No Boundaries: Can Photons Coexist in One State?

PHYSICXION: Photons, unlike us, don't care about personal space. They can sit in the same "seat"—or quantum state—without any issues, happily coexist.

No Boundaries: Can Photons Coexist in One State?

Photons, unlike us, don't care about personal space. They can sit in the same "seat"—or quantum state—without any issues, happily coexisting without bumping elbows. In the quantum world, this ability is part of what makes them so unique. Governed by the laws of Bose-Einstein statistics, photons are bosons, which means they’re inherently social particles, perfectly at ease piling up in the same quantum state. This strange yet fundamental behavior leads to phenomena like lasers, where countless photons act in unison, or the glow of the cosmic microwave background, a testament to their cooperative nature.

But why can they do this? And how does this ability distinguish photons from their fermion counterparts, like electrons, who absolutely refuse to share? To explore this fascinating quantum trait, we’ll dive into the mathematics of identical particles, Bose-Einstein condensation, and the extraordinary world of photons—two “friends” sharing the same seat in perfect harmony.

What is Photon?

A photon is a truly fascinating and unique entity. It’s not just a particle or a wave—it’s both, depending on how we observe it. At its core, a photon is a quantum of light or electromagnetic radiation, meaning it’s the smallest indivisible unit of energy for that radiation. Photons are what make the universe visible to us, yet they carry a kind of elegance and mystery that goes far beyond simple visibility.

Here’s what makes a photon so unique,

1. The Building Block of Light

Photons are the fundamental particles that makeup light and all forms of electromagnetic radiation, from the warm glow of a candle to the high-energy gamma rays from distant stars. Unlike matter particles (like electrons or protons), photons have no mass—they are purely energetic traveling at the speed of light, about

3 \times 10^8 \, \text{m/s}

2. Dual Nature: Particle and Wave

Photons defy classical definitions. They behave like particles when we detect them (e.g., they hit a screen one at a time), yet they act like waves when they spread out and interfere in experiments like the famous double-slit experiment. This dual nature is a cornerstone of quantum mechanics and a reminder that the universe doesn’t always fit neatly into our intuitive categories.

3. Messengers of Energy and Information

Each photon carries energy, which depends on its frequency or wavelength. High-energy photons, like X-rays and gamma rays, have short wavelengths, while low-energy photons, like radio waves, have long wavelengths. Their energy is given by the famous equation:

E = h ν

where

E

is the photon’s energy,

h

is Planck’s constant, and

ν

is the frequency of the wave.

In this way, photons act as messengers, carrying energy and information across vast distances—from the surface of the Sun to your eyes in just 8 minutes!

4. No Mass, No Rest

One of the most fascinating things about photons is that they have no rest mass. This is why they can travel at the speed of light, which is the ultimate speed limit of the universe. Unlike most particles that slow down or stop, photons are always in motion. They’re the cosmic sprinters, constantly zipping around, powering life on Earth through sunlight and enabling technologies like lasers and fiber optics.

5. Born from the Universe Itself

Photons are created in countless ways: when charged particles accelerate, when atoms transition between energy levels, during nuclear reactions, or even from the annihilation of matter and antimatter. Every photon you see is a glimpse of some underlying process in the universe, whether it’s the warm glow of a lamp or the ancient light from the Big Bang that still travels as cosmic microwave background radiation.

6. Quantum Oddities

Photons are quantum entities, which means they break a lot of classical rules. For example:

Superposition: A photon can exist in multiple states until observed.
Entanglement: Two photons can share a connection across vast distances, so that measuring one instantly affects the other—a phenomenon Einstein called “spooky action at a distance.”
No Pauli Exclusion Principle: Photons, as bosons, can share the same state, allowing for phenomena like laser light, where trillions of photons act in perfect unison.

7. Timeless Travelers

Because photons move at the speed of light, time effectively stops for them in their own frame of reference. From the photon’s perspective, it exists in a kind of eternal present, traveling from one end of the universe to another without experiencing time.

8. Life and Technology Depend on Them

Without photons, life as we know it wouldn’t exist. They power photosynthesis, provide warmth, and allow us to see the world. Beyond nature, photons have revolutionized technology—from lasers and solar panels to quantum computing and medical imaging.

