De Broglie Wavelength

11 Key excerpts on "De Broglie Wavelength"

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  101 Quantum Questions

  What You Need to Know About the World You Can't See

  • Kenneth W. Ford, Paul Hewitt, Kenneth William Ford(Authors)
  • 2011(Publication Date)
  • Harvard University Press

    (Publisher)

  In these early experiments, the wavelength of the electrons was comparable to the spacing between atoms in a solid. Later, experimenters learned how to slow down neutrons so much that their wavelength greatly exceeded the spacing of atoms in a solid. (I will explain this inverse connection between speed and wavelength in answer to the next question.) As a result, a neutron drifting lazily through a material “reaches out” via its wavelength to interact simultaneously with many atoms, behavior hardly expected of a particle smaller than a single atomic nucleus.

  65. What is the de Broglie equation? What is its significance?

  By chance, the young French Prince Louis-Victor de Broglie (pronounced, roughly, “Broy” ) received his bachelor’s degree in physics in 1913, the same year as Niels Bohr’s groundbreaking work on the quantum theory of the hydrogen atom. I don’t know if quantum physics was yet on de Broglie’s mind at that time, but later, in his Nobel Prize address of 1929, he did speak of his attraction to “the strange concept of the quantum, [which] continued to encroach on the whole domain of physics.”

  FIGURE

  42 Experimental results of Davisson and Germer showing that electrons of 54 eV, after striking a nickel crystal, emerge mostly in a certain direction because of diffraction and interference of the electron waves. Image adapted from Nobel Lectures, Physics (Amsterdam: Elsevier, 1965).

  Following service in World War I, de Broglie took up graduate study in physics and, in 1924, as part of his doctoral dissertation at the University of Paris, he offered the deceptively simple but powerful equation that now bears his name. The de Broglie equation is written

  λ = h/p

  On the left, the symbol λ (lambda) stands for wavelength—evidently a wave property. The p in the denominator on the right stands for momentum—clearly a particle property. Linking the two is Planck’s constant h, which appears in every equation of quantum physics. This equation was known to be true for photons—if you believed in photons. De Broglie asserted it to be true for electrons and all particles. When Davisson, Germer, and Thomson measured the wavelengths of electrons a few years later, they found that indeed their measurements conformed to de Broglie’s equation. The de Broglie equation has stood the test of time and remains a pillar of quantum physics. It is as simple in appearance and, in its way, as powerful as Einstein’s E = mc 2 .*

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  Electronic Structure Modeling

  Connections Between Theory and Software

  • Carl Trindle, Donald Shillady(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  However, for subatomic particles, and in particular the electron, the De Broglie Wavelength is comparable in size to chemical bonds. * De Broglie arrived at this relationship by consideration of an oddity in relativity theory. [http: == www.davis-inc.com = physics = ]. It of course seems magical. Consider that the circular Bohr orbit with circumference 2 p r accommodates one wavelength. Then the momentum is p ¼ h = (2 p r ) and the kinetic energy T is h 2 = [2 m (2 p r ) 2 ]. The potential V is Z = r . The minimum energy T þ V ¼ m (2 p Z ) 2 = 2 h 2 is in agreement with Bohr’s fit. Hey presto! 1 Only 3 years after the De Broglie hypothesis was published, Schro ¨ dinger [6] published a series of papers defining the wave mechanics and solving the harmonic oscillator and the hydrogen atom. Here is a way to express a link between the De Broglie waves and the Schro ¨ dinger mechanics. The general form of a wave can be described as a complex exponential with amplitude A , frequency v , and wavelength l . c ( x , t ) ¼ A exp 2 p i x l v t h i The first derivative of c with respect to x is simple d c dx ¼ 2 p i l A exp 2 p i x l v t h i ¼ 2 p ip h c ¼ ip ¯ h c and returns the momentum as the eigenvalue. The second derivative produces the kinetic energy. d 2 c dx 2 ¼ 2 p l 2 A exp 2 p i x l v t h i ¼ 2 p h 2 p 2 c ¼ 2 mT ¯ h 2 c ¼ 2 m ( E V ) ¯ h 2 c Use of the De Broglie relation already incorporates reference to relativity. We can rearrange this further to obtain the standard form of the Schro ¨dinger equation ¯ h 2 2 m d 2 dx 2 þ V # c ¼ H c ¼ E c The Schro ¨ dinger equation is thus a consequence of relativistic ideas. The operator which produces the energy eigenvalue is called the Hamilto-nian, after the formulation of the total energy in classical mechanics. One may also recover the energy by evaluation of the time derivative @ c @ t ¼ 2 p i v A exp 2 p i x l v t h i ¼ i E ¯ h c Now we are ready to form a strategy for using ‘‘wave mechanics.’’ 1.

