Schodinger Equation Example

11 Key excerpts on "Schodinger Equation Example"

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  eBook - ePub

  Quantum Mechanics I

  The Fundamentals

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

    (Publisher)

  2 Schrödinger Equation and Wave Function

  DOI: 10.1201/9781003172178-2

  2.1 Introduction

  Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger proposed wave mechanics; Heisenberg developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie's matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. Physically,

  | ψ ( X , t )

  | 2

  , where

  ψ ( X , t )

  is the solution of the Schrödinger equation, is interpreted as position probability density. That is,

  | ψ ( X , t )

  | 2

  is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities. Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.

  There are several interesting features of the wave function ψ. The probability interpretation of

  | ψ

  | 2

  imposes certain conditions on meaningful ψ. Further, the

  | ψ

  | 2

  satisfies a conservation law, an equation analogous to the continuity equation of flow in hydrodynamics. To set the total probability unity the ψ must satisfy the normalization condition

  ∫

  − ∞

  ∞

  |

  ψ

  | 2

  d τ = 1

  . Knowing ψ we can compute expectation values of variables such as position, momentum, etc. In quantum mechanics the experimentally measurable variables are no longer dynamical variables but they become operators. The outcomes of experiments are the eigenvalues of the operators of the observables. An interesting result is that the equations of the time evolution of expectation values of position and momentum operators (called Ehrenfest's theorem) obey the same equation of motion of these variables in classical mechanics. On the other hand, the equation for the phase of the ψ

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  Introduction to Quantum Mechanics

  • Henrik Smith(Author)
  • 1991(Publication Date)
  • WSPC

    (Publisher)

  106 Introduction to Quantum Mechanics 4 THE SCHRODINGER EQUATION In the following we discuss some properties of the time-dependent Schrodinger equation, which is the fundamental equation of motion in the quantum theory, corresponding to Newton's equations in classical mechanics. Like Newton's equations the validity of the Schrodinger equation is limited to situations where relativistic effects may be neglected. We shall see how the time-dependent Schrodinger equation is reduced to the eigenvalue equation for the energy under stationary conditions. The eigen-states that are solutions to this equation are called stationary states, since the associated probability densities are independent of time. By forming superpo-sitions of stationary states we are able to describe time-dependent phenomena and compare with the result of solving the classical equations of motion. Al-though a given state may not be an eigenstate for the energy operator, it may be an eigenstate for other operators associated with, say, momentum or an-gular momentum. As we shall see in Example 4 below, the state of a system may also be characterized by being an eigenstate for a non-Hermitian operator. The special feature of the eigenstates of the energy operator is the simplicity of their development in time, as demonstrated below. We consider a single particle with mass m moving in the potential V(r). As shown in Section 3.3.3 of the previous chapter, the time-dependent Schrodinger equation has the form - ^V 2 V-(r,<) + VW(r,t) = ift^^M (4.1) where the left-hand side is Hip with H being the Hamiltonian. In general the wave function ip(r,t) is not necessarily an eigenstate of the Hamiltonian and may therefore not be labelled by an energy eigenvalue. Its physical interpretation is that of a probability amplitude. Thus |V(r,t)| 2 dr (4.2) is the probability that a measurement of the position of the particle yields a result in the volume element dr(= dxdydz) at r.

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  A Student's Guide to the Schrödinger Equation

  • Daniel A. Fleisch(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 The Schr¨ odinger Equation If you’ve worked through Chapters 1 and 2 , you’ve already seen several references to the Schr¨ odinger equation and its solutions. As you’ll learn in this chapter, the Schr¨ odinger equation describes how a quantum state evolves over time, and understanding the physical meaning of the terms of this powerful equation will prepare you to understand the behavior of quantum wavefunctions. So this chapter is all about the Schr¨ odinger equation, and you can read about the solutions to the Schr¨ odinger equation in Chapters 4 and 5 . In the first section of this chapter, you’ll see a “derivation” of several forms of the Schr¨ odinger equation, and you’ll learn why the word “derivation” is in quotes. Then, in Section 3.2 , you’ll find a description of the meaning of each term in the Schr¨ odinger equation as well as an explanation of exactly what the Schr¨ odinger equation tells you about the behavior of quantum wavefunctions. The subject of Section 3.3 is a time-independent version of the Schr¨ odinger equation that you’re sure to encounter if you read more advanced quantum books or take a course in quantum mechanics. To help you focus on the physics of the situation without getting too bogged down in mathematical notation, the Schr¨ odinger equation discussed in most of this chapter is a function of only one spatial variable ( x ). As you’ll see in later chapters, even this one-dimensional treatment will let you solve several interesting problems in quantum mechanics, but for certain situations you’re going to need the three-dimensional version of the Schr¨ odinger equation. So that’s the subject of the final section of this chapter ( Section 3.4 ). 63 64 3 The Schr¨ odinger Equation 3.1 Origin of the Schr¨ odinger Equation If you look at the introduction of the Schr¨ odinger equation in popular quantum texts, you’ll find that there are several ways to “derive” the Schr¨ odinger equation.