Photon as a member of the Boson family

In particle physics, the photon is classified as a gauge boson, a specific type of boson that mediates one of the fundamental forces of nature. To fully understand its classification, let’s break down the broader context of particle physics and how photons fit into the framework of bosons.

1. Bosons and Their Role in Nature

Particles in the universe are broadly categorized into fermions and bosons based on their quantum spin:

Fermions: These have half-integer spin ( $,$ ) and obey the Pauli Exclusion Principle, meaning no two fermions can occupy the same quantum state.
Bosons: These have integer spin ( $0, 1, 2, \dots,$ ) and do not obey the exclusion principle, allowing multiple bosons to occupy the same quantum state.

Bosons are responsible for mediating forces between particles, making them essential for the interactions that govern the universe. Photons fall into this bosonic category but have unique properties that place them in the subcategory of gauge bosons.

2. The Photon as a Gauge Boson

Gauge bosons are a special class of bosons that mediate the fundamental forces of nature, as described by the Standard Model of particle physics. These forces arise from the concept of gauge symmetries, which are mathematical symmetries underlying the interactions of particles. The photon is the gauge boson associated with the force of electromagnetism, one of the four fundamental forces.

Electromagnetism and the Photon

The electromagnetic force governs interactions between charged particles, such as the repulsion between two electrons or the attraction between an electron and a proton.
The photon is the "carrier" or mediator of this force. When charged particles interact, they exchange photons, which transfer energy and momentum between them.
This exchange happens in quantum field theory, where photons are the quanta (smallest discrete units) of the electromagnetic field.

3. Properties of the Photon as a Gauge Boson

The photon’s classification as a gauge boson gives it specific characteristics that distinguish it from other particles:

Spin: The photon has a spin of $1$ , making it a vector boson (a type of gauge boson with spin-1).
Massless: Photons are unique among gauge bosons in that they are completely massless. This property allows them to travel at the speed of light ( $c$ ) in a vacuum.
Charge: The photon is electrically neutral, meaning it doesn’t directly interact with itself (unlike gluons in the strong force).
Long Range: Because photons are massless, the electromagnetic force they mediate has an infinite range, enabling interactions over vast distances (e.g., sunlight reaching Earth from the Sun).
Gauge Symmetry: The photon is the gauge boson of the U(1) gauge symmetry in the Standard Model, corresponding to the electromagnetic field.

4. Comparison to Other Gauge Bosons

The photon shares similarities with other gauge bosons in the Standard Model but also has key differences:

Weak Force Bosons ( $W^\pm$ and $Z^0$ ): These bosons mediate the weak nuclear force but are massive, giving the weak force a very short range.
Gluons: Gluons mediate the strong nuclear force between quarks. Unlike photons, they are massless but interact with each other due to their color charge.
Graviton (Hypothetical): The graviton, if it exists, would be the gauge boson for gravity and would share the photon’s massless property, though it would have a spin of $2$ .

5. Beyond the Standard Model

In modern extensions of the Standard Model, the photon is still treated as a fundamental gauge boson. However, new theories, like quantum gravity and string theory, suggest it might have additional interactions or emerge from deeper principles of symmetry and geometry in higher-dimensional spaces.

For example:

In supersymmetry (SUSY), photons may have a corresponding superpartner called the photino.
In grand unified theories (GUTs), photons are unified with other gauge bosons at extremely high energy scales.

6. Role of Photons in Quantum Electrodynamics (QED)

The photon is central to Quantum Electrodynamics (QED), the quantum field theory describing electromagnetic interactions. In QED:

Photons are exchanged as virtual particles in Feynman diagrams, representing electromagnetic forces between charged particles.
The theory’s precision in describing phenomena like the Lamb shift and the magnetic moment of the electron showcases the photon’s critical role.

7. Unique Status of the Photon

What makes the photon so special among bosons is its massless nature and its universal role in mediating the electromagnetic force. Unlike other forces, electromagnetism governs the behavior of charged particles across all scales—from the atomic level to cosmic scales.

How the law of occupying space is different?

The Pauli Exclusion Principle is a fundamental concept in quantum mechanics, but it applies only to fermions (particles with half-integer spin, like 1/2,3/2 ,… ).