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  Physical Properties of Materials for Engineers

  • Daniel D. Pollock(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Thus, when the group velocity equals the phase velocity the wave packet may be treated like a particle. The phase velocity of a particle has no analog in classical mechanics. Under the condition of Equation 2-20, however, this equivalence reduces waves to their classical counterparts. Such “particles” can be treated classically if they obey Equation 2-20. If they do not obey this equation, they spread out as they move like ripples on a pond and vanish; they cannot be treated as particles. This applies to matter waves associated with a particle of matter, with a quantum of energy, or with the corpuscular properties of electromagnetic radiation.

  Thus, it has been established that electromagnetic radiation can be treated either as particles or as matter waves. The same is true for electrons or other quantum particles. The ability to be treated in either of these ways is known as duality, a situation arising not from the electrons or the light waves themselves, but from the fundamental inability to describe their behavior more precisely.

  It is interesting to note again that this duality is implicit in Einstein’s fourth assumption (Section 1.4 ) that photons behave like waves of corresponding frequency.

  In general, the longer the wavelength, the more difficult it is to show the corpuscular nature of the radiation. The wave-like character is not pronounced because of the low energies of the quanta. As the wavelength becomes shorter the corpuscular properties become more evident.

  2.3 THE De Broglie Wavelength

  Use was made of the de Broglie equation (Equation 2-5) in the preceding section prior to proving it. This important contribution, anticipated by Einstein, is derived in this section.

  The relationship between particle- and wave-like behavior was given by de Broglie in 1924. In effect, he showed that a wavelength could be associated with a particle and that wavelength is related to the momentum of the particle, p. This is expressed as

  λ  = h p

  (2-21)

  This emphasizes the dual behavior

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  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  He made the explicit proposal that the wavelength l of a particle is given by the same relation (Equation 29.6) that applies to a photon: De Broglie Wavelength l 5 h p (29.8) where h is Planck’s constant and p is the magnitude of the relativistic momentum of the particle. Today, l is known as the De Broglie Wavelength of the particle. Confirmation of de Broglie’s suggestion came in 1927 from the experiments of the American physicists Clinton J. Davisson (1881–1958) and Lester H. Germer (1896–1971) and, independently, those of the English physicist George P. Thomson (1892–1975). Davisson and Germer directed a beam of electrons onto a crystal of nickel and observed that the electrons exhibited a diffraction behavior, analogous to that seen when X-rays are diffracted by a crystal (see Section 27.9 for a discussion of X-ray diffraction). The wavelength of the electrons revealed by the diffraction pattern matched that predicted by de Broglie’s hypothesis, l 5 h/p. More recently, Young’s double-slit experiment has been performed with electrons and reveals the effects of wave interference illustrated in Figure 29.1. Particles other than electrons can also exhibit wave-like properties. For instance, neutrons are sometimes used in diffraction studies of crystal structure. Figure 29.12 9. The speed of a particle is much less than the speed of light. Thus, when the particle’s speed doubles, the particle’s momentum doubles, and its kinetic energy becomes four times greater. However, when the momentum of a photon doubles, its energy becomes (a) two times greater (b) four times greater (c) one-half as much (d) one-fourth as much. 10. Review Conceptual Example 4 as background for this question. The photograph shows a device called a radiometer. The four regular panels are black on one side and shiny like a mirror on the other side. In bright light, the panel arrangement spins around in a direction from the black side of a panel toward the shiny side.

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  Problems and Solutions in University Physics

  Optics, Thermal Physics, Modern Physics

  • Fuxiang Han(Author)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  πr for the electron in the hydrogen atom, we can not treat the hydrogen atom classically.

  11.4 Thermal energy and De Broglie Wavelength of molecules.

  The typical kinetic energy of particles, such as the molecules in a gas, at temperature T in Kelvins, is kB T, where kB is the Boltzmann constant.

