ħω, P = ħk (equation 10)

Where, ħ is the Dirac constant with ħ = հ/2π = 1.0546 Χ 10^-34 J s

Considering equations (1) and (6), we get three different cases of mass, energy, and wavelength.

Case I: For particle moving with non-relativistic speed and energy E,

λ = հ/ (2mE) ^1/2 (equation 11)

Case II: For charged particles with charge q,

λ = հ/ (2mqV) ^1/2 (V is the voltage measured in volts) (equation 12)

Case III: For thermally excited particle

λ = հ/ (3mkT) ^½ (equation 13)

Here, T is the temperature and k is the Boltzmann constant whose value is 1.38 Χ 10^-23 Jk^-1.

Relativistic Perspectives

Special relativity explains the mass of a body in motion, often known as relativistic mass. Mathematically,

m = ϒm₀ (equation 14)

Where ϒ = 1 / (1 – v²/c²)^1/2, v is the velocity of a relative observer. Also relativistic energy is given by E = ϒm₀c² When compared to the velocity of light, v is insignificant or equal to zero.. Therefore, ϒ will be 1 for photons, and we will have m = m₀. As a result, we have the energy Furthermore, the relativistic energy-momentum expression is written as

E² = p²c² + m²c⁴ (equation 15)

According to special relativity theory, a particle traveling at the speed of light must have a rest mass of zero. Therefore, m = 0 for photons and equation (15) becomes,

E = pc

OR

p = E/c

Consequently, relativity demonstrates that even a massless particle may gain momentum.

Photon Interactions with Matter

Photons do not interact within themselves, so they are not magnetic. On the contrary, they interact otherwise in matter than charged particles. Unlike charged particles, photons do not drop energy constantly as they move through matter. Charged particles, often electrons, accomplish energy when photons encounter them. The charged particles afterwards lose energy along secondary interactions, primarily ionization. The photon interactions with matter are random. The photon energy, atomic number, and electron count of the matter that absorbs them determine the likelihood of a photon interaction. We focus on three predominant photon interactions below. 

Photoelectric Effect

The photoelectric effect for photon interactions was explained by Einstein’s 1905 postulation of light packet energy. The argument states that electrons are likely liberated from a metal surface when light of a particular frequency strikes it. The intensity of light doesn’t affect this electron emission. In order for electrons to exit a metal with a work function φ, հν must be greater than the work function (հν > φ). When the electrons escape, their maximal kinetic energy, E_e, is such that 

E_e = հν – φ (equation 16)

Ee is hence reliant on radiation frequency “v.” This is known as the Einstein photoelectric effect, which is in obedience with the available evidence.

Compton Effect

In 1924, Compton proved that radiation is a particle by examining how free or weakly bound electrons scatter X-rays. When a photon with a certain energy and momentum hits a stationary electron, some of the energy is imparted to the electron that is loosely bounded. The photon shifts in direction and gives up energy. 

Let the initial momentum of a photon be p₁ and the energy p₁c that interacts with an electron of mass m and momentum p_e that is at rest. Following the collision, the photon acquires momentum p₂ and energy p₂c. By the momentum conservation, p₁ = p₂ + p_e, we get

P_e² = p₁² + p₂² – 2p₁p₂cosθ (equation 17)

where θ represents the angle between the scattered photon and the incident photon’s direction. Using energy conservation,

p₁ + mc = p₂ + (p_e² + m²c²)^1/2 (equation 18)

Excluding p_e² from (17) and (18), we get

mc (p₁ – p₂) = 2p₁p₂sin²1/2θ (equation 19)

We derive the Compton formula in reference with equation (9)

∆λ = λ₂ – λ_{1 =} (2հ/mc) sin² 1/2θ (equation 20)

Equation (20) is completely consistent with the experiments. The wavelength shift caused by scattering is determined by θ. This outcome emerges from defining the photon as a particle with momentum and energy, verifying wave-particle duality.

Pair production

Photons interact with the electric field of an atom’s nucleus in the pair production. The photon vanishes entirely, leaving only an electron and a positron. In this method, high-energy gamma rays are absorbed in materials. To produce an electron-positron pair, the energy of a photon must be a minimum equal to the mass of two electrons, as electrons and positrons have the same mass. A single electron has a mass equivalent to 0.511 MeV of energy E as calculated from the equation E = mc². In order to generate two electrons, the photon energy must be no less than 1.022 MeV.  Consequently, those electrons and positrons can vary in their kinetic energy, which they will give off by ionization or excitation.. Thus produced electrons and positrons may differ in their kinetic energy, which they will give up via ionization, excitation, or bremsstrahlung. The positron eventually annihilates itself and an electron, perhaps after losing all kinetic energy. This produces two 0.511 MeV annihilation photons.  Once the positron comes to rest before annihilation, it gives off two photons that are opposite directions (180° apart)

Conservation Laws in Photon Processes

Conservation laws have been thoroughly researched and widely implemented. In a similar fashion, photons obey conservation principles since they cannot be achieved or shattered down. Given that mass and energy are analogous, interactions can occur in which energy is turned into mass, as in pair production, where photon energy is converted to particles (electron and positron).  The converse is also valid. Uranium disintegration creates particles with different masses than the parent uranium atom since some of the mass has already been turned into energy. The Compton experiment revealed energy and momentum conservation. After interacting with an electron, a photon with a certain energy and momentum encounters a drop in energy and momentum, which is transmitted to matter.