To understand its relationship with bosons, we need to explore the distinction between fermions and bosons and why bosons are not subject to this principle.

What is the Pauli Exclusion Principle?

The Pauli Exclusion Principle states that no two identical fermions can occupy the same quantum state simultaneously in a quantum system. This principle is a consequence of the antisymmetric nature of the wave function for fermions.

Wavefunctions of Fermions:

The total wavefunction

Ψ

for a system of identical fermions must be antisymmetric under the exchange of any two particles. Mathematically, this is expressed as:

Ψ (x_{1}, x_{2}) = - Ψ (x_{2}, x_{1})

If two fermions occupy the same quantum state, swapping their positions does not change the system, and the wavefunction would equal zero:

\Psi(x_1, x_1) = 0

This means it’s impossible for two fermions to have the same set of quantum numbers.

What About Bosons?

Bosons, unlike fermions, have integer spin (

0

,

1

,

2

,

\dots

) and follow entirely different quantum statistics: the Bose-Einstein statistics. They are not subject to the Pauli Exclusion Principle because their wavefunctions are symmetric under particle exchange.

Wavefunctions of Bosons:

For bosons, the total wavefunction

Ψ

must be symmetric:

Ψ (x_{1}, x_{2}) = Ψ (x_{2}, x_{1})

This symmetry means that not only can bosons occupy the same quantum state, but they also have a tendency to do so, especially at low temperatures. This behavior is at the core of phenomena like Bose-Einstein condensation.

Why the Difference?

The difference arises from the quantum mechanical property called spin:

Fermions: Have half-integer spin (e.g., electrons, protons, neutrons). Their antisymmetric wavefunctions lead to the exclusion principle.
Bosons: Have integer spin (e.g., photons, gluons, Higgs bosons). Their symmetric wavefunctions allow multiple bosons to exist in the same state.

The spin-statistics theorem in quantum field theory explains why particles with half-integer spin are fermions and obey the exclusion principle, while particles with integer spin are bosons and do not.

Key Implications for Bosons and Fermions:

Fermions and the Structure of Matter: The Pauli Exclusion Principle is why matter has structure. For example:
- Electrons in an atom cannot all occupy the same energy state, leading to the formation of distinct atomic orbitals and the periodic table of elements.
- Neutron stars resist gravitational collapse due to the degeneracy pressure from the exclusion principle applied to neutrons.
Bosons and Force Mediation: Bosons, such as photons and gluons, mediate the fundamental forces of nature. Their ability to share quantum states enables them to act collectively:

In lasers, photons amplify coherently because they can occupy the same state.
In Bose-Einstein condensates, a large number of bosons occupy the same lowest-energy state, resulting in macroscopic quantum phenomena.

Why no boundaries?

Identical Particles and Wavefunctions

In quantum mechanics, photons are indistinguishable particles, meaning that their identities cannot be tracked individually. When describing identical particles like photons, their collective quantum state is represented by a symmetrized wavefunction. This symmetrization means the wavefunction remains unchanged when you swap two photons, which mathematically allows the particles to coexist in the same place and quantum state.

For bosons:

Ψ (x_{1}, x_{2}) = Ψ (x_{2}, x_{1})

This symmetry ensures that photons can "pile up" in the same state, unlike fermions, whose wavefunction is antisymmetric:

\Psi(x_1, x_2) = -\Psi(x_2, x_1)

Spatial Wavefunction Overlap

Photons are described by quantum wavefunctions. The spatial wavefunction for a single photon in a quantum state

∣

ψ

⟩

is given by:

ψ (x) = ⟨ x ∣ ψ ⟩

For two photons, if they occupy the same state

∣

ψ

⟩, their total wavefunction is:

Ψ (x_{1}, x_{2}) = ψ (x_{1}) ψ (x_{2})

Here:

$x_1$ and $x_2$ are the positions of the two photons.
The product $\psi(x_1)\psi(x_2)$ allows the photons to occupy the same position $x_1 = x_2$ , since there is no exclusion principle preventing it.

