Fermions vs Bosons: What Is the Difference?

Physics sorts every particle in the universe into just two families. Electrons, protons, neutrons and quarks are fermions. Photons, gluons and the Higgs boson are bosons. The split is not about mass or charge. It is about how identical particles behave in groups, and it explains why atoms have structure, why stars can resist collapse, and why lasers work. Our Satyendra Nath Bose sticker honours the physicist whose name is on one half of the divide. This post builds on our explainer Why Can’t Two Electrons Share a State?.

What is the difference between fermions and bosons?

The short answer is spin. OpenStax’s university textbook puts it simply: fermions have half-integral spin (½ħ, ³⁄₂ħ, …) and bosons have integral spin (0, ħ, 2ħ, …). “Familiar examples of fermions are electrons, protons, and neutrons. A familiar example of a boson is a photon” (OpenStax University Physics, §11.1^[1]).

Here ħ (h-bar) is the reduced Planck constant, about 1.055 × 10⁻³⁴ J·s, the natural unit of angular momentum in quantum mechanics. Spin is intrinsic angular momentum, a built-in property of the particle, not a ball literally rotating. We explain where the strange half comes from in Why Do Electrons Have Spin ½?.

	Fermions	Bosons
Spin	½ħ, ³⁄₂ħ, …	0, ħ, 2ħ, …
Swap two identical particles	ψ → −ψ	ψ → +ψ
Particles per quantum state	0 or 1	any number
Statistics	Fermi–Dirac	Bose–Einstein
Examples	electron, quarks, proton, neutron	photon, gluon, W and Z, Higgs

What happens when you swap two identical particles?

Two electrons are perfectly identical. Nothing labels one as “first”. So if you swap them, every measurable quantity must stay the same. Measurable quantities depend on |ψ|², so the wavefunction can change at most by a phase factor. Swapping twice must bring back the original state, so in three dimensions that factor is either +1 or −1. OpenStax states the two options: ψ → −ψ for indistinguishable fermions and ψ → +ψ for indistinguishable bosons. Fermions are “antisymmetric on exchange” and bosons are “symmetric on exchange” (OpenStax University Physics, §11.1^[1]).

The one-line proof of the exclusion principle

Put particle 1 and particle 2 into single-particle states a and b. Write φₐ(1) for “particle 1 in state a”. The correctly symmetrised two-particle wavefunctions are

ψ± = (1/√2) [φₐ(1) φ_b(2) ± φₐ(2) φ_b(1)]

The plus sign is for bosons and the minus sign for fermions. The factor 1/√2 keeps the total probability equal to one. Now try to put both fermions in the same state, a = b. The two terms become identical and subtract:

ψ₋ = (1/√2) [φₐ(1) φₐ(2) − φₐ(2) φₐ(1)] = 0

A wavefunction that is zero everywhere means probability zero: the state cannot exist. That is Pauli’s exclusion principle, which OpenStax describes as “a consequence of exchange symmetry of fermions” (OpenStax University Physics, §11.1^[1]). For bosons the two terms add instead, and nothing forbids a = b.

Why do fermions make matter solid?

Because no two electrons can share a quantum state, electrons in an atom fill successive shells rather than all dropping into the lowest one. That filling pattern produces the periodic table (OpenStax University Physics, §8.4^[2]). In a metal, OpenStax explains, classical physics would put all the electrons in the ground state at T = 0 K, “however, this violates Pauli’s exclusion principle”; instead they stack up to the Fermi energy, and the probability that a given state is filled is called the Fermi factor (OpenStax University Physics, §9.4^[3]).

The same stacking produces degeneracy pressure, which our explainer shows can hold up a dead star. Our Exclusor tee turns that refusal to share into a character, and the Wolfgang Pauli sticker says it in three words: one per state.

Why do bosons pile together?

For bosons the symmetric combination adds, so many particles can occupy one state. Quanta Magazine gives the everyday example: “A laser, for instance, consists of many photons synchronized at the same wavelength, together in a single beam” (von Hippel, Quanta^[4]).

Taken to the extreme, a gas of bosonic atoms cooled close to absolute zero can drop into a single quantum state, a Bose–Einstein condensate. Eric Cornell, Wolfgang Ketterle and Carl Wieman won the 2001 Nobel Prize in Physics “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms” (NobelPrize.org^[5]). Note that whole atoms can be bosons even though they are built from fermions: an even total number of fermions gives integer total spin. Quanta mentions helium atoms as composite particles that behave like bosons (von Hippel, Quanta^[4]).

