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PHOTON

For other uses, see Photon (disambiguation).

In modern physics, the photon is the elementary particle responsible for electromagnetic interactions and light. The photon is massless[1], has no electric charge and does not decay spontaneously in empty space. A photon can have two different helicities (polarization states) and has three continuous parameters, the components of its wave vector. Photons are emitted in many natural processes, e.g., when a charge is accelerated, when an atom or a nucleus jumps from a higher to lower energy level, or when a particle and its antiparticle are annihilated. Photons are absorbed in the time-reversed processes, e.g., in the production of particle–antiparticle pairs or in an atomic or nuclear transition to a higher energy level.

Contents

Nomenclature and symbols

The concept of the photon was developed gradually (1905–1917) by Albert Einstein, who called it a "light quantum" [Lichtquantum]. The name "photon" derives from the Greek word φῶς, "phōs" (meaning light) and was coined by the distinguished physical chemist Gilbert N. Lewis, who published a speculative theory in which photons were "uncreatable and indestructible" (Lewis 1926). Although Lewis' theory was never accepted (being contradicted by many experiments), his new name, photon, was adopted immediately by most physicists.

A photon is usually given the symbol γ, the Greek letter gamma, although in nuclear physics this symbol refers only to photons of the highest energy (a gamma ray). In chemistry, photons are sometimes symbolized by hν, which is the amount of energy of each photon.

Energy, momentum, angular momentum and mass

Main article: Special relativity

The prevailing Standard Model of physics predicts that the photon is massless, which is consistent with experiment. Hence, the photon moves at c \! (the speed of light in empty space), and its energy E \! and momentum \mathbf{p} are related by E = c \, p \!, where p \! is the magnitude of the momentum. For comparison, the corresponding equation for particles with an invariant mass m \! would be E^{2} = c^{2} p^{2} + m^{2} c^{4} \!, as shown in special relativity.

As an illustration of photon momentum, the classical pressure of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change in momentum per unit time.

The energy and momentum of a photon depend only on its frequency \nu \! or, equivalently, its wavelength \lambda \!

E = h\nu \!
p = \frac{h}{\lambda} = \frac{h\nu}{c}

where h \! is Planck's constant. Hence, the four-momentum of a photon is p_{\mu} \equiv (-\frac{E}{c}, \mathbf{p}) = \hbar (-\frac{\omega}{c}, \mathbf{k}) where \hbar \equiv h/2\pi is Planck's reduced constant, \omega = 2\pi\nu \! is the angular frequency and \mathbf{k} is the wave vector of magnitude k \equiv 2\pi/\lambda \! that points in the direction of the photon's propagation. The photon also carries a fixed amount \hbar of angular momentum that does not depend on its frequency.

For illustration, these formulae show that the annihilation of a particle with its antiparticle must result in at least two photons, not one. In the center of mass frame, the colliding antiparticles have no net momentum, whereas a single photon always has momentum. Hence, conservation of momentum requires that at least two photons are created, with zero net momentum. The energy of the two photons (and, hence, their frequency) may be determined from conservation of four-momentum. The reverse process, pair production, is the dominant mechanism by which high-energy photons (such as gamma rays) lose energy while passing through matter.

Historical development

Main article: Light

With a few notable exceptions, most theories up to the eighteenth century hypothesized that light was composed of particles, although such models could not easily account for the refraction and diffraction of light. Several wave theories of light were proposed in the seventeenth century by René Descartes (1637), Robert Hooke (~1665), and Christian Huygens (1678), but particle models remained the dominant theory, owing primarily to the support of Isaac Newton. In the early nineteenth century, particle models of light were abandoned and wave models became the dominant theory, primarily because of experiments showing interference. James Clerk Maxwell's prediction (1873) that light was an electromagnetic wave — which was confirmed experimentally by Heinrich Hertz's detection of radio waves (1888) — seemed to be the final blow to particle models of light.

Nevertheless, the Maxwell wave theory does not account for all properties of light. The Maxwell theory predicts that the energy of a light wave depends only on its intensity, not on its frequency; nevertheless, several independent types of experiments show that the energy imparted by light to atoms depends only on the light's frequency, not on its intensity. For example, some chemical reactions can be provoked only by light of frequency higher than a certain threshold; light of lower frequency, no matter how intense, is incapable of exciting the reaction. Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the photoelectric effect); the energy of the ejected electron is proportional only to the light's frequency, not to its intensity.

