Wave Packet

9 Key excerpts on "Wave Packet"

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  Modern Physics

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

    (Publisher)

  4.5 Wave PacketS In Section 4.3, we described measurements of the wavelength or frequency of a Wave Packet, which we consider to be a finite group of oscillations of a wave. That is, the wave amplitude is large over a finite region of space or time and is very small outside that region. Before we begin our discussion, it is necessary to keep in mind that we are discussing traveling waves, which we imagine as moving in one direction with a uniform speed. (We’ll discuss the speed of the Wave Packet later.) As the Wave Packet moves, individual locations in space will oscillate with the frequency or wavelength that characterizes the Wave Packet. When we show a static picture of a Wave Packet, it doesn’t matter that some points within the packet appear to have positive displacement, some have negative displacement, and some may even have zero displacement. As the wave travels, those locations are in the process of oscillating, and our drawings may “freeze” that oscillation. What is important is the locations in space where the overall Wave Packet has a large oscillation amplitude and where it has a very small amplitude. ∗ In this section, we discuss how to build a Wave Packet by adding waves together. A pure sinusoidal wave is of no use in representing a particle—the wave extends from −∞ to +∞, so the particle could be found anywhere. We would like the particle to be represented by a Wave Packet that describes how the particle is localized to a particular region of space, such as an atom or a nucleus. The key to the process of building a Wave Packet involves adding together waves of different wavelength. We represent our waves as A cos kx, where k is ∗ By analogy, think of a radio wave traveling from the station to your receiver. At a particular instant of time, some points in space may have instantaneous electromagnetic field values of zero, but that doesn’t affect your reception of the signal. What is important is the overall amplitude of the traveling wave.

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  Atomic And Free Electrons In A Strong Light Field

  • Mikhail V Fedorov(Author)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  CHAPTER 6 Wave PacketS 6.1. Free-electron Wave Packets In all the previous chapters, the two main alternative approaches used to describe an electron were either (1) a purely classical description in which an electron was considered as a localized particle of a very small and constant size r 0 = ^/mc 2 « 2.5xl0~ 13 cm, or (2) a version of the quantum-mechanical description with an electron wave function approximated by a plane wave homogeneously spread over the entire space and showing no localization at all. Surprisingly enough, sometimes these two models give coinciding results. However, this is not the general rule. There is a class of problems in which experimentally measurable parameters depend strongly on the electron state and, in particular, on the degree of the electron localization in space. Some such problems are considered below in this and the following three chapters. Of course, in reality electrons, as well as any other particles, have some degree of localization, and their size is not infinitely large. On the other hand, this size is usually not as small as a classical electron radius ro- The most general pure quantum-mechanical state of an electron free from any interactions is given by a superposition of plane waves where the coefficients C p are constant. Quantum states of the form 278 Wave Packets 279 (6.1.1) are known as Wave Packets, and probably for the first time this concept was introduced by Schrodinger. 185 Wave Packets describe nonstationary states of a particle.

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  Quantum Mechanics I

  The Fundamentals

  • S. Rajasekar, R. Velusamy(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  The behaviour of Coulomb Wave Packets on circular [ 14 ] and elliptical orbits [ 15 ] are tested experimentally on Rydberg atom systems [ 16, 17 ]. Study of Wave Packet is the subject of much current research in atomic, molecular, chemical and condensed matter physics. They are well suited for studying the classical limit of quantum mechanical systems. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron Wave Packets. Solved Problem 2: Find the relation between the phase and group velocities of a Wave Packet obtained by superposing the three waves given by ψ 1 = A e i (k x − ω t), ψ 2 = A 2 e i [ (k + d k) x − (ω + d ω) t ] and ψ 3 = A 2 e i [ (k − d k) x − (ω − d ω) t ]. We. obtain ψ = ψ 1 + ψ 2 + ψ 3 = A e i (k x − ω t) 1 + 1 2 e i (x d k − d ω t) + 1 2 e − i (x d k − d ω t) = A e i (k x − ω t) [ 1 + cos (x d k − d ω t) ] = A e i (k[--=PL. GO-SEPARATOR=--]x − ω t) 1 + cos d k x − d ω d k t. The envelope moves with the velocity d ω / d k and is the group velocity v g. 10.3 Wave Packets and Uncertainty Principle In quantum mechanics, a particle is described by a Wave Packet ψ (x, t). The Wave Packet surrounds the position of the classical particle. The particle may be found anywhere within the region with a probability density | ψ (x, t) | 2. This implies that the position of the particle is indeterminate within the domain of the Wave Packet. A similar argument for the momentum wave function ϕ (p, t) says that the momentum of the particle is also indeterminate within the limits of ϕ (p, t). Since ψ (x, t) and ϕ (p, t) are related by Fourier transforms the squaring of the Wave Packet in one domain will elongate in the other domain as space domain and momentum domain have inverse relationship

