Quantization of Energy

11 Key excerpts on "Quantization of Energy"

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  College Physics

  • Paul Peter Urone, Roger Hinrichs(Authors)
  • 2012(Publication Date)
  • Openstax

    (Publisher)

  In later chapters, we will examine many areas, such as atomic and nuclear physics, in which quantum mechanics is crucial. 29.1 Quantization of Energy Planck’s Contribution Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy. While most of us are familiar with the quantization of matter into lumps called atoms, molecules, and the like, we are less aware that energy, too, can be quantized. Some of the earliest clues about the necessity of quantum mechanics over classical physics came from the Quantization of Energy. 1144 Chapter 29 | Introduction to Quantum Physics This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Figure 29.3 Graphs of blackbody radiation (from an ideal radiator) at three different radiator temperatures. The intensity or rate of radiation emission increases dramatically with temperature, and the peak of the spectrum shifts toward the visible and ultraviolet parts of the spectrum. The shape of the spectrum cannot be described with classical physics. Where is the Quantization of Energy observed? Let us begin by considering the emission and absorption of electromagnetic (EM) radiation. The EM spectrum radiated by a hot solid is linked directly to the solid’s temperature. (See Figure 29.3.) An ideal radiator is one that has an emissivity of 1 at all wavelengths and, thus, is jet black. Ideal radiators are therefore called blackbodies, and their EM radiation is called blackbody radiation. It was discussed that the total intensity of the radiation varies as T 4 , the fourth power of the absolute temperature of the body, and that the peak of the spectrum shifts to shorter wavelengths at higher temperatures.

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  An Introduction to Quantum Optics

  Photon and Biphoton Physics

  • Yanhua Shih(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  ℏω must be carried by the wavefront of the subfield and thus uniformly distributed on the big sphere. Einstein then asked again: suppose that photon is annihilated by a point-like photon counting detector located on the surface of the big sphere, how long does it take for the energy on the other side of the big sphere to arrive at the detector? Two years? Bohr provided a famous answer to this question: the “wavefunction collapses” instantaneously! Why does the wavefunction need to “collapse”? Bohr did not explain. Nevertheless, Bohr has passed an important message to us: quantum mechanical picture of photon is different from Einstein’s granularity, or EM subfield. Although we still have questions regarding the wave-particle duality of a photon, we have to accept the experimental effect that the energy of the electromagnetic field is quantized in nature. We may have to face the truth that a new theory of radiation with quantization is necessary.

  2.4     FIELD QUANTIZATION AND THE LIGHT QUANTUM

  In blackbody radiation, the atoms on the walls of the cavity box continuously radiate electromagnetic waves into the cavity. In general, there are two fundamental principles governing the physical process of the radiation and determining the physical properties of the radiation field. The Schrödinger equation determines the quantized atomic energy level, and the govern the behavior of the radiation field. The interaction between the field and the atom results in a quantized electromagnetic field. The energy and the frequency of the emitted photon are determined by the quantized energy levels of the atom, ℏω = E2 − E1 . On the other hand, any excited electromagnetic field must satisfy the Maxwell equations which determine the harmonic mode structure and the superposition.

  In the quantum theory of light, the radiation field is treated as a set of harmonic oscillators. The energy of each mode is quantized in a similar way as that of a harmonic oscillator. To quantize the field, we will follow the standard procedure. First, we proceed to link the Hamiltonian of the free electromagnetic field to a set of independent harmonic oscillators. The quantum mechanical results of harmonic oscillators are then adapted to the quantized radiation field. Notice, here, free field means no “sources” or “drains” of the radiation field in the chosen volume of V = L3 that covers the field of interest. The energy of the free field is given by

  H =

  1 2

  ∫ V

  d 3

  r

  [

  ϵ 0

  E 2

  (

  r , t

  )

  +

  1

  μ 0

  B 2

  (

  r , t

  )

  ]