At the same time, careful investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers culminated in Planck's law of black-body radiation, which hypothesized that the energy of any system that absorbs or emits electromagnetic radiation of frequency \nu \! must be an integer multiple of a frequency-dependent energy quantum E = h\nu \!. If this energy quantization is not adopted, there can be no thermal equilibrium between matter and radiation, as shown by Einstein in 1905.

Since the Maxwell theory of light allows for all possible energies of electromagnetic radiation, most physicists assumed initially that the energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. Einstein was the first to propose that the energy quantization was a property of electromagnetic radiation itself. In his 1905 paper A Heuristic Model of the Creation and Transformation of Light, Einstein accepts provisionally the validity of Maxwell's wave theory, but points out that the energy of a light wave may be localized into points that move independently of one another, even if the wave itself is spread continuously over space. In his 1909 work The Development of Our Views on the Composition and Essence of Radiation and two later works (1916b and 1917 references below), Einstein goes further and shows that, if Planck's law of black-body radiation is accepted, the energy quanta must also carry momentum, making them full-fledged particles. This photon momentum was observed experimentally by Arthur Compton in 1922, for which he received the Nobel Prize in 1927. The pivotal question was then: how to unify Maxwell's wave theory of light with its experimentally observed particle nature? The answer to this question occupied Albert Einstein for the rest of his life, and was solved in quantum electrodynamics and its successor, the Standard Model.

Early objections to the photon hypothesis

Before Compton's famous experiment in 1922, most physicists were very reluctant to believe that electromagnetic radiation itself might be particulate. This reluctance is understandable, given the success and plausibility of Maxwell's electromagnetic wave model of light. Therefore, most physicists assumed rather that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation.

Even after Compton's experiment, Bohr, Kramers and Slater made one last attempt to preserve the Maxwell's continuous electromagnetic field model of light, the so-called BKS model of 1924. To account for the then-available data, two drastic hypotheses had to be made:

  • Energy and momentum are conserved only on the average in interactions between matter and radiation, not in elementary processes such as absorption and emission. This allows one to reconcile the discontinuously changing energy of the atom (jump between energy states) with the continuously release of energy into radiation.

Refined Compton experiments showed, however, that energy-momentum is conserved extraordinarily well in elementary processes; and also that the jolting of the electron and the generation of a new photon in Compton scattering obey causality to within 10 ps. Accordingly, Bohr and his co-workers gave their model "as honorable a funeral as possible".

Wave–particle duality

Main articles: Wave–particle duality and Squeezed coherent state

The dual wave–particle nature of photons is difficult to visualize, and warrants further discussion. On the one hand, the photon seems like a point-like particle, since it can be absorbed or emitted as a whole by systems much smaller than its wavelength, such as an atomic nucleus (≈10–15 m across) or even the pointlike electron. Nevertheless, the photon displays diffraction and interference (wave phenomena) on the length scale of its wavelength. For example, a single photon passing through a double-slit experiment would land on the screen with a probability distribution determined by its interference pattern determined by Maxwell's equations. According to quantum electrodynamics (see below), the photon is not a point-like particle whose trajectory is shaped probabilistically by the electromagnetic field (as conceived by Einstein), but rather a quantum of an electromagnetic mode that stretches over all space, a quantum that is responsible for the electromagnetic field in the first place.

The quantum mechanics of material particles features an uncertainty principle that forbids the simultaneous measurement of the position and momentum of a particle in the same direction. An analogous principle for photons forbids the simultaneous measurement of the number of photons (see Fock state and below) in an electromagnetic wave and the phase of that wave (see coherent state and squeezed coherent state).

Bose–Einstein model of a photon gas

Main article: Bose–Einstein statistics

In 1924, Satyendra Nath Bose derived Planck's law of black-body radiation without using any electromagnetism at all, but rather a modification of coarse-grained counting of phase space. In 1924–1925, Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction", now understood as the requirement for a symmetric quantum mechanical wave function. This work led to the concept of coherent states and the development of the laser. In the same papers, Einstein extended Bose's formalism to material particles (bosons) and predicted that they would condense into their lowest quantum state at low enough temperatures (Bose–Einstein condensation). In 1926, Paul Dirac showed that the number of photons is not conserved.