Applications of Photon Energy and Momentum

As light is the essential source of energy, photons are the essential sources of all existent energy forms. The daylight can be changed into different forms of modern technologies. A few of the most noticeable employments are sstated below.

Solar cells and photovoltaic

A sun oriented cell, moreover known as a photovoltaic cell (PV), changes daylight into an electric current when uncovered to radiation specifically. It is a non-mechanical, semiconductor-based machine. When photons reach a PV cell, they bounce off it, travel through it, or are ingested by the semiconductor fabric. The ingested photons contribute energy for the generation of electricity. When sufficient light is collected on the semiconductor, electrons stream out from its atoms.

The motion of electrons toward the front surface of the PV cell produces a rise of electrical charge between the cell’s front and rear surfaces, resulting in a potential difference equivalent to the negative and positive terminals of a battery. Electrons are absorbed by the PV cell’s electrical conductors. As the conductors of an electrical circuit are coupled to a power supply, electricity flows through them.

Lasers and optical technologies

A laser formation process involves the energy absorption by electrons in the atoms from an electrical current or a light. Thus, increased energy excites the electrons, causing them to migrate from a lower energy level to a higher energy level around the nucleus. A solid medium, such as glass or gas, is employed to transport excited electrons to the coherent photon beam. Light’s wavelength is governed by the amount of energy produced when an excited electron loses energy to a lower energy level. A preferred color beam is generated based on the energy level, which may be customized to the material in the optical absorption medium. 

A mirror on the other side of the optical material reflects the photon back and forth to the electrons. The gap between mirrors is chosen so that the photon required for the specific type of optical gain media is returned to the medium, stimulating the emission of a nearly identical copy of that photon. They each proceed in the same direction and speed, rebounding against another mirror on the other side to continue the procedure. Photons multiply until they are intensified sufficiently to pass through mirrors and optical device with ideal balance.

Radiation pressure and solar sails

Despite the pressure of electromagnetic radiation being extremely poor, a sufficiently huge surface may generate pressure due to the momentum of radiation that strikes on it. This radiation pressure is employed for spaceship propelling, also known as solar sailing. A key factor for practical applications is that such solar sails are constructed of lightweight yet robust elements.

The radiation pressure factor can potentially be used to determine the optical power of an intense laser beam. Several laser cooling technologies also utilize light pressures on atoms of matter.

Experimental Evidence

Before Einstein and Compton, Thomas Young conducted an experiment in 1801 that demonstrated an interference pattern of light, which was further improved for every single photon. That well-known double-slit experiment demonstrated that photons have both wave-like and particle-like features. Davisson and Germer’s experiment on electron diffraction, conducted in 1927, demonstrated that matter had wave qualities. 

Today several techniques have been developed to detect single and multiple photons precisely. Photoelectric cells were early photon detectors that measured light intensity. Tragically, they were missing both sensibility and amplification capability. In the 20th century, photomultiplier tubes (PMT) were discovered for the specific photon measurement, which is based on the photoelectric effect. The photosensitive cathode of the tube has been optimized so that the electrons emitted from the light interaction are amplified by an ongoing series of dynodes, allowing for the detection of single photons. Advanced photodiodes adopt the same strategy to detect photons. Superconducting Nanowires are also single-photon detectors that function at frigid temperatures.

Photons are diligently detected using a broad spectrum of quantum techniques. Quantum Efficiency Measurement is a technique for evaluating the efficiency of photon detection in various materials, which is vital to enhance detectors. The quantum state of a photon can possibly be reconstructed by taking a sequence of measurements in unique bases using tomography. Photons are detected using quantum dot sensors, which allow for extremely precise and adaptable light measurements.

Certain interferometry approaches, such as Michelson interferometry and Twiss interferometry, are utilized for exploring phase shift and the stochastic character of photons. Meanwhile, tactics such as time-correlated single photon counting and Hanbury Brown algorithms are employed to investigate photons’ statistical characteristics. There also exist photon number resolution methodologies that can distinguish amidst multiple numbers of photons. 

Photons in Quantum Mechanics

The experimental recognition of photons is a cornerstone of quantum mechanics. All of quantum theory’s proposals rely on the notion of wave-particle duality. A photon’s state can neither be detected nor localized, but it may be correlated, resulting in the entanglement hypothesis, which is also employed in quantum computing. The contact of a photon with matter causes in the annihilation and multiplication of matter. This combines the classical concepts of electric and magnetic fields with quantum physics of radiation.

 In the realm of particle physics, photons are one of the elementary subatomic particles that transport force throughout the cosmos. Photons have been accepted as the intermediaries of electromagnetic forces between charged particles in quantum electrodynamics. The electromagnetic interaction on a massive surface exerts a force following radiation pressure. It assumes that the force is quantized and shifted away by photons, also known as gauge bosonic particles.

Concluding Remarks

We must acknowledge that photons constitute the universe’s principal energy source. Photons hold a profound effect on particles ranging from subatomic to galactic. The photon-matter interaction is a prerequisite for the breakdown of energy and particle formation. The uncertainty in the quantum state of particles assists the existence of distinct energies at multiple positions in the exact time. However, the conceptual framework is constrained by the inability to pinpoint the particles’ exact position.

We are gradually entering an unprecedented time when quantum physics is governing through the study of photons. The dual nature of photons is being investigated for teleportation and telepathy, and the outcome will herald in an entirely novel evolution of science and technology. On the contrary, the experimental results may be deferred due to the insensitivity of the detecting equipment. The improvement in measurement sensibility may be substantial.

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