  ,

  (2.4.1)

  where V is the total volume of the field of interest. The volume is usually, but not necessarily, treated as a large finite cubic cavity of L

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  Basic Physical Chemistry

  The Route to Understanding

  • E Brian Smith(Author)
  • 2012(Publication Date)
  • ICP

    (Publisher)

  A most important difference between the new and the old physics is the recognition that energy is not continuous. 3.2 Basic ideas of quantum mechanics It was found that many of the failures of classical physics described above could be explained if the following bold assumption was made: energy comes in discrete packets and is not continuous . Max Planck, in 1900, first introduced this radical new idea, quantisation , to explain the colour of hot objects (black-body radiation). Albert Einstein extended its application to electromagnetic radiation by suggesting that the radiation itself consisted of small packets of energy, photons , with energy related to their frequency, ν , by E = hν, where h is Planck’s constant, 6 . 62 × 10 − 34 J s. Red light consists of low energy photons whereas blue or ultraviolet radiation is comprised of photons of higher energy. This idea provides a direct explanation of the photoelectronic effect. If the radiation is of too low a frequency, each photon has insufficient energy to dislodge electrons from the metal. A critical minimum energy, and a related frequency, ν c , given by E c = hν c , is necessary. The photons that comprise electromagnetic radiation in the blue or ultraviolet part of the electromagnetic spectrum correspond to radiation with a frequency on the order of magnitude 10 15 s − 1 . Their energy is E = hν = 6 . 62 × 10 − 34 J s × 10 15 s − 1 . The energy transferred if a mole of photons of this frequency is absorbed by a material is E = 6 . 62 × 10 − 19 J × 6 . 02 × 10 23 mol − 1 = 399 kJ mol − 1 . Electromagnetic radiation of this frequency can be absorbed by electrons in atoms or molecules and the energy absorbed is comparable with the strength of many chemical bonds. Ultraviolet light is often used to dissociate molecules and initiate chemical reactions. The new assumptions provided an explanation of the spectra of atoms.

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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  A system with a particular energy is in that energy state, and the energy state changes when the system assumes a different allowed energy value. The energy–time uncertainty relation can be stated as follows: Heisenberg’s Energy-Time Uncertainty Principle Let E Δ be the uncertainty in the energy of a system that is in a particular energy state. If t Δ is the uncertainty in the time during which the particle is in the state, then E t h 4π Δ Δ ≥ (29.5.2) According to Equation 29.5.2, a small value of t Δ corresponds to a large value of E Δ — that is, a state that exists for only a short time does not have a well-defined energy. Practice Problem 29.5.2 gives you practice using Equation 29.5.2. SUMMARY 29.1 Blackbody Radiation and Quantization A blackbody is a perfect emitter and absorber of radiation. The walls of a blackbody cavity continuously emit and absorb radiation, and the intensity of the radiation within the cavity depends only on the temperature of the cavity walls. German physicist Max Planck (1858–1947) discovered that he could describe blackbody radiation by imagining that atoms in the walls of the cavity behaved as tiny oscillators that emitted and absorbed electromagnetic radiation. Moreover, he assumed that the oscillator energy E may assume only discrete values for a given oscillator frequency f, as follows: … = = E nhf n 0, 1, 2, (29.1.1) The constant h is called Planck’s constant and has a value of = × ⋅ - h 6.63 10 J s 34 It follows from the conservation of energy and Equation 29.1.1 that the atomic oscillators that form the walls of the container can emit and absorb electromagnetic waves only in discrete energy packets, or quanta. I N T E R A C T I V E F E A T U R E I N T E R A C T I V E F E A T U R E 818 | Chapter 29 29.2 The Photon and the Photoelectric Effect Albert Einstein proposed that light consists of bundles of energy called photons, whose energy is E hf = (29.2.1) where f is the photon frequency and h is Planck’s constant.