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  Quantum Mechanics for Tomorrow's Engineers

  • Junichiro Kono(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Electron Waves and Schrödinger’s Equation Learning objectives: • Developing an understanding of when and how classical par- ticles start behaving as quan- tum mechanical waves. • Deriving the most fundamen- tal equation in quantum me- chanics, the Schrödinger equa- tion, which determines the states and dynamics of quan- tum particles. • Understanding the probabilis- tic meaning of the wavefunc- tion, ψ, and learn how to cal- culate the expectation values of observable quantities. • Solving the Schrödinger equa- tion for example problems in- volving electrons in simple po- tential energy landscapes. Quantum mechanics is currently the most fundamental theory in use in many disciplines of science and engineering. It is particu- larly important when one is dealing with nanoscale and atomic-scale systems. However, many phenomena and properties that occur at atomic scales are strange and nonintuitive. There are a number of concepts that simply do not exist in the macroscopic world where we live. Wave–particle duality is one of them. In this chapter, we exam- ine how and when classical particles start behaving as quantum me- chanical waves, derive the most important wave equation that quan- tum particles obey, Schrödinger’s equation, and solve it for the ele- mentary problems of electron waves in given potential energy land- scapes. We will also learn how to calculate the expectation values of observables when the wavefunction is known. Schrödinger’s equa- tion will be extensively used throughout the rest of this textbook. More complicated potential energy problems, particularly those rel- evant to materials and devices, will be dealt with in Chapters 5 and 7, building upon the formulations developed in this chapter. 2.1 Wave Equation and Wavefunction One of the striking consequences of quantum theory is that light be- haves as particles under certain circumstances and electrons behave as waves under certain circumstances.

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  Introduction to Quantum Mechanics with Applications to Chemistry

  • Linus Pauling, E. Bright Wilson, E. Bright Wilson(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Chap. XV ), but for nearly all purposes the wave equation is a convenient and sufficient starting point.

  9a. The Wave Equation Including the Time. —Let us first consider a Newtonian system with one degree of freedom, consisting of a particle of mass m restricted to motion along a fixed straight line, which we take as the x axis, and let us assume that the system is further described by a potential-energy function V (x ) throughout the region – ∞ < × < + ∞. For this system the Schrödinger wave equation is assumed to be

  In this equation the function Ψ(x , t ) is called the Schrödinger wave function including the time , or the probability amplitude function. It will be noticed that the equation is somewhat similar in form to the wave equations occurring in other branches of theoretical physics, as in the discussion of the motion of a vibrating string. The student facile in mathematical physics may well profit from investigating this similarity and also the analogy between classical mechanics and geometrical optics on the one hand, and wave mechanics and undulatory optics on the other.1 However, it is not necessary to do this. An extensive previous knowledge of partial differential equations and their usual applications in mathematical physics is not a necessary prerequisite for the study of wave mechanics, and indeed the study of wave mechanics may provide a satisfactory introduction to the subject for the more physically minded or chemically minded student.

  The Schrödinger time equation is closely related to the equation of classical Newtonian mechanics

  which states that the total energy W is equal to the sum of the kinetic energy T and the potential energy V and hence to the Hamiltonian function H (

  px

  , x ). Introducing the coordinate x and momentum

  px

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  Electrons, Neutrons and Protons in Engineering

  A Study of Engineering Materials and Processes Whose Characteristics May Be Explained by Considering the Behavior of Small Particles When Grouped Into Systems Such as Nuclei, Atoms, Gases, and Crystals