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  Interpreting Quantum Mechanics

  A Realistic View in Schrodinger's Vein

  • Lars-Göran Johansson(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  But a Wave Packet can transmit information. The mathematically simplest form to handle is the minimum uncertainty Wave Packet. The signal velocity is the group velocity of the Wave Packet and it is defined as v g = 2 π (d f d k) k = k 0 (4.24) where k 0 refers to the average wave number in the Wave Packet (McGervey, 1971, p. 116). For a matter Wave Packet this becomes v g = (d E d p) p = p 0 = p 0 c 2 E 0 = v 1 ⁢ (4.25) which is just as it should be. The phase velocity of equations (4.20) – (4.23) is the average phase velocity in the Wave Packet and it must be assumed that the spread around this value is small in order to get a Wave Packet. The frequencies f 0 and f 1 in equations (4.18) – (4.22) are the average frequencies, in different frames, of the Wave Packet. The picture is that of rather 'amorphous' object made up of an indefinite number of vibrations of slightly different frequencies. The reason why this object is usually referred to as a particle is that it interacts as one inseparable whole at a rather well-localised places in space. The wave description is appropriate when and only when discussing its motion, whereas in interactions we use the particle picture. As the Wave Packet is composed of many plane waves with infinite extension the one-dimensional case can be expressed as Ψ (x, t) = ∫ − ∞ ∞ G (k) exp [ − i k (x - v p t) ] d k ⁢ (4.26) where G (k) is the envelope function. If, for the sake of simplicity, we chose a Gaussian function for G (k) we will get Ψ (x, t) = ∫ − ∞ ∞ exp ⁡ (- (k - k 0) 2 a) exp [ − i k (x - v p t) ] d k ⁢ (4.26) where k 0 is the mean wave number in the Wave Packet. The rms-width of this Wave Packet is proportional to √a

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  Introduction to Modern Physics

  • John Mcgervey(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  7, and hence it does not have a single, precise wavelength. However, a localized wave, or Wave Packet, can be built up by adding together infinite waves which interefere destructively everywhere except in one small region. The procedure for doing this was worked out long ago. Although it is purely a mathematical 14 One method of observing (he electron as it passes a point would be to detect the charge induced by the electron as it passes a conductor; the sudden appearance of the induced charge produces a pulse which can be observed on an oscilloscope. A sufficiently large pulse can be produced if one has a group of electrons traveling together; such a group can be obtained from an electron accelerator. This method of measuring electron velocity has been used in a demonstration that electrons obey the relativistic relation between energy and speed. (See the Physical Science Study Committee film The Ultimate Speed produced by William Bertozzi.) 4.3 Wave PacketS 113 problem, the superposition of waves is so fundamental in quantum theory that it is worthwhile at this point to go into the details (in a nonrigorous way). Fourier Series. We begin with the Fourier series, which enables one to express a periodic function as a sum of sine and cosine functions. Suppose 5b Fig. 7. A Wave Packet representing a particle whose position at t —0 lies approximately between χ = + b and χ = —b, and whose momentum is approximately h/λο. At a later time, when the center of the packet has moved to χ = +5b, the packet is broader, because the various component waves move with different speeds. Notice that the crest marked X moves only half as far as the Wave Packet as a whole; the phase velocity is one-half of the group velocity. (See Problems 17 and 20 at the end of the chapter.) we are given φ(χ), a function of x, which is periodic, with period 2a.