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  Quantum Concepts in Physics

  An Alternative Approach to the Understanding of Quantum Mechanics

  • Malcolm Longair(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  (3.12) The average energy of the quanta is closely related to the mean kinetic energy per particle in the black-body enclosure, 3 2 kT . So far, Einstein has stated that the radiation ‘behaved as though’ it consisted of a number of independent particles. Is this just another ‘formal device’? The last sentence of Sect. 6 of his paper leaves the reader in no doubt: ‘the next obvious step is to investigate whether the laws of emission and transformation of light are also of such a nature that they can be interpreted or explained by considering light to consist of such energy quanta.’ Einstein considers three phenomena which cannot be explained by classical electromag- netic theory. 1. Stokes’ rule is the observation that the frequency of photoluminescent emission is less than the frequency of the incident light. This is explained as a consequence of the conservation of energy. If the incoming quanta each have energy h ν 1 , the re-emitted quanta can at most have this energy. If some of the energy of the quanta is absorbed by the material before re-emission, the emitted quanta of energy h ν 2 must have h ν 2 ≤ h ν 1 . 2. The photoelectric effect. This is the most famous result of the paper because Einstein made a definite quantitative prediction on the basis of the theory expounded above. Ironically, the photoelectric effect had been discovered by Hertz in 1887 in the same experiments which fully validated Maxwell’s equations. Perhaps the most remarkable feature of the effect was L´ en´ ard’s discovery that the energies of the electrons emitted from the metal surface are independent of the intensity of the incident radiation (L´ en´ ard, 1902). Einstein’s proposal provided an immediate solution to this problem. Radiation of a given frequency consists of quanta of the same energy h ν . If one of these is absorbed by the material, the electron may receive sufficient energy to remove it from the surface against the forces which bind it to the material.

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  Quantum Mechanics

  A Paradigms Approach

  • David H. McIntyre(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  107 C H A P T E R 5 Quantized Energies: Particle in a Box In the first part of this book we used the spin system to illustrate the basic concepts and tools of quan- tum mechanics. With a firm foundation in how quantum mechanics works, we are ready to address the central question that quantum mechanics was designed to answer: How do we explain the structure of the microscopic world? All around us are nuclei, atoms, molecules, and solids with unique properties that cannot be explained with classical physics but require quantum mechanics. For example, quantum mechanics can tell us why sodium lamps are yellow, why laser diodes have a unique color, and why uranium is radioactive. The key to understanding the structure of microscopic systems lies in the energy states that the systems are allowed to have. Each microscopic system has a unique set of energy levels that gives that system a “fingerprint” that sets it apart from other systems. With the tools of quantum mechanics, we can build a theoretical model for the system, predict that fingerprint, and compare it to the experimen- tal measurement. Our goal in this chapter and the ones that follow is to learn how to predict this energy fingerprint. In this chapter we will study a particularly simple model system that exhibits most of the important features that are shared by all microscopic systems. 5.1  SPECTROSCOPY The energy fingerprint of a system not only identifies that system uniquely, but the allowed energies determine the time evolution of the system through the Schrödinger equation, as we learned in Chapter 3. One of the primary experimental techniques for measuring the energy fingerprint of a system is spectros- copy. We saw a hint of this in the magnetic resonance example of Section 3.4: absorption and emission of photons causes transitions between quantized energy levels of the system only when the photon energy matches the spacing between the energy eigenstates.

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  Physics : Imagination And Reality