  (1) What is the typical kinetic energy of such particles at room temperature (taken as 300 K), measured in electron volts?

  (2) Estimate the De Broglie Wavelength of nitrogen molecules in air at room temperature.

  (1) At T = 300 K, the kinetic energy of a molecule is given by

  (2) Because EK is very small, we can make use of Newtonian mechanics to compute the momentum of a nitrogen molecule and have

  with m the mass of a nitrogen molecule, m = 28.014 u = 4.652 × 10−26 kg ≈ 26094.875 MeV/c2 . The De Broglie Wavelength of a nitrogen molecule is then given by

  11.5 Width of a spectral line. We have seen that the hydrogen atom and other atoms possess states of various energies and that the higher-energy excited states decay spontaneously to lower-energy states, emitting photons in the process. We can compute the wavelength of the emitted photons from the difference in energies of the initial and final states and hence predict the spectral lines in the emission spectrum of hydrogen and other elements. Atoms do not stay in higher-energy states for a long time before they decay to lower-energy states. Suppose a certain excited state lives for just 10−7 s. In this case we know the time when the atom is in the excited state quite accurately – to within Δt = 10−7 s. That means there is a significant uncertainty ΔE in the energy of the state.

  (1) Estimate the size of the uncertainty in the energy of the state.

  (2) Estimate the range Δf of frequencies that will be measured for the photons emitted when the state decays. This results in a broadened spectral line that, when inspected closely, does not consist of just a single frequency but a narrow range of frequencies.

  (3) We can also turn the calculation around. A particular spectral line in hydrogen has wavelength λ = 121.5 nm and is observed to have a line width (in terms of wavelength) of about 10−7 λ. Estimate the lifetime of the corresponding excited state.

  (1) From the time-energy uncertainty relation ΔEΔt ≥ ħ/2, the uncertainty in the energy of the state is given by

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  Modern Physics

  • Kenneth S. Krane(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Is this dual particle-wave nature a property only of light or of material objects as well? In a bold and daring hypothesis in his 1924 doctoral dissertation, de Broglie chose the latter alternative. Examining Eq. 3.18, E = hf , and Eq. 3.20, p = h∕λ, we find some difficulty in applying the first equation in the case of particles, for we cannot be sure whether E should be the kinetic energy, total energy, or total relativistic energy (all, of course, are identical for light). No such difficulties arise from the second relationship. De Broglie suggested, lacking any experimental evidence in support of his hypothesis, that Meggers Gallery / AIP / Science Source Louis de Broglie (1892–1987, France). A member of an aristo- cratic family, his work contributed substantially to the early devel- opment of the quantum theory. ∗ De Broglie’s name should be pronounced “deh-BROY” or “deh-BROY-eh,” but it is often said as “deh-BROH-lee.” 4.1 De Broglie’s Hypothesis 107 associated with any material particle moving with momentum p there is a wave of wavelength λ, related to p according to λ = h p (4.1) where h is Planck’s constant. The wavelength λ of a particle computed accord- ing to Eq. 4.1 is called its De Broglie Wavelength. Example 4.1 Compute the De Broglie Wavelength of the following: (a) A 1000-kg automobile traveling at 100 m∕s (about 200 mi∕h). (b) A 10-g bullet traveling at 500 m∕s. (c) A smoke particle of mass 10 −9 g moving at 1 cm∕s. (d ) An electron with a kinetic energy of 1 eV. (e) An elec- tron with a kinetic energy of 100 MeV. Solution (a) Using the classical relation between velocity and momentum, λ = h p = h mv = 6.6 × 10 −34 J ⋅ s (10 3 kg) (100 m∕s) = 6.6 × 10 −39 m (b) As in part (a), λ = h mv = 6.6 × 10 −34 J ⋅ s (10 −2 kg) (500 m∕s) = 1.3 × 10 −34 m (c) λ = h mv = 6.6 × 10 −34 J ⋅ s (10 −12 kg) (10 −2 m∕s) = 6.6 × 10 −20 m (d ) The rest energy (mc 2 ) of an electron is 5.1 × 10 5 eV.

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  An Introduction to Quantum Physics

  • A.P. French(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  Eq. 2-11 ).

