Free Particle in Quantum Mechanics

3 Key excerpts on "Free Particle in Quantum Mechanics"

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  eBook - ePub

  Quantum Mechanics, Volume 1

  Basic Concepts, Tools, and Applications

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

    (Publisher)

  iii) It is worth noting that, unlike photons, which can be emitted or absorbed during an experiment, material particles can neither be created nor destroyed. The electrons emitted by a heated filament for example already existed in the filament. In the same way, an electron absorbed by a counter does not disappear; it becomes part of an atom or an electric current. Actually, the theory of relativity shows that it is possible to create and annihilate material particles: for example, a photon having sufficient energy, passing near an atom, can materialize into an electron-positron pair. Inversely, the positron, when it collides with an electron, annihilates with it, emitting photons. However, we pointed out in the beginning of this chapter that we would limit ourselves here to the non-relativistic quantum domain, and we have indeed treated time and space coordinates asymmetrically. In the framework of non-relativistic quantum mechanics, material particles can neither be created nor annihilated. This conservation law, as we shall see, plays a role of primary importance. The need to abandon it is one of the important difficulties encountered when one tries to construct a relativistic quantum mechanics.

  C. Quantum description of a particle. Wave packets

  In the preceding paragraph, we introduced the fundamental concepts necessary for the quantum description of a particle. In this paragraph, we are going to familiarize ourselves with these concepts and deduce from them several very important properties. We start with the very simple case of a free particle.

  C-1. Free particle

  Consider a particle whose potential energy is zero (or has a constant value) at every point in space. The particle is thus not subjected to any force; it is said to be free.

  When V(r, t) = 0, the Schrödinger equation becomes:

  (C-1 )

  This differential equation is obviously satisfied by solutions of the form:

  (C-2 )

  (where A is a constant), on the condition that k and ω satisfy the relation:

  (C-3 )

  Observe that, according to the de Broglie relations [see (B-2 )], condition (C-3 ) expresses the fact that the energy E and the momentum p of a free particle satisfy the equation, which is well-known in classical mechanics:

  (C-4)

  We shall come back later (§ C-3) to the physical interpretation of a state of the form (C-2 ). We already see that, since

  (C-5 )

  a plane wave of this type represents a particle whose probability of presence is uniform throughout all space (see comment below).

  The principle of superposition tells us that every linear combination of plane waves satisfying (C-3 ) will also be a solution of equation (C-1 ). Such a superposition can be written:

  (C-6 )

  (d3 k represents, by definition, the infinitesimal volume element in k-space: dkx dky dkz ). g(k), which can be complex, must be sufficiently regular to allow differentiation inside the integral. It can be shown, moreover, that any square-integrable solution can be written in the form (C-6

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  eBook - ePub

  Introduction to Nanoelectronic Single-Electron Circuit Design

  • Jaap Hoekstra(Author)
  • 2016(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  We inspect the various descriptions. 4.3.1 Electron as a Particle Before we consider the quantum mechanical wavefunction for a free electron we briefly look at the electron as a particle in classical mechanics. The dynamic state of an electron is determined by the forces acting on it and by the electron’s total energy. The electron’s momentum p is related to the force F on it by Newton’s equation (we assume the velocity of the electron to be sufficiently far below the speed of light) F = d p d t (4.13) and the electron’s total energy is just the sum of its kinetic and potential energy E = p 2 2 m + E P (x). (4.14) The velocity of the electron in the classical description becomes v cl = p m (4.15) and can be used to calculate the mean velocity in the definition of the classical current density J : if a charge distribution consists of electrons with charge e and moves with a mean velocity v, then if N is the number of electrons per unit. volume J = N e v. (4.16) 4.3.2 Electron as a Wave In quantum mechanics the laws of conservation of momentum and energy remain valid. In the case of a free electron the potential energy is zero and the 1D Schrödinger’s equation becomes − ℏ 2 2 m d 2 ψ (x) d x 2 = E ψ (x). (4.17) For a free electron, E = p 2 /2 m. Setting p = ℏ k, we have E = ℏ 2 k 2 2 m (4.18) and Eq. 4.17 can be written as d 2 ψ (x) d x 2 + k 2 ψ (x) = 0. (4.19) Direct substitution shows that the solutions ψ (x) = e j k x and ψ (x) = e − j k x are admitted

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  eBook - ePub

  Surprising Quantum Bounces

  • Valery Nesvizhevsky, Alexei Voronin(Authors)
  • 2015(Publication Date)
  • ICP

    (Publisher)

  wave . We are going to use this momentum–space relation further on.

  The practical advice of quantum mechanics for any concrete problem is to solve a wave-equation in order to find a wave-function, then to predict the probability of measuring any physical value of the particle using this wave-function.

  The famous Schrödinger equation 2 , the quantum equation of motion, allows us to calculate a wave-function in any point of space at any moment of time, and thus it provides us with the full description of a quantum system.

  In order to get a feeling how such an equation could be discovered in a more general case, try to derive it yourself by means of solving Problem 2.2 .

  Problem 2.2. Guess what the form of equation is for the wave-function of a freely moving particle .

  Note: Take into account that the solution of this equation is given in the expression (2.3)

  .

  Note: Take also into account that the equation, which has to be found, should not change while being written in a rotated frame of reference as it should follow from the invariance of physical laws under the operation of rotation

  .

  Again, we write down the solution for this problem immediately as we need it to have known explicitly in order to continue:

  Using this very brief outline of principal statements of quantum mechanics, we underline that the wave character of motion of a quantum particle imposes that the particle has to be associated with a wave-function, which (generally speaking) should be defined in every point of space at any moment of time. Such a wave-function provides a complete self-consistent description of quantum motion.

  In order to understand what is in fact motion in quantum case, as well as to analyze what physical characteristics of quantum motion could be measured, let us think of another “imaginary Pisa tower” experiment, but now with a falling object in the form of a quantum particle. At the moment we do not care what kind of physical particle it is, although below we are going to discuss its eventual nature in detail, with references to real realizations and experiments.

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