Physics
Momentum of a Photon
The momentum of a photon refers to the quantity of motion it possesses due to its energy and momentum. Photons, which are massless particles of light, carry momentum as a result of their wave-particle duality. This momentum can be transferred to other objects upon collision, contributing to various physical phenomena such as the pressure of sunlight in space.
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10 Key excerpts on "Momentum of a Photon"
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
KEY IDEAS • Although it is massless, a photon has momentum, which is related to its energy E, frequency f, and wavelength by p = hf c = h . • In Compton scattering, x-rays scatter as particles (as photons) from loosely bound electrons in a target. • In the scattering, an x-ray photon loses energy and momentum to the target electron. • The resulting increase (Compton shift) in the photon wavelength is Δ = h mc (1 − cos ) , where m is the mass of the target electron and is the angle at which the photon is scattered from its initial travel direction. • Photons: when light interacts with matter, the interaction is particle-like, occurring at a point and transferring energy and momentum. • Wave: when a single photon is emitted by a source, we interpret its travel as being that of a probability wave. • Wave: when many photons are emitted or absorbed by matter, we interpret the combined light as a classical electromagnetic wave. Photons have momentum FIGURE 38.3 Compton’s apparatus. A beam of x-rays of wavelength = 71.1 pm is directed onto a carbon target T. The x-rays scattered from the target are observed at various angles to the direction of the incident beam. The detector measures both the intensity of the scattered x-rays and their wavelength. Incident x-rays Collimating slits λ T Scattered x-rays λ' Detector ϕ In 1916, Einstein extended his concept of light quanta (photons) by proposing that a quantum of light has linear momentum. For a photon with energy hf, the magnitude of that momentum is p = hf c = h (photon momentum), (38.7) where we have substituted for f from equation 38.1 (f = c/). Thus, when a photon interacts with matter, energy and momentum are transferred, as if there were a collision between the photon and matter in the classical sense (as in chapter 9). Pdf_Folio:924 924 Fundamentals of physics In 1923, Arthur Compton at Washington University in St Louis showed that both momentum and energy are transferred via photons.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
With these substitutions, our expression for the Momentum of a Photon becomes p = E __ c = hf __ f λ = h __ λ (29.6) Momentum of incident photon Momentum of scattered photon Momentum of recoil electron 29.4 The Momentum of a Photon and the Compton Effect 931 Using Equations 29.4, 29.5, and 29.6, Compton showed that the difference between the wavelength λʹ of the scattered photon and the wavelength λ of the incident photon is related to the scattering angle θ by λ ′ − λ = h ___ mc (1 − cos θ) (29.7) In this equation m is the mass of the electron. The quantity h/(mc) is called the Compton wavelength of the electron and has the value h/(mc) = 2.43 × 10 −12 m. Since cos θ varies between +1 and −1, the shift λʹ − λ in the wavelength can vary between zero and 2h/(mc), depending on the value of θ, a fact observed by Compton. The photoelectric effect and the Compton effect provide compelling evidence that light can exhibit particle-like characteristics attributable to energy packets called photons. But what about the interference phenomena discussed in Chapter 27, such as Young’s double-slit experiment and single-slit diffraction, which demonstrate that light behaves as a wave? Does light have two distinct personalities, in which it behaves like a stream of particles in some experiments and like a wave in others? The answer is yes, for physicists now believe that this wave–particle duality is an inherent property of light. Light is a far more interesting (and complex) phenomenon than just a stream of particles or an electromagnetic wave. In the Compton effect the electron recoils because it gains some of the photon’s momentum. In principle, then, the momentum that photons have can be used to make other objects move. Conceptual Example 5 considers a propulsion system for an inter- stellar spaceship that is based on the Momentum of a Photon.