Stimulated and spontaneous emission

Main article: Stimulated emission

In 1916, Einstein showed that Planck's quantum hypothesis E = hν could be derived from a kinetic rate equation. Given a cavity filled with electromagnetic radiation and systems that can emit and absorb that radiation, Einstein assumed that emission/absorption of radiation of frequency ν causes a well-defined change in a system's energy and further hypothesized that the transition rate Rij from energy Ei to energy Ej is given by

R_{ij} = N_{i} A_{ij} + N_{i} B_{ij} \rho(\nu) \!

where Ni represents the number of molecules of energy Ei and ρ(ν) is the density of radiation at frequency ν within the cavity. Aij and Bij are rate constants; Aij is the rate constant for emitting radiation spontaneously, whereas Bij is the rate constant for absorbing radiation from the cavity (if Ei < Ej) or emitting it in response to the ambient radiation (if Ei > Ej, induced or stimulated emission). Planck's energy quantization E = hν is a necessary consequence of this hypothesized rate equation and the basic requirements that the ambient radiation be in thermal equilibrium with the emitting/absorbing systems and independent of their material composition.

Einstein was also able to show that Bij = Bji and, perhaps even more remarkably,

A_{ij} = \frac{8 \pi h \nu^{3}}{c^{3}} B_{ij}

Einstein noted that the rate constants Aij and Bij should be derivable from a "mechanics and electrodynamics modified to accommodate the quantum hypothesis". This prediction was borne out in quantum mechanics and quantum electrodynamics respectively; both are required to derive Einstein's rate constants from first principles (see Dirac 1927 references below).

Semiclassical approach to radiation

Main article: Perturbation theory (quantum mechanics)

As shown by Dirac in 1926, Einstein's coefficients Bij for induced transitions can be computed semiclassically, i.e., by treating the electromagnetic radiation classically and the material system quantum mechanically. The incoming radiation is treated as a sinusoidal electric field applied to the material system, with an small (perturbative) interaction energy H = -2 \mathbf{d} \cdot \mathbf{\mathcal{E}_{0}} \cos \omega t, where \mathbf{d} is the material system's electric dipole moment and where \mathbf{\mathcal{E}_{0}} and ω represent the electric field and angular frequency of the incoming radiation, respectively. The probability per unit time wji of the radiation inducing a transition between discrete energy levels Ei and Ej may be computed using time-dependent perturbation theory

w_{ji} = \frac{2\pi}{\hbar^{2}} \left| \langle \phi_{j} | \mathbf{d} \cdot \mathbf{\mathcal{E}_{0}} | \phi_{i} \rangle \right|^{2} \delta(\omega_{ij} - \omega)

where ωij is defined by \omega_{ij} \equiv \left( E_{i} - E_{j} \right)/\hbar, and where φi and φj represent the unperturbed eigenstates of energy Ei and Ej, respectively. Assuming that the polarization vector \mathbf{\mathcal{E}_{0}} of the incoming radiation is oriented randomly relative to the dipole moment \mathbf{d} of the material system, the corresponding Bij rate constants can be computed

B_{ji} = \frac{8\pi^{2}}{3\hbar^{2}} \left| \langle \phi_{j} | \mathbf{d} | \phi_{i} \rangle \right|^{2}

from which Bji = Bij. Thus, if the two states φi and φj do not result in a net dipole moment (i.e., if \langle \phi_{j} | \mathbf{d} | \phi_{i} \rangle = 0), the absorption and induced emission are said to be "disallowed".

However, this semiclassical approach does not yield the coefficients Aij for spontaneous emission from first principles, although they can be calculated using the correspondence principle and the classical (low-frequency) limit of Planck's law of black body radiation (the Rayleigh-Einstein-Jeans law). Nor does this semiclassical approach require the introduction of photons per se, although their energy law E=\hbar \omega is used. A true derivation from first principles was developed by Dirac that required the quantization of the electromagnetic field itself; in this approach, photons are the quanta of the field. This approach is called second quantization or quantum field theory; the earlier quantum mechanics (the quantization of material particles moving in a potential) represents the "first quantization".

Second quantization

Main article: Quantum field theory

In 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption. He correctly decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of h\nu \!, where \nu \! is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of blackbody radiation, derived by Einstein in 1909.

In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way. The electromagnetic Fourier modes (defined by their wave vector \mathbf{k} and polarization state) are equivalent to a set of uncoupled simple harmonic oscillators, as may be shown classically. Treated quantum mechanically, the energy levels of such oscillators are known to be E = nh\nu \!, where \nu \! is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy E = nh\nu \! as a state with n \! photons, each of energy h\nu \!. This approach gives the correct energy fluctuation formula.

In 1927, Dirac took this one step further. He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A_{ij} \! and B_{ij} \! coefficients from first principles, and showed that photons automatically obey Bose–Einstein statistics. The second-order perturbation theory can involve virtual photons, transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories, the probability of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy E = p \, c \!, and can exist in three polarization states (instead of the two states observed physically). Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can make infinite contributions, a problem that was overcome in quantum electrodynamics by using renormalization.