In the Standard Model, CERN summarises the division of labour: matter particles are fermions, and the fundamental forces are carried by bosons, with the gluon for the strong force, the photon for electromagnetism and the W and Z bosons for the weak force (CERN^[6]).

Who worked this out?

• 1924, Bose. Satyendra Nath Bose derived Planck’s radiation law by a new way of counting photons, published as “Plancks Gesetz und Lichtquantenhypothese” in Zeitschrift für Physik (Bose 1924^[7]). According to Quanta, Bose wrote to Albert Einstein for help submitting the result to a German journal, then collaborated with him to flesh out the idea, which is why the statistics carries both names (von Hippel, Quanta^[4]).
• 1926, Fermi and Dirac. Enrico Fermi worked out the statistics of a gas whose particles obey the exclusion principle (Fermi 1926^[8]), and Paul Dirac reached the same conclusion working independently that year (Dirac 1926^[9]): electrons are not bosons. Dirac later coined both names, “boson” and “fermion”, in a 1945 speech, after Bose and Fermi (von Hippel, Quanta^[4]).
• 1940, Pauli. Wolfgang Pauli showed that relativistic quantum theory requires “Fermi-Dirac statistics for particles with arbitrary half-integral spin” and “Einstein-Bose statistics for particles with arbitrary integral spin” (Pauli 1940^[10]). This link between spin and statistics is the subject of our post The Spin-Statistics Theorem.

FAQ

Is a proton a fermion or a boson?

A fermion. It has spin ½ and obeys the exclusion principle, like the neutron and the electron.

Are there particles that are neither?

In three dimensions, every particle is one or the other. In effectively two-dimensional systems, physicists describe quasiparticles called anyons, which Quanta notes are possible because particle paths behave differently on a flat surface (von Hippel, Quanta^[4]).

Is the Higgs a boson?

Yes. CERN notes that “the Higgs boson is the only particle to have a spin of zero”, and zero is an integer, so it is a boson (CERN^[11]).

References

1. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §11.1 “Introduction to Particle Physics”. https://openstax.org/books/university-physics-volume-3/pages/11-1-introduction-to-particle-physics
2. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §8.4 “The Exclusion Principle and the Periodic Table”. https://openstax.org/books/university-physics-volume-3/pages/8-4-the-exclusion-principle-and-the-periodic-table
3. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §9.4 “Free Electron Model of Metals”. https://openstax.org/books/university-physics-volume-3/pages/9-4-free-electron-model-of-metals
4. M. von Hippel, “Matter vs. Force: Why There Are Exactly Two Types of Particles,” Quanta Magazine (23 June 2025). https://www.quantamagazine.org/matter-vs-force-why-there-are-exactly-two-types-of-particles-20250623/
5. NobelPrize.org, The Nobel Prize in Physics 2001 (Cornell, Ketterle, Wieman). https://www.nobelprize.org/prizes/physics/2001/summary/
6. CERN, “The Standard Model”. https://home.cern/science/physics/standard-model
7. S. N. Bose (1924), “Plancks Gesetz und Lichtquantenhypothese,” Zeitschrift für Physik 26, 178–181 (DOI 10.1007/BF01327326; metadata verified via Crossref). https://doi.org/10.1007/BF01327326
8. E. Fermi (1926), “Zur Quantelung des idealen einatomigen Gases,” Zeitschrift für Physik 36, 902–912 (DOI 10.1007/BF01400221; metadata verified via Crossref). https://doi.org/10.1007/BF01400221
9. P. A. M. Dirac (1926), “On the Theory of Quantum Mechanics,” Proc. R. Soc. Lond. A 112, 661–677 (DOI 10.1098/rspa.1926.0133; metadata verified via Crossref). https://doi.org/10.1098/rspa.1926.0133
10. W. Pauli (1940), “The Connection Between Spin and Statistics,” Phys. Rev. 58, 716–722. https://journals.aps.org/pr/abstract/10.1103/PhysRev.58.716
11. CERN, “The Higgs boson”. https://home.cern/science/physics/higgs-boson

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.