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  Computational Physics

  Problem Solving with Python

  • Rubin H. Landau, Manuel J Páez, Cristian C. Bordeianu, Manuel J. Páez(Authors)
  • 2015(Publication Date)
  • Wiley-VCH

    (Publisher)

  In quantum mechanics, where we represent a particle by a Wave Packet, this means that an interference pattern should be formed when a particle passes through a small slit. Pass a Gaussian Wave Packet of width 3 through a slit of width 5 (Figure 22.3), and look for the resultant quantum interference. 22.4 Wave Packet–Wave Packet Scattering 2) We have just seen how to represent a quantum particle as a Wave Packet and how to compute the interaction of that Wave Packet/particle with an external potential. 1) For reference sake, note that the constants in the equation change as the dimension of the equation changes; that is, there will be different constants for the 3D equation, and therefore our constants are different from the references! 2) This section is based on the Master of Science thesis of Jon Maestri. It is included in his memory. 519 22.4 Wave Packet–Wave Packet Scattering Although external potentials do exist in nature, realistic scattering often involves the interaction of one particle with another, which in turn would be represented as the interaction of a Wave Packet with a different Wave Packet. We have already done the hard work needed to compute Wave Packet–Wave Packet scattering in our implementation of Wave Packet–potential scattering even in 2D. We now need only to generalize it a bit. Two interacting particles are described by the time-dependent Schrödinger equation in the coordinates of the two particles i  ψ(x 1 , x 2 , t )  t = − 1 2m 1  2 ψ(x 1 , x 2 , t )  x 2 1 − 1 2m 2  2 ψ(x 1 , x 2 , t )  x 2 2 + V (x 1 , x 2 )ψ(x 1 , x 2 , t ) . (22.26) where, for simplicity, we assume a one-dimensional space and again set ℏ = 1. Here m i and x i are the mass and position of particle i = 1, 2. Knowledge of the two- particle wavefunction ψ(x 1 , x 2 , t ) at time t permits the calculation of the proba- bility density of particle 1 being at x 1 and particle 2 being at x 2 : ρ(x 1 , x 2 , t ) = |ψ(x 1 , x 2 , t )| 2 .

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  Physical Acoustics V5

  Principles and Methods

  • Warren P. Mason(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Wave Packet Propagation and Frequency-Dependent Internal Friction M. E L I C E S Laboratorio Central de Ensayo de Materiales de Construction Madrid, Spain and F. G A R C I A -M O L I N E R Instituto * 'Rocasolano'' C.S.I.C., Madrid, Spain I. Introduction 163 II. The Physical Nature of the Waves 166 A. Electromagnetic and Stress Waves 166 B. Introduction of the Medium (Phenomenological) 168 III. The Propagation of W a v e Packets: Morphological Analysis 170 A. Quasimonochromatic Signals: Power Series Expansion 172 B. Pulse-Type Signals: Decomposition in Quasimonochromatic Packets . . . . 176 C. Pulse-Type Signals: Asymptotic Expansion Methods 180 IV. The Propagation of Wave Packets: Energetic Analysis 202 A. The Energy Density 202 B. The Velocity of Energy Propagation 204 V. Application to the Theory of Internal Friction 209 A. Stress Waves in Imperfect Media: Models of Frequency-Dependent Internal Friction 209 B. The Analysis of Experimental Data. Concluding Remarks 212 References 217 I. Introduction The purpose of this chapter is to review the theory of Wave Packet propaga-tion through real media from the standpoint of the theoretical analysis of experimental data within the framework of linear causal responses. By Wave Packet we mean a signal of finite duration in time, whose frequency spectrum is therefore nonmonochromatic. To be specific, if ωο is the driving frequency, the signal is a packet of frequencies with a central peak at ωο. 163 —4— 164 Μ . Elices and F. Garcia-Moliner By real medium we mean one which exhibits dispersion and absorption (with or without dissipation) determined by the interaction between the wave and the medium. The information concerning this is contained in the res-ponse function, which can be evaluated once the physical model has been chosen.