In modern physics notation, the quantum state of the electromagnetic field is written as a Fock state, a tensor product of the states for each electromagnetic mode

|n_{k_0}\rangle\otimes|n_{k_1}\rangle\otimes\dots\otimes|n_{k_n}\rangle\dots

where |n_{k_i}\rangle represents the state in which \, n_{k_i} photons are in the mode \, k_i. In this notation, the creation of a new photon in mode \, k_i (e.g., emitted from an atomic transition) is written as |n_{k_i}\rangle \rightarrow |n_{k_i}+1\rangle. This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics.

The photon as a gauge boson

Main article: Gauge theory

The electromagnetic field can be understood as a gauge theory, i.e., as a field that results from requiring that a symmetry hold independently at every position in spacetime. For the electromagnetic field, this symmetry is the U(1) symmetry of a complex number, i.e., the ability to vary the phase of a complex number without affecting real numbers constructed from it (such as the energy or the Lagrangian).

The quanta of a gauge field must be massless, chargeless bosons, as long as the symmetry is not broken. Hence, the photon is predicted to be massless, chargeless and have integer spin (boson). In particular, the form of the electromagnetic interaction specifies that the photon must have spin ±1, i.e, its spin (not orbital) angular momentum measured along its direction of motion must be \pm \hbar. These two components correspond to the classical concepts of right-handed and left-handed circularly polarized light. However, the transient virtual photons of quantum electrodynamics may also adopt unphysical polarization states.

Experimental limits on mass

Since the photon is a gauge boson, most physicists believe that its intrinsic mass is exactly zero. This theoretical prediction has been checked experimentally to high accuracy. For example, satellite measurements of planetary magnetic fields were carried out by the Charge Composition Explorer spacecraft and used to derive an upper limit of 6x10-16 eV (1x10-51 kg) on the mass of the photon. An improved upper limit of 6x10-17 eV (1x10-52 kg) was obtained in 1998 by Roderic Lakes using a Cavendish balance. The same upper limit is given by the Particle Data Group, based on the magnetohydrodynamics of the solar wind.[2]. Studies of galactic magnetic fields suggest an even better upper limit of 3x10-27 eV (5x10-63 kg) but the validity of this method has been questioned. Although further experiments to tighten the error bars on the photon mass are expected, all data hitherto are consistent with the photon having zero mass.

References:

  1. Eric Adelberger, Gia Dvali, Andrei Gruzinov, "Photon Mass Bound Destroyed by Vortices", arXig.org
  2. Goldhaber, Alfred S., and Nieto, Michael Martin, "Terrestrial and Extraterrestrial Limits on The Photon Mass", Rev. Mod. Phys. vol.43 #3 pp.277–296, 1971 [1]
  3. E. Fischbach et al., Physical Review Letters, 73, 514–517 25 July 1994.
  4. Official particle table http://pdg.lbl.gov/2005/tables/gxxx.pdf
  5. L. Davis, A. S. Goldhaber, and M. M. Nieto, Phys. Rev. Lett. 35, 1402 (1975)
  6. Roderic Lakes, "Experimental Limits on the Photon Mass and Cosmic Magnetic Vector Potential", Phys. Rev. Lett. 80, 1826 (1998) [2]
  7. J. Luo et al., Phys. Rev. D 59, 042001 (1999)
  8. B. E. Schaefer, Phys. Rev. Lett. 82, 4964 (1999)
  9. J.Luo et al., Physical Review Letters, (28 February 2003)

Contributions of photons to the invariant mass of a system

Main article: Mass in special relativity

Although the photon is itself massless, it adds to the invariant mass of any system to which it belongs; this is true for every form of energy, as predicted by the special theory of relativity. For example, the invariant mass of a system that emits a photon is decreased by an amount E / c2 upon emission (where E is the energy of the photon in the frame of the emitting system). Similarly, the invariant mass of a system that absorbs a photon is increased by a corresponding amount based on the energy of the photon in the frame of the absorbing system.

This concept is applied in a key prediction of quantum electrodynamics (QED), the quantum theory of electromagnetic interactions begun by Dirac (described above). QED is able to predict the magnetic dipole moment of leptons (such as that of the electron) to extremely high accuracy (16 decimal places); experimental measurements of these magnetic dipole moments have agreed with these predictions perfectly. The predictions, however, require counting the contributions of virtual photons to the invariant mass of the lepton. Another example of such contributions verified experimentally is the QED prediction of the hyperfine structure (the Lamb shift) of bound lepton pairs, such as muonium and positronium.