  • P R Wallace(Author)
  • 1991(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 13 F U N D A M E N T A L P R I N C I P L E S OF Q U A N T U M T H E O R Y The accumulation of spectroscopic evidence that atomic radia-tion showed frequency peaks, along with Einstein's introduction of the light quantum (which was first called a photon only in 1926), was strong evidence that atoms existed in discrete energy states. The hydrogen atom, for example, had strong peaks in the emission and absorption of light at frequencies proportional to where n x and n 2 were integers. This could be interpreted thus: hy-drogen atoms existed in energy states —R/n 2 , and when a hydrogen atom in a state n 2 made a transition to a state rii , it would emit a light quantum of energy and hence of frequency equal to this quantity divided by h, the Planck constant. But how could this be? The hydrogen atom might be likened to a sort of solar system held together by electric rather than grav-itational forces, the negatively charged electron playing the role of a planet orbiting around a positively charged proton sun. But planets in solar systems can have arbitrary energy. Furthermore, it 336 1 «r i „ / « ( ' i v? i 'n) Fundamental Principle! of Quantum Theory 337 was known from Maxwell's electromagnetic theory that accelerating charges emitted a continuous spectrum of electromagnetic radiation. Worse than that, radiation was predicted to be so strong that the electron lost energy very rapidly, and would in fact spiral in to the nucleus in a minute fraction of a second. Nothing in a world built in this way would be stable! Even if one could think of a mechanism for transitions between atomic states in which the atom suddenly lost a discrete quantity of energy, other questions remained. What determined when the atom would make such transitions, or in what direction the atom would emit its quantum of radiation? The new quantum ideas seemed to pose a myriad of new and seemingly unfathomable puzzles.

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  Quantum Theory and Measurement

  • John Archibald Wheeler, Wojciech Hubert Zurek, John Archibald Wheeler, Wojciech Hubert Zurek, John Wheeler, Wojciech Zurek(Authors)
  • 2014(Publication Date)
  • Princeton University Press

    (Publisher)

  The statistical result of position determinations will always be the same, whatever the phase of the incident radiation. We may assume that experiments with radiation, the theory of which has not yet been developed, will give the same results about phase relations between atoms and incident radiation. Finally let us examine the connection* of equation (2), E 1 J 1 ~ h, with a complex of problems which Ehrenfest and other investigators have discussed on the basis of Bohr's correspondence principle in two important papers^ Ehrenfest and Tolman speak of "weak quantization" when the quantized periodic motion is interrupted through quantum jumps or rather perturbations in intervals of time which can be regarded as not very long compared to the periods of the system. These cases should reveal not only the exact quantum energy values but also energy values which do not differ too much from the quantum values, and these with a smaller and qualitatively predictable a priori probability. In quantum mechanics this behavior is to be interpreted in these terms. As the energy is really changed by external perturbations or quantum jumps, every energy measurement, insofar as it is to be unique, must be done in the time between two perturbations. In this way an upper bound is specified for I 1 in the sense of §1. Therefore we measure the energy value E0 of a quantized state also only within a spread E1 ~ Hft1. Here the question is meaningless even in principle whether the system "really" takes on with the correspondingly lower statistical weight such energy values E as deviate from E0, or whether their experimental realization is to be attributed only to the inaccuracy of the measurement. If J1 is smaller than the period of the system then it is no longer meaningful to speak of discrete stationary states or discrete energy values. * W. Pauli drew my attention to this connection. f P. Ehrenfest and G. Breit (Zeits. f. Physik, 9, 207 [1922]) and P.

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  An Introduction to Quantum Optics

  Photon and Biphoton Physics

  • Yanhua Shih, Yanhua Shih(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  The field quantization is necessary. 8.3 The Light Quantum and the Field Quantization In blackbody radiation, the atoms on the walls of the cavity box continu-ously radiate electromagnetic waves into the cavity. In general, there are two fundamental principles governing the physical process of the radiation and determining the physical properties of the radiation field. The Schrödinger equation determines the quantized atomic energy level, and the Maxwell equations govern the behavior of the radiation field. The interaction of the field and the atom results in a quantized electromagnetic field. The energy and the frequency of the emitted photon are determined by the quantized energy levels of the atom, ω = E 2 − E 1 . On the other hand, any excited elec-tromagnetic field must satisfy the Maxwell equations, which determine the harmonic mode structure and the superposition. In the quantum theory of light, the radiation field is treated as a set of harmonic oscillators. The energy of each mode is quantized in a sim-ilar way as that of a harmonic oscillator. To quantize the field, we will follow the standard procedure. First, we proceed to link the Hamiltonian of the free electromagnetic field to a set of independent harmonic oscillators. The quantum mechanical results of harmonic oscillators are then adapted to the quantized radiation field. Notice, here, free field means no “sources” or “drains” of the radiation field in the chosen volume of V = L 3 that covers the field of interest. The energy of the free field is given by H = 1 2 V d 3 r 0 E 2 ( r , t ) + 1 μ 0 B 2 ( r , t ) , (8.16) where V is the total volume of the field of interest. The volume is usually, but not necessarily, treated as a large finite cubic cavity of L 3 to simplify the mathematics. We will rewrite Equation 8.16 in the following form to link it with the Hamiltonian of a set of independent harmonic oscillators