  • J. R. Eaton(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  For example, if ψ is a + jb, then ψ* is a — jb, and Uis a 2 + b 2 . 11.3. THE SCHRÖDINGER EQUATION Probably the most important single relation of wave mechanics is the Schrö-dinger equation, a relation which applies over a wide range of conditions ex-tending from particles in the microscopic system to the large scale objects of laboratory dimensions with which we are most familiar. When applying to ob-jects of large dimensions the Schrödinger equation, as might be expected, leads to relations which are equivalent to Newton's Laws. The Schrödinger equation in rectangular co-ordinance is J^(^ + ^ + £ï_ E JL^ÊÏ. (n. 2)t %n 2 m dx 2 dy 2 dz 2 ) j 2π dt where E p is the potential energy of the particle under study. The behavior of ψ is limited by the following restrictions: ψ must be finite, continuous, and single-valued at all points in space and time; (11.3) the rate of change of ψ must be finite and continuous at all points in space. (11.4) The reasons for these restrictions will be pointed out later in regard to a one-dimensional problem. The Schrödinger equation in general form is a second-order partial differen-tial equation involving space co-ordinates and time, and contains both real and imaginary terms. The potential energy E p may also vary in time and space. t In this text j = yj — 1. 156 ELECTRONS, NEUTRONS AND PROTONS IN ENGINEERING In attempting to apply the Schrödinger equation to a practical problem one finds, as would be expected, that the simplest situation is that of the hydrogen atom. In attempting to apply the Schrödinger equation to more complicated atoms, to molecules, and to crystals, exact mathematical solutions are, in many cases, impossible and the investigator is forced to resort to methods of ap-proximation. 11.4. A P P L I C A T I O N TO A ONE-DIMENSIONAL PROBLEM.

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  Basic Molecular Quantum Mechanics

  • Steven A. Adelman(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  3 The Schrodinger Equation and the Particle-in-a-Box

  In Chapters 4 –6 , we lay out the full formal basis of quantum mechanics, and in the succeeding chapters, we describe increasingly advanced applications of this basis. However, in order to convey some feeling for quantum mechanics before developing its full formal machinery in this chapter, we introduce the most useful quantum equation, the time-independent Schrodinger equation, and then apply it to one of the simplest quantum systems, the instructive one-dimensional particle-in-a-box system.

  The time-independent Schrodinger equation, as we will show in Section 3.2 , may be derived from the more fundamental time-dependent Schrodinger equation, already touched on in Section 1.5 . So we begin with the time-dependent Schrodinger equation. This equation cannot be derived from anything more fundamental. Rather, like Newton’s equation of motion, it is best viewed as a postulate that is accepted because it successfully predicts a vast range of phenomena.

  While the time-dependent Schrodinger equation cannot be derived, several non-rigorous plausibility arguments for its form exist. We next give one of these.

  3.1 A Heuristic “Derivation” of the Time-Dependent Schrodinger Equation

  In our discussion of the photoelectric effect in Section 1.1 , we noted that Einstein discovered that light exhibits a wave–particle duality, namely that a light wave of frequency υ or wavelength

  λ =

  c υ

  ,

  where c is the speed of light, could also be viewed as a stream of particles called photons each with an energy E and momentum

  p .

  Einstein postulated that the particle properties of light E and p were related to its wave properties υ and λ as follows:

  E = h υ     a n d   p =

  h λ

  .

  (3.1)

  We further noted in Section 1.4 that de Broglie later hypothesized that ordinary particles also exhibit a wave–particle duality. Namely, de Broglie hypothesized that associated with a particle is a wave with de Broglie wavelength

  λ .

  In analogy to Einstein’s photon relation

  p =

  h λ

  , de Broglie postulated that the wavelength of the matter wave associated with a particle of momentum p

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  Quantum Mechanics

  Foundations and Applications

  • Donald Gary Swanson(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  2 The Schr¨ odinger Equation in One Dimension In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dy-namical quantity F ( x, p, t ) → F (ˆ x, ˆ p, t ) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schr¨ odinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angu-lar momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics. 2.1 The Free Particle As our first example of solving the Schr¨ odinger equation, we choose the free particle in one dimension, which means that the particle is free of any forces. It thus has a constant potential, so the time-independent Schr¨ odinger equation in one dimension is written as ˆ H ψ = - 2 2 m d 2 ψ d x 2 + V ψ = Eψ , (2.1) where ψ = ψ ( x ) is the spatial part of the wave function, and the complete wave function is Ψ( x, t ) = ψ ( x ) exp ( -i Et/ ). Just as the time-dependent equation was easily solved in the previous chapter, so is the solution of Equation (2.1) easily solved, with solution ψ ( x ) = A exp ± i 2 m ( E -V ) x/ . (2.2) Problem 2.1 Show that the solution of the 1-dimensional Schr¨ odinger equa-tion for a free particle can be written as ψ ( x ) = A exp ( ± i kx ) where k is the classical momentum of a free particle of energy E . 41