Symmetric Wavefunction and Particle Overlap

Now, let's consider the symmetrization explicitly. The two-particle wavefunction for bosons must satisfy:

Ψ (x_{1}, x_{2}) = \frac{1}{\sqrt{2}} [ψ_{1} (x_{1}) ψ_{2} (x_{2}) + ψ_{1} (x_{2}) ψ_{2} (x_{1})]

If both photons are in the same state, say

ψ

(

x

)

, then:

Ψ (x_{1}, x_{2}) = \frac{1}{\sqrt{2}} [ψ (x_{1}) ψ (x_{2}) + ψ (x_{2}) ψ (x_{1})] = ψ (x_{1}) ψ (x_{2})

This result shows that the photons' wavefunctions fully overlap, meaning the two photons can occupy the same spatial location and quantum state simultaneously.

Statistical Mechanics Perspective

From the viewpoint of statistical mechanics, Bose-Einstein statistics describe the distribution of bosons over energy states at a given temperature. Unlike classical particles that have independent states, bosons exhibit a tendency to cluster into the same state when possible, particularly at low temperatures. This phenomenon leads to effects like Bose-Einstein condensation, where a large number of bosons occupy the same quantum state.

The Bose-Einstein distribution function describes how bosons are distributed among quantum states. For a system of bosons at thermal equilibrium, the average number of particles in a quantum state with energy

ϵ

is given by:

n (ϵ) = \frac{1}{e^{(ϵ - μ) / k_{B} T} - 1​}

where:

$\epsilon$ is the energy of the state,
$\mu$ is the chemical potential (for photons, $\mu = 0$ ),
$k_B$ is Boltzmann's constant, and
$T$ is the temperature.

Since the denominator can approach zero as

(

ϵ

-

μ

)

\to

0,

n

(

ϵ

)

can grow arbitrarily large. This shows that there is no limit to how many bosons (e.g., photons) can occupy the same quantum state.

For photons, this clustering means they can occupy the same place in space while sharing identical properties, such as energy, momentum, and polarization.

Photon Statistics and Bose-Einstein Condensation

When multiple photons occupy the same quantum state, they exhibit collective behavior. For example, in a laser:

All photons occupy the same quantum state ( $x, k, E, etc.$ ).
The photon occupation number $n (ϵ)$ becomes macroscopically large for a single quantum state.

This is a direct consequence of Bose-Einstein statistics.

Superposition and Constructive Interference

If the photons share the same wavefunction

ψ

(

x

)

, their probability distributions also overlap.

The total probability density is:

∣ Ψ (x_{1}, x_{2}) ∣^{2} = ∣ ψ (x_{1}) ψ (x_{2}) ∣^{2}

For

x

1

=

x

2

, this is nonzero and constructive, allowing two photons to coexist at the same position.

Physical Manifestations

The ability of photons to occupy the same quantum state has practical and observable consequences:

Laser Light: In a laser, photons are all in the same quantum state, resulting in coherent light where all photons share the same phase, direction, and energy.
Superposition of Waves: In classical terms, the superposition principle of electromagnetic waves (which photons compose) explains how photons in the same place and state constructively interfere.

Why This Does Not Violate Physics

For bosons like photons:

The concept of "same place" refers to overlapping spatial wavefunctions. Their quantum mechanical nature allows their probability distributions to coexist without contradiction.
Energy, momentum, and other quantum numbers of the photons do not violate conservation laws when they share a state.

In contrast, fermions (e.g., electrons) cannot share the same state due to the Pauli exclusion principle, a restriction absent for bosons.

Simplified mathematical demonstration with two photons just as an example of the wonderful phenomena

Let’s work through an example showing how identical bosons, such as photons, can coexist in the same quantum state. We'll focus on a particle-in-a-box scenario in quantum mechanics, which is a simple yet powerful framework.

Here’s the plan:

Derive the energy levels for a single particle in a 1D box of length $a$ .
Show how two photons (bosons) can occupy the same quantum state.
Calculate the ground state energy and the total energy when two photons occupy the ground state.
Extend the calculation to the first and second excited states.

1. Single Particle in a 1D Box: Energy Levels

For a single particle in a 1D box of length

a

, the energy levels are given by:

E_{n} = \frac{n^{2} π^{2} ℏ^{2}}{2 m a^{2​}}

where:

$n = 1, 2, 3, \dots$ is the quantum number,
$m$ is the mass of the particle,
$a$ is the length of the box,
$\hbar$ is the reduced Planck constant.