- Andrews, David L., Univ. of East Anglia, Bradshaw, David S.(Authors)
- 2018(Publication Date)
7 Chapter 2 Properties of a Photon In this chapter, we examine some of the principal attributes of electromagnetic radiation that feature in photons. At the outset, it is worth asserting that the photon concept owes its legitimacy to the quantum theory of light. As we shall see in later chapters, the photon is formally regarded as a quantum of excitation of any specific optical mode. In the literature, one can find occasional mentions of photons qualified by an adjective, such as dipole photons, electric photons, ballistic photons, etc. 56,* These should be regarded as potentially misleading 57 because each alludes to a property that is more correctly associated with a particular kind of phenomenon. Even if the distinction between “real” and “virtual” photons represents one of the most widely used descriptors, it does not signify a clear-cut difference. 58 We thus begin by listing the following photon properties, all of which can be involved in specific forms of optical interaction, with the exception of the first. 2.1 Intrinsic Photon Properties 1. Mass. Photons are elementary particles with zero mass, necessarily so because no particle with a finite mass can move at the speed of light. This is a conclusion that follows from special relativity. Occasional representations of theory introduce a contrived “effective mass of the photon” for certain applications—for example, in connection with superconductivity and polariton propagation—but even then it is a concept that is neither necessary nor especially commendable. 2. Velocity. Photons propagate with a velocity whose absolute magnitude is normally quoted as the speed c in vacuum, with refractive corrections to be applied as appropriate for passage through material media. In free space, light travels in a straight line, and the direction of its well-defined velocity is usually denoted by the unit vector k ̂ .
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
This follows from the definition of relativistic momentum p = m 0 u ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 − u 2 / c 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . If we consider the limit m 0 → 0 while u → c , then p can remain finite. Evidently a particle without mass can carry momentum, provided that it travels at the speed of light. From Eq. ( 13.12 ), E 2 = ( pc ) 2 + ( m 0 c 2 ) 2 , 490 RELATIVISTIC DYNAMICS and if we take m 0 = 0, then we have, denoting photon energy by the symbol , 2 = ( pc ) 2 , = pc . (13.13) We have taken the positive square root because the negative solution would predict that in an isolated system the Momentum of a Photon could increase without limit as its energy dropped. Combining Eq. ( 13.13 ) with Einstein’s relation = h ν , we find that a photon possesses momentum p of magnitude p = h ν c . (13.14) The direction of the momentum vector is along the direction of travel of the light wave. Einstein’s quantum hypothesis was designed to solve a theoretical dilemma—the spectrum of blackbody radiation—but its first application was to a totally di ff erent problem—the photoelectric e ff ect. Example 13.4 The Photoelectric Effect In 1887 Heinrich Hertz discovered that metals can give o ff electrons when illuminated by ultraviolet light. This process, the photoelectric e ff ect , represents the direct conversion of light into mechanical energy (here, the kinetic energy of the electron). Einstein predicted that the energy a single electron absorbs from a beam of light at frequency v is exactly the energy of a single photon, h ν. For the electron to escape from the surface it must overcome the energy barrier that confines it to the surface. The electron must expend energy W = e Φ to escape from the surface, where e is the charge of the electron and Φ is an electric potential known as the work function of the material, typically a few volts. The maximum kinetic energy of the emitted electron is therefore K = h ν − e Φ .
- Martinus J G Veltman(Author)
- 2003(Publication Date)
- World Scientific(Publisher)
Momentum is, for any particle at low speeds, defined as velocity times the mass of that particle. At higher speeds the relationship is more complicated such that the momentum becomes infinite if the 119 E N E R G Y, M O M E N T U M A N D M A S S -S H E L L velocity approaches the speed of light. Clearly, velocity is not con-served in any process: if you shoot a small object, for example a pea, against a billiard ball at rest then the billiard ball will after the collision move very slowly compared to the pea before the collision. It will however move in the same direction as the pea before the collision. Whatever energy the pea transfers to the billiard ball will have relatively little effect, as that ball is much heavier than the pea. Thus if we are looking for a conservation law the mass must be taken into account, and this is the reason why one considers the product of mass and velocity, i.e. the momentum. So, at the end the pea will be smeared all over the billiard ball, and that ball will have a speed that is the speed of the pea scaled down by the mass ratio, but in the same direction. Here is a most important observation. For any object, in par-ticular a particle, momentum and energy are not indepen-dent. If the momentum of a particle of given mass is known then its velocity and thus its energy are also known . This is really the key point of this Chapter. In the following the relation-ship between momentum and energy will be considered in some more detail. Furthermore, the theory of relativity allows the exist-ence of particles of zero mass but arbitrary energy, and that must be understood, as photons (and perhaps neutrinos) are such mass-less particles. Also for massless particles the above statement re-mains true: if the momentum of a particle is known then its energy is known. Some readers may remember this from their school days: if the velocity of a particle is v then the momentum (called p ) of the particle is mv .