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  Theoretical Concepts in Physics

  An Alternative View of Theoretical Reasoning in Physics

  • Malcolm S. Longair(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  I now knew for a fact that the elementary quantum of action played a far more significant part in physics than I had originally been inclined to suspect and this recognition made me see clearly the need for the introduction of totally new methods of analysis and reasoning in the treatment of atomic problems. 12 Indeed, it was not until after about 1908 that Planck fully appreciated the quite fundamental nature of quantisation, which has no counterpart in classical physics. His original view was that the introduction of energy elements was 383 15.5 Planck and the Physical Significance of h a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result. 13 This quotation is from a letter by Planck to R.W. Wood written in 1931, 30 years after the events described in this chapter. I find it a rather moving letter and it is worthwhile reproducing it in full. October 7 1931 My dear colleague, You recently expressed the wish, after our fine dinner in Trinity Hall, that I should describe from a psychological viewpoint the considerations which had led me to propose the hypothesis of energy quanta. I shall attempt herewith to respond to your wish. Briefly summarised, what I did can be described as simply an act of desperation. By nature I am peacefully inclined and reject all doubtful adventures. But by then I had been wrestling unsuccessfully for six years (since 1894) with the problem of equilibrium between radiation and matter and I knew that this problem was of fundamental impor- tance to physics; I also knew the formula that expresses the energy distribution in the normal spectrum. A theoretical interpretation therefore had to be found at any cost, no matter how high. It was clear to me that classical physics could offer no solution to this problem and would have meant that all energy would eventually transfer from matter into radiation.

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  This is an absurd result because the energy emitted at any finite temperature should be finite. Neither the Rayleigh–Jeans formula nor Wien’s displacement formula could explain the blackbody radiation phenomenon completely over the whole range of wavelengths. Only the quantum theory of radiation can explain blackbody radiation! To understand the chemical properties of matter, as summarized in the periodic table of elements, the fact that not all the states of electrons permitted by the classical model are feasible realistically must be taken into account. Even cosmological and astrophysical phenomena are not completely explainable by the laws of classical physics. The failure of classical concepts to explain the physical phenomena completely forced the scientific community to search for the missing link. Their attempts to search for the missing link gave birth to quantum physics and the mathematical modelling of quantum physics is quantum mechanics. The finer laws of quantum physics are not far away from the reality of the macro-world or are only means to explain microscopic phenomena; they are the true laws of nature. Actually, all of physics is quantum physics. All the laws of quantum physics reduce to the laws of classical physics under certain circumstances. If quantum physics is a super set, then classical physics is a sub-set. To be specific, lim Quantum Physics = Classical Physics n →∞ In this relation, n is the principal quantum number. 7.3 Particles and Waves Everything in the world is a wave; every thing in the world is a particle. They are the manifestation of the same thing in different forms. The physical reality we perceive has Elementary Concepts of Quantum Physics 551 its roots in the world of elementary particles. Electron has mass and charge like a particle and obeys the laws of particle mechanics as in a CRO tube or picture tube of a television set but it behaves like a perfect wave in case of an electron microscope.

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