The wavefunction for the particle is:

ψ_{n} (x) = \sqrt{\frac{2}{a}} \sin (\frac{n π x}{a})

This describes the allowed standing waves in the box.

2. Two Identical Photons in a Box

Since photons are bosons, their total wavefunction is symmetric under particle exchange, and multiple photons can occupy the same quantum state. When two photons are in the same state

n

, their total energy is:

E_{total} = 2 E_{n} = 2 \cdot \frac{n^{2} π^{2} ℏ^{2}}{2 m a^{2​}}

Ground State (Both Photons in $n = 1$ ):

If both photons occupy the ground state (

n

=

1

), the energy of the ground state is:

E_{1} = \frac{π^{2} ℏ^{2}}{2 m a^{2​}}

The total energy of the system is:

E_{total, ground} = 2 E_{1} = 2 \cdot \frac{π^{2} ℏ^{2}}{2 m a^{2}} = \frac{π^{2} ℏ^{2}}{m a^{2​}}

First Excited State:

For the first excited state, the two photons can either:

Both occupy the $n = 2$ state, or
Occupy different states, e.g., one in $n = 1$ and the other in $n = 2$ .

Case 1: Both in $n = 2$

The energy of the

n

=

2

state is:

E_{2} = \frac{4 π^{2} ℏ^{2}}{2 m a^{2​}}

The total energy is:

E_{total, excited (1)} = 2 E_{2} = 2 \cdot \frac{4 π^{2} ℏ^{2}}{2 m a^{2}} = \frac{8 π^{2} ℏ^{2}}{2 m a^{2​}}

Case 2: One in $n = 1$ , One in $n = 2$

The total energy is:

E_{total, excited (2)} = E_{1} + E_{2} = \frac{π^{2} ℏ^{2}}{2 m a^{2}} + \frac{4 π^{2} ℏ^{2}}{2 m a^{2}} = \frac{5 π^{2} ℏ^{2}}{2 m a^{2​}}

Second Excited State:

For the second excited state, the two photons can either:

Both occupy $n = 3$ ,
Occupy $n = 2$ and $n = 3$ , or
Occupy $n = 1$ and $n = 3$ .

Case 1: Both in $n = 3$

The energy of the

n

=

3

state is:

E_{3} = \frac{9 π^{2} ℏ^{2}}{2 m a^{2​}}

The total energy is:

E_{total, excited (3)} = 2 E_{3} = 2 \cdot \frac{9 π^{2} ℏ^{2}}{2 m a^{2}} = \frac{18 π^{2} ℏ^{2}}{2 m a^{2​}}

Case 2: One in $n = 2$ , One in $n = 3$

The total energy is:

E_{total, excited (4)} = E_{2} + E_{3} = \frac{4 π^{2} ℏ^{2}}{2 m a^{2}} + \frac{9 π^{2} ℏ^{2}}{2 m a^{2}} = \frac{13 π^{2} ℏ^{2}}{2 m a^{2​}}

Case 3: One in $n = 1$ , One in $n = 3$

The total energy is:

E_{total, excited (5)} = E_{1} + E_{3} = \frac{π^{2} ℏ^{2}}{2 m a^{2}} + \frac{9 π^{2} ℏ^{2}}{2 m a^{2}} = \frac{10 π^{2} ℏ^{2}}{2 m a^{2​}}

illustration of Possible energy states of two photons in 1-D box on website physicxion

Illustration of possible energy states of 2 photons in 1-D box
from ground state to second excited state showing that they can coexist in single quantum state

Summary of Energies

Here is a summary of the total energies for the two-photon system in different states:

Ground State: $E_{\text{total, ground}} = \frac{\pi^2 \hbar^2}{m a^2}$
First Excited States:
- Case 1: $E_{\text{total, excited (1)}} = \frac{8 \pi^2 \hbar^2}{2m a^2}$
- Case 2: $E_{\text{total, excited (2)}} = \frac{5 \pi^2 \hbar^2}{2m a^2}$
Second Excited States:
- Case 1: $E_{\text{total, excited (3)}} = \frac{18 \pi^2 \hbar^2}{2m a^2}$
- Case 2: $E_{\text{total, excited (4)}} = \frac{13 \pi^2 \hbar^2}{2m a^2}$
- Case 3: $E_{total, excited (5)} = \frac{10 π^{2} ℏ^{2}}{2 m a^{2}}$

Small yet powerful enough to challenge the nature of reality?