Photons in matter

When photons pass through matter, different frequencies are transmitted at different speeds; this is called dispersion. A similar phenomenon occurs in reflection where surfaces can reflect photons of various frequencies at different angles.

In a material, photons couple to the excitations of the medium and behave differently. These excitations can often be described as quasi-particles (such as phonons and excitons); that is, as quantized wave- or particle-like entities propagating though the matter. "Coupling" means here that photons can transform into these excitations (that is, the photon gets absorbed and medium excited, involving the creation of a quasi-particle) and vice versa (the quasi-particle transforms back into a photon, or the medium relaxes by re-emitting the energy as a photon). However, as these transformations are only possibilities, they are not bound to happen and what actually propagates through the medium is a polariton; that is, a quantum-mechanical superposition of the energy quantum being a photon and of it being one of the quasi-particle matter excitations.

According to the rules of quantum mechanics, a measurement (here: just observing what happens to the polariton) breaks this superposition; that is, the quantum either gets absorbed in the medium and stays there (likely to happen in opaque media) or it re-emerges as photon from the surface into space (likely to happen in transparent media).

Matter excitations have a nonlinear dispersion relation; that is, their momentum is not proportional to their energy. Hence, these particles propagate slower than the vacuum speed of light. (The propagation speed is the derivative of the dispersion relation with respect to momentum.) This is the formal reason why light is slower in media (such as glass) than in vacuum. (The reason for diffraction can be deduced from this by Huygens' principle.) Another way of phrasing it is to say that the photon, by being blended with the matter excitation to form a polariton, acquires a nonzero effective mass, which means that it cannot travel at c, the speed of light in a vacuum.

See also

Footnotes

  1. ^ The intrinsic or "invariant" mass of the photon is believed to be exactly zero, based on experiment and theoretical considerations, as described above. This is the standard definition of "mass" among physicists. However, some popularizations of physics have ascribed to the photon a relativistic mass, defined as E/c2, where E represents the photon's energy. See mass in special relativity for a discussion of the relationship between invariant mass and relativistic mass.
  2. ^ 2006 PDG listing for photon

References

  • Pais A. (1982) Subtle is the Lord: The Science and the Life of Albert Einstein, Oxford University Press, pp. 364-388, 402-415.
  • Pais A. (1986) Inward Bound: Of Matter and Forces in the Physical World, Oxford University Press.
  • Lewis GN. "The conservation of photons." Nature 1926, 118, 874-875.
  • Planck M. (1901) "Über das Gesetz der Energieverteilung im Normalspectrum", Annalen der Physik, 4, 553-563.
  • Debye P. (1910) "Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung", Annalen der Physik, 33, 1427-1434.
  • Einstein A. (1916a) Verh. Deutsch. Phys. Ges., 18, 318.
  • Einstein A. (1916b) Mitt. Phys. Ges. Zürich., 16, 47.
  • Einstein A. (1917) "Zur Quantentheorie der Strahlung", Phys. Zeitschr., 18, 121-128.
  • Compton AH. (1923) "A Quantum Theory of the Scattering of X-rays by Light Elements", Physical Review, 21, 483-502.
  • Bohr N, Kramers HA, and Slater JC (1924) "The Quantum Theory of Radiation", Philosophical Magazine, 47, 785-802.
  • Bose SN. (1924) "Plancks Gesetz und Lichtquantenhypothese", Zeitschrift für Physik, 26, 178-181.
  • Einstein A. (1924) Sitzungsberichte der Preussischen Akademie der Wissenschaften, p. 261.
  • Einstein A. (1925) Sitzungsberichte der Preussischen Akademie der Wissenschaften, p. 3.
  • Born M, Heisenberg W, and Jordan P. (1925) "Quantenmechanik II", Zeitschrift für Physik, 35, 557-615.
  • Dirac PAM. (1926) "On the Theory of Quantum Mechanics", Proc. Roy. Soc. A, 112, 661-677.
  • Dirac PAM. (1927) "The Quantum Theory of the Emission and Absorption of Radiation", Proc. Roy. Soc. A, 114, 243-265.
  • Dirac PAM. (1927) "The Quantum Theory of Dispersion", Proc. Roy. Soc. A, 114, 710-728.

External links

Particles in physics - elementary particles

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Fermions: Quarks: (Up · Down · Strange · Charm · Bottom · Top) | Leptons: (Electron · Muon · Tau · Neutrinos)
Gauge bosons: Photon | W and Z bosons | Gluons
Not yet observed: Higgs boson | Graviton | Other hypothetical particles