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  Quantum Mechanics, Volume 1

  Basic Concepts, Tools, and Applications

  • Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë(Authors)
  • 2020(Publication Date)
  • Wiley-VCH

    (Publisher)

  For example, when one is dealing with a macroscopic particle (and the example of the dust particle discussed in Complement B I shows how small it can be), the uncertainty relation does not introduce an observable limit on the accuracy with which its position and momentum are known. This means that we can construct, in order to describe such a particle in a quantum mechanical way, a Wave Packet whose characteristic widths Δx and Δp are negligible. We would then speak, in classical terms, of the position x M (t) and the momentum p 0 of the particle. But then its velocity must be. This is indeed what is implied by formula (C-32), obtained in the quantum description: in the cases where Δx and Δp can both be made negligible, the maximum of the Wave Packet moves like a particle that obeys the laws of classical mechanics. Comment: We have stressed here the motion of the center of the free Wave Packet. It is also possible to study the way in which its form evolves in time. It is then easy to show that, if the width Δp is a constant of the motion, Δx varies over time and, for sufficiently long times, increases without limit (spreading of the Wave Packet). The discussion of this phenomenon is given in Complement G I, where the special case of a Gaussian Wave Packet is treated. D. Particle in a time-independent scalar potential We have seen, in § C, how the quantum mechanical description of a particle reduces to the classical description when Planck’s constant h can be considered to be negligible. In the classical approximation, the wavelike character does not appear because the wavelength associated with the particle is much smaller than the characteristic lengths of its motion. This situation is analogous to the one encountered in optics. Geometrical optics, which ignores the wavelike properties of light, constitutes a good approximation when the corresponding wavelength can be neglected compared to the lengths with which one is concerned

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  Waves in Complex Media

  • Luca Dal Negro(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  96 Wave Propagation and Diffraction is the propagated temporal spectrum at an arbitrary distance z. A direct consequence of the scaling property of the Fourier transform 1 is that the duration of an pulse is inversely proportional to the range of frequencies involved in the mixture, or the fre- quency spread of the harmonic wave components. Moreover, the wavenumber is gen- erally frequency dependent in a dispersive linear medium, e.g., k(ω) = ω  (ω)μ 0 , where we neglect the magnetic response. As a result, a Wave Packet travels across a nondispersive medium with the same shape (no distortion). On the other hand, truly dispersive media characterized by a nonlinear dispersion relation k = k(ω) that will generally alter the shape of propa- gating Wave Packets because different frequency components in the spectrum of the initial pulse will travel at different speeds, thus “dispersing” its original shape. 4.1.1 Velocity of Wave Packets In order to better understand the physical meaning of the propagation velocity of a general Wave Packet, we first consider the case of a simple packet created by the interference of two monochromatic plane waves. In particular, we study waves that propagate along the z-direction with equal amplitudes and slightly different frequen- cies (ω 1 ≈ ω 2 = ω) in a nondispersive medium. In this case, the total electric field of the superposition is as follows: E T (z,t ) = E 0 [cos(k 1 z − ω 1 t ) + cos(k 2 z − ω 2 t )]. (4.3) Using the well-known trigonometric identity for the sum of two cosine functions, we can express the previous field in the following product form: E T (z,t ) = 2E 0 cos(k p z − ω p t ) cos(k g x − ω g t ) (4.4) This equation describes a beat phenomenon and gives rise to a train of localized Wave Packets created by the product of a fast-oscillating carrier wave and a slow-oscillating envelope function.

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