Does quantum mechanics provide a more fundamental description of reality than classical physics? What are the implications of this for our understanding of the universe and our place within it? This small yet significant incident is powerful enough to put a big question mark on our existence. It challenges our classical notions of individuality and distinctness. If two photons can "be" in the same state, does it mean they lose their individual identities and become indistinguishable? Does this blur the line between individual existence and collective behavior? How does the act of measurement affect the state of two photons that were previously in the same quantum state? Does the measurement process somehow "individualize" them? Does the fact that indistinguishable particles can share a quantum state imply a deeper level of interconnectedness in the universe? Does it suggest that individual existence may be an illusion at the fundamental level? what is more fundamental individuality or coexistence or both according to the universe? When really do they become indistinguishable from individuals and coexist at what boundaries, are there really any boundaries?

Conclusion

The photon is much more than a particle of light. It’s a massless, spin-1 gauge boson that mediates the electromagnetic force, a cornerstone of the universe’s structure. Its unique properties, including its ability to occupy the same state as other photons, arise from its nature as a boson and its adherence to Bose-Einstein statistics.

Mathematically, the symmetric wavefunction of photons and the Bose-Einstein distribution reveal why photons can coexist in the same space without limitation. Beyond its role in the Standard Model, the photon connects us to deeper theories and phenomena, from quantum field interactions to cosmological events. It’s a timeless traveler, a carrier of energy, and an essential piece of the cosmic puzzle, bridging the quantum and the infinite with elegance and mystery. Deciphering the nature of photons clearly is still a great puzzle.

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No Boundaries: Can Photons Coexist in One State?

No Boundaries: Can Photons Coexist in One State?

What is Photon?

1. The Building Block of Light

2. Dual Nature: Particle and Wave

3. Messengers of Energy and Information

4. No Mass, No Rest

5. Born from the Universe Itself

6. Quantum Oddities

7. Timeless Travelers

8. Life and Technology Depend on Them

Photon as a member of the Boson family

1. Bosons and Their Role in Nature

2. The Photon as a Gauge Boson

Electromagnetism and the Photon

3. Properties of the Photon as a Gauge Boson

4. Comparison to Other Gauge Bosons

5. Beyond the Standard Model

6. Role of Photons in Quantum Electrodynamics (QED)

7. Unique Status of the Photon

How the law of occupying space is different?

What is the Pauli Exclusion Principle?

Wavefunctions of Fermions:

What About Bosons?

Wavefunctions of Bosons:

Why the Difference?

Key Implications for Bosons and Fermions:

Why no boundaries?

Identical Particles and Wavefunctions

Spatial Wavefunction Overlap

Symmetric Wavefunction and Particle Overlap

Statistical Mechanics Perspective

Photon Statistics and Bose-Einstein Condensation

Superposition and Constructive Interference

Physical Manifestations

Why This Does Not Violate Physics

Simplified mathematical demonstration with two photons just as an example of the wonderful phenomena

1. Single Particle in a 1D Box: Energy Levels

2. Two Identical Photons in a Box

Ground State (Both Photons in n=1n = 1):

First Excited State:

Case 1: Both in n=2n = 2

Case 2: One in n=1n = 1, One in n=2n = 2

Second Excited State:

Case 1: Both in n=3n = 3

Case 2: One in n=2n = 2, One in n=3n = 3

Case 3: One in n=1n = 1, One in n=3n = 3

Summary of Energies

Small yet powerful enough to challenge the nature of reality?

Conclusion

Join the conversation

Ground State (Both Photons in $n = 1$ ):

Case 1: Both in $n = 2$

Case 2: One in $n = 1$ , One in $n = 2$

Case 1: Both in $n = 3$

Case 2: One in $n = 2$ , One in $n = 3$

Case 3: One in $n = 1$ , One in $n = 3$