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
38.3.7 In terms of photons, explain the double-slit experiment in the standard version, the single-photon version, and the single-photon, wide-angle version. Additional examples, video, and practice available at WileyPLUS 1231 38.3 PHOTONS, MOMENTUM, COMPTON SCATTERING, LIGHT INTERFERENCE Key Ideas ● Although it is massless, a photon has momentum, which is related to its energy E, frequency f, and wavelength by p = hf _ c = h _ λ . ● In Compton scattering, x rays scatter as particles (as photons) from loosely bound electrons in a target. ● In the scattering, an x-ray photon loses energy and momentum to the target electron. ● The resulting increase (Compton shift) in the photon wavelength is Δλ = h _ mc (1 − cos ϕ), where m is the mass of the target electron and ϕ is the angle at which the photon is scattered from its initial travel direction. ● Photons: When light interacts with matter, the inter- action is particle-like, occurring at a point and transfer- ring energy and momentum. ● Wave: When a single photon is emitted by a source, we interpret its travel as being that of a probability wave. ● Wave: When many photons are emitted or absorbed by matter, we interpret the combined light as a classi- cal electromagnetic wave. Photons Have Momentum In 1916, Einstein extended his concept of light quanta (photons) by proposing that a quantum of light has linear momentum. For a photon with energy hf, the magnitude of that momentum is p = hf _ c = h _ λ (photon momentum), (38.3.1) where we have substituted for f from Eq. 38.1.1 ( f = c/λ). Thus, when a photon interacts with matter, energy and momentum are transferred, as if there were a collision between the photon and matter in the classical sense (as in Chapter 9). In 1923, Arthur Compton at Washington University in St. Louis showed that both momentum and energy are transferred via photons. He directed a beam of x rays of wavelength λ onto a target made of carbon, as shown in Fig.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The quantity h/(mc) is called the Compton wavelength of the electron and has the value h/(mc) 5 2.43 3 10 212 m. Since cos u varies between 11 and 21, the shift l9 2 l in the wavelength can vary between zero and 2h/(mc), depending on the value of u, a fact observed by Compton. The photoelectric effect and the Compton effect provide compelling evidence that light can exhibit particle-like characteristics attributable to energy packets called photons. But what about the interference phenomena discussed in Chapter 27, such as Young’s double-slit experiment and single-slit diffraction, which demonstrate that light behaves as a wave? Does light have two distinct personalities, in which it behaves like a stream of particles in some experiments and like a wave in others? The answer is yes, for physicists now believe that this wave–particle duality is an inherent property of light. Light is a far more interesting (and complex) phenomenon than just a stream of particles or an electromagnetic wave. In the Compton effect the electron recoils because it gains some of the photon’s mo- mentum. In principle, then, the momentum that photons have can be used to make other objects move. Conceptual Example 4 considers a propulsion system for an interstellar spaceship that is based on the Momentum of a Photon. E v MATH SKILLS Equation 29.5 is a relationship between the momentum p B of the incident photon, the momentum p B ¿ of the scattered photon, and the mo- mentum p B electron of the recoil electron in Figure 29.10. These momenta are vector quantities. Therefore, Equation 29.5 is equivalent to two equations; one is for the x components of the vectors and one for the y components of the vectors (see Sections 1.7 and 1.8).
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
38.3.7 In terms of photons, explain the double-slit experiment in the standard version, the single-photon version, and the single-photon, wide-angle version. where m is the mass of the target electron and ϕ is the angle at which the pho- ton is scattered from its initial travel direction. 5. Photons: When light interacts with matter, the interaction is particle-like, occurring at a point and transferring energy and momentum. 6. Wave: When a single photon is emitted by a source, we interpret its travel as being that of a probability wave. 7. Wave: When many photons are emitted or absorbed by matter, we interpret the combined light as a classical electromagnetic wave. Photons Have Momentum In 1916, Einstein extended his concept of light quanta (photons) by proposing that a quantum of light has linear momentum. For a photon with energy hf, the magnitude of that momentum is p = hf _ c = h _ λ (photon momentum), (38.3.1) where we have substituted for f from Eq. 38.1.1 ( f = c/λ). Thus, when a photon interacts with matter, energy and momentum are transferred, as if there were a collision between the photon and matter in the classical sense (as in Chapter 9). In 1923, Arthur Compton at Washington University in St. Louis showed that both momentum and energy are transferred via photons. He directed a beam of x rays of wavelength λ onto a target made of carbon, as shown in Fig. 38.3.1. An x ray is a form of electromagnetic radiation, at high frequency and thus small wavelength. Compton measured the wavelengths and intensities of the x rays that were scattered in various directions from his carbon target. Figure 38.3.2 shows his results. Although there is only a single wavelength (λ = 71.1 pm) in the incident x-ray beam, we see that the scattered x rays con- tain a range of wavelengths with two prominent intensity peaks.