  Fig. 2-2 De Broglie Wavelength as a function of kinetic energy. For electrons, λo = 0.0243 Åand E

  o

  = 0.511 Me V; for protons and neutrons, λ

  o

  ≈ 1.32 × 10−5 Å = 1.32 F and E

  0

  ≈ 939 MeV.

  In Table 2-1 we list a few examples of the numerical values of De Broglie Wavelengths. See the end of the chapter for practice problems involving such calculations.

  TABLE 2-1 De Broglie Wavelengths

  Particle	Value of λa
  Electrons of kinetic energy	1 eV	12.2 Å
  100 eV	1.2 Å
  104 eV	0.12 Å
  Protons of kinetic energy	1 keV	0.009 Å (= 900 F)
  1 MeV	28.6 F
  1 GeV	0.73 F
  Thermal neutrons (at 300 K)	1.5 Å (average)
  Neutrons of kinetic energy	10 MeV	9.0 F
  He atoms at 300 K	0.75 Å (average)

  a 1 Å = 10−8 cm; 1 F = KT−13 cm = 10−5 Å

  2-4  THE DAVISSON-GERMER EXPERIMENTS

  In 1925 C. J. Davisson and L. H. Germer at Bell Telephone Laboratories were making studies of electron scattering by crystal surfaces using polycrystals of nickel when they had a famous disaster that led them to triumph. The vacuum system broke open to the air while the nickel target was hot, there by oxidizing the target. They tried to change the nickel oxide back to nickel by heating it in hydrogen and then in vacuum. In the process they incidentally changed it from a polycrystalline aggregate into a few large crystals. The character of the electron scattering was drastically changed, showing new strong reflections at particular angles. Taking the hint, they then deliberately used single crystals as targets, and proceeded to discover just the kind of behavior that de Broglie’s hypothesis of particle waves predicted (although they had not known of de Broglie’s theory at the time they made the initial observations).8

  The basic features of the experimental arrangement are shown schematically in Figure 2-3a . An electron beam was directed perpendicularly onto a face of a nickel crystal. A collector accepted electrons coming off at some angle ϕ to the normal (or π − ϕ

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Today, l is known as the De Broglie Wavelength of the particle. Confirmation of de Broglie’s suggestion came in 1927 from the experiments of the American physicists Clinton J. Davisson (1881–1958) and Lester H. Germer (1896–1971) and, independently, those of the English physicist George P. Thomson (1892–1975). Davisson and Germer directed a beam of electrons onto a crystal of nickel and observed that the electrons exhibited a diffraction behavior, analogous to that seen when X-rays are diffracted by a crystal (see Section 27.9 for a discussion of X-ray diffraction). The wavelength of the electrons revealed by the diffraction pattern matched that predicted by de Broglie’s hypothesis, l 5 h/p. More recently, Young’s double-slit experiment has been performed with electrons and reveals the effects of wave interference illustrated in Figure 29.1. Particles other than electrons can also exhibit wave-like properties. For instance, neutrons are sometimes used in diffraction studies of crystal structure. Figure 29.12 9. The speed of a particle is much less than the speed of light. Thus, when the particle’s speed doubles, the particle’s momentum doubles, and its kinetic energy becomes four times greater. However, when the momentum of a photon doubles, its energy becomes (a) two times greater (b) four times greater (c) one-half as much (d) one-fourth as much. 10. Review Conceptual Example 4 as background for this question. The photograph shows a device called a radiometer. The four regular panels are black on one side and shiny like a mirror on the other side. In bright light, the panel arrangement spins around in a direction from the black side of a panel toward the shiny side. Do photon collisions with both sides of the panels cause the observed spinning? Question 10 Figure 29.12 (a) The neutron diffraction pattern and (b) the X-ray diffraction pattern for a crystal of sodium chloride (NaCl).

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  Quantum Mechanics

  • T Y Wu, T D Lee;C N Yang;;(Authors)
  • 1986(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3 Wave Mechanics: Early Developments Einstein, in 1905, proposed the quantum theory of radiation in which electromagnetic waves having the characteristic properties of wave length, frequency, diffraction and interference are ascribed the particle properties of energy and momentum. In 1923, Louis de Broglie (1892- ) proposed the converse, namely, a particle is associated with wave properties thereby completing the particle-wave symmetry, or duality and, for a while, further deepening the dilemma confronting physics. Fortunately the new ideas of de Broglie led to the development of wave mechanics by Schrodinger in 1926, and by 1927, to a new, complete, at least self-consistent, system of physics, including the matrix mechanics of Heisenberg, Born and Jordan described in the preceding chapter. 1. de Broglie's ideas (1923-1924) de Broglie's ideas—namely, an even closer, more basic relation between particle and wave than that suggested by Einstein for the photon theory— had been suggested in some observations of W. R. Hamilton (1805-1865) as early as between 1828-1837. Hamilton noted that there is a close formal similarity between the rays in optics and the trajectories in particle dynamics. For a light ray in a medium whose index of refraction n is a function of spatial coordinates n — n(x, y, z), Fermat gives the equation in the form of a varia-tional principle—known as the principle of least time (1657) B ds — = 0, u = phase velocity, 4 U which states that the light ray from a point A to B follows a path such that the time of transit is minimum. As vX — u and u = c[n, the equation can be expressed in the form i A Hx,y,z) 1 ds = 0. (IIM) On the other hand, for a particle in a potential field, Maupertuis gave the principle of least action (1744)

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  Physics 1922 – 1941

  Including Presentation Speeches and Laureates' Biographies

  • Sam Stuart(Author)
  • 2013(Publication Date)
  • Elsevier

    (Publisher)

  The lines of the grating will appear to us much closer together than they actually are. For X-rays impinging at this almost skimming incidence on the grating the effect will be as if the lines were very closely set and diffraction 256 I 9 2 9 L. DE B R O G L I E phenomena analogous to those of light will occur. This is what the above-mentioned physicists confirmed. But then, since the electron wavelengths are of the order of X-ray wavelengths, it must also be possible to obtain diffraction phenomena by directing a beam of electrons on to an optical diffraction grating at a very low angle. Rupp succeeded in doing so and was thus able to measure the wavelength of electron waves by comparing them directly with the spacing of the mechanically traced lines on the grating. Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity ; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discov-ered. And it is on this concept of the duality of waves and corpuscles in Na-ture, expressed in a more or less abstract form, that the whole recent devel-opment of theoretical physics has been founded and that all future devel-opment of this science will apparently have to be founded. Biography Prince Louis-Victor de Broglie of the French Academy, Permanent Secretary of the Academy of Sciences, and Professor at the Faculty of Sciences at Paris University, was born at Dieppe (Seine Inférieure) on 15th August, 1892, the son of Victor, Duc de Broglie and Pauline d'Armaillé. After studying at the Lycée Janson of Sailly, he passed his school-leaving certificate in 1909. He ap-plied himself first to literary studies and took his degree in history in 1910.

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  Foundations of Modern Physics

  • Steven Weinberg(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  Whether or not this story is true, the idea was a good one. According to Eq. (5.1.2), the wavelength of a non-relativistic electron with kinetic energy E e  m e c 2 is given by λ = 2π ¯ h/p e = 2π ¯ h/  2m e E e = 12.26 × 10 −8 cm [E e (eV)] −1/2 . (5.1.7) Hence we only need electrons with energy a bit larger than 10 eV to get wave- lengths nearly as small as a typical lattice spacing, about 10 −8 cm. This is no coincidence. In de Broglie’s interpretation of the Bohr quantization assumption, the wavelength of an electron with an energy of a few eV, which is typical of atomic binding energies, must fit a few times around an atomic orbit, and therefore must be similar to the size of the atom, which is similar to the spacing of atoms in crystals. Several physicists tried and failed to observe the diffraction of electron waves, until it was finally measured in 1927 by Clinton Davisson (1881–1958) and Lester Germer (1896–1971) at the old Bell Telephone Laboratories building on West Street in Manhattan. 5 (It was also measured at about the same time at the University of Aberdeen by George Paget Thomson (1892–1975), a son of J. J. Thomson.) They used a beam of electrons with kinetic energy 54 eV, incident on a single crystal of nickel with a spacing of lattice planes d = 0.91 × 10 −8 cm (already known from measurements using X-ray diffrac- tion). Electrons are reflected not only from the surface of the crystal, but from numerous planes within the nickel. At certain angles θ between the incident and reflected waves all these reflected waves go off with the same phase and therefore add constructively, leading to enhanced reflection at these angles.

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