eBook - PDF- Norman Gray(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
Since the photon’s velocity 4-vector is null, the photon’s 4-momentum must be also (since it is defined above to be pointing in the same direction). Thus we must have ⋅ = 0, thus ⋅ = 2 , recovering the = 0 version of Eq. (7.9), 2 = 2 (massless particle), (7.10) so that even massless particles have a non-zero momentum. In quantum mechanics, we learn that the energy associated with a quantum of light – a photon – is = ℎ, where ℎ is Planck’s constant, 132 7 Dynamics P 1 P 2 P 3 Figure 7.2 The momenta involved in a collision. ℎ = 6.626 × 10 −34 J s (or 2.199 × 10 −42 kg m in natural units), so that = (ℎ, ℎ, 0, 0) (photon). (7.11) Thinking back to the approach of Section 7.1.1, we realise that Eq. (7.11) is the conclusion we must come to if we speculate that the zeroth com- ponent of a photon’s 4-momentum is its energy, = ℎ, and then demand that that 4-vector is null. 7.3 Relativistic Collisions and the Centre-of-Momentum Frame Consider two particles, of momenta 1 and 2 , which collide and produce a single particle of momentum 3 – you can think of this as describing either two balls of relativistic putty, or an elementary particle collision which produces a single new outgoing particle (this example is adapted from the characteristically excellent discussion in Taylor & Wheeler (1992, §8.3)); note also that the subscripts denote the different particle momenta, but the superscripts denote the different momentum components, so that (for example) 1 2 is component 1 (the ‘-component’) of vector 2 . Recalling Eq. (7.1), the particles have momenta = (1, ), (7.12) where the three particles have velocities , and ≡ ( ). From momentum conservation, we also know that 1 + 2 = 3 . (7.13)
eBook - PDF- Dietrich Marcuse(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
The interaction of electrons with other charged particles or photons, on the other hand, can be described without second quantization of the field equa-tions of the electron, because creation and annihilation of electrons require en-ergies that are not usually available in processes of interest to quantum elec-tronics. In this chapter we show that Maxwell's equations assume the form of the 448 Maxwell's Theory as Quantum Theory of the Photon CHAPTER TEN Schrödinger equation of the photon when they are written in momentum repre-sentation. However, Planck's constant must be introduced as a multiplicative factor whose value remains undetermined by Maxwell's theory. The expressions for the photon momentum and the photon spin follow naturally if the classical ex-pressions for the momentum and angular momentum of the electromagnetic field are interpreted as quantum-mechanical expectation values. The material in this chapter is not usually found in books on quantum theory or quantum electronics. It has been covered in the excellent book Quantum Elec-trodynamics by Akhiezer and Berestetskii [1]. Our treatment deviates from this reference in the use of MKS units instead of the so-called natural units, which cause the speed of light c and the quantum constant h to assume the value unity. Natural units completely obscure the way in which Planck's constant enters the theory. In addition to changing units we try to explain some of the complicated mathematics in more detail and give more physical insight into the meaning of certain quantum states of the photon. 1 0 . 2 MAXWELL'S EQUATIONS IN MOMENTUM REPRESENTATION Electromagnetic theory is based on Maxwell's equations (2.3-2) and (2.3-3) that describe the interrelation between the electric field vector and the magnetic field vector in the absence of currents and charges in an isotropic, nondisper-sive medium: V X H = e ^ (10.2-1) at and V = - (10.2-2) dt The electric permittivity is e and the magnetic permeability is .
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