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  Modern Physics

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

    (Publisher)

  162 Chapter 5 The Schrödinger Equation and especially in the hydrogen atom in Chapter 7, we find that the effects of degeneracy become more significant; in the case of atomic physics, the degeneracy is a major contributor to the structure and properties of atoms. 5.5 THE SIMPLE HARMONIC OSCILLATOR Another situation that can be analyzed using the Schrödinger equation is the one-dimensional simple harmonic oscillator. The classical oscillator is an object of mass m attached to a spring of force constant k. The spring exerts a restoring force F = −kx on the object, where x is the displacement from its equilibrium position. Using Newton’s laws, we can analyze the oscillator and show that it has a (circular or angular) frequency  0 = √ k∕m and a period T = 2 √ m∕k. The maximum distance of the oscillating object from its equi- librium position is x 0 , the amplitude of the oscillation. The oscillator has its maximum kinetic energy at x = 0; its kinetic energy vanishes at the turning points x = ±x 0 . At the turning points the oscillator comes to rest for an instant and then reverses its direction of motion. The motion is, of course, confined to the region −x 0 ≤ x ≤ +x 0 . Why analyze the motion of such a system using quantum mechanics? Although we never find in nature an example of a one-dimensional quantum oscillator, there are systems that behave approximately as one—a vibrating diatomic molecule, for example. In fact, any system in a smoothly varying potential energy well near its minimum behaves approximately like a simple harmonic oscillator. A force F = −kx has the associated potential energy U = 1 2 kx 2 , and so we have the Schrödinger equation: − − h 2 2m d 2  dx 2 + 1 2 kx 2  = E (5.46) (Because we are working in one dimension, U and  are functions only of x.) There are no boundaries between different regions of potential energy here, so the wave function must fall to zero for both x → +∞ and x → −∞.

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  Modern Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  With those two skills in mind, you can see where we are in this chapter and where we need to go. In Section 5.2 (and most of the rest of this chapter) the time-independent Schrödinger equation is used to find the energy eigenstates associated with a given potential function. In this section we will present a general rule that you can use to predict how any wavefunction will evolve over time, based on knowing its energy eigenstates. In Chapter 6 we will present the time-dependent Schrödinger equation, the fundamental axiom of quantum mechanics, and show how it leads to that time-evolution rule. This section relies on the math from Section 5.5. Make sure you are comfortable with complex numbers, modulus, complex conjugates, complex exponentials, and Euler’s formula, or you’ll miss all the fun. 5.6.1 Discovery Exercise: Time Evolution of a Wavefunction In each question below we’ll give you a complex number and ask you to calculate its modulus squared. Assume x is real, and simplify your answers as much as possible. You should be able to write each answer in terms of all real quantities (no i). 1. z 1 = e ix , so |z 1 | 2 = 2. z 2 = e 2ix so |z 2 | 2 = 3. z 3 = z 1 + z 2 = e ix + e 2ix so |z 1 + z 2 | 2 = Hint : The answer is not the sum of the previous two answers, |z 1 | 2 + |z 2 | 2 . 5.6.2 Explanation: Time Evolution of a Wavefunction You have a particle with a known wavefunction in a known potential energy field. How will that wavefunction evolve over time? This section will build up a process for answering that question through three scenarios of increasing complexity. First, though, we need to introduce some notation. Some Notation It’s common in quantum mechanics to write a wavefunction at one particular time as ψ(x), and to write the wavefunction at all times as (x, t ). (Those symbols are the lowercase and uppercase versions of the Greek letter psi.) So if we say a particle’s wavefunction started out

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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)

  I did not succeed in doing this with the matrix form of quantum mechanics, but did with the Schrodinger formulation. According to Schrodinger, the atom in its nth quantum state is a vibration of a state function of fixed frequency W°/h spread over all of space. In particular, an electron moving in a straight line is such a vibratory phenomenon which corre- sponds to a plane wave. When two such waves interact, a complicated vibration arises. However, one sees immediately that one can determine it through its asymptotic behavior at infinity. Indeed one has nothing more than a "diffraction problem" in which an incoming plane wave is refracted or scattered at an atom. In place of the boundary conditions which one uses in optics for the description of the diffraction diaphragm, one has here the potential energy of interaction be- tween the atom and the electron. The task is clear. We have to solve the Schrodinger wave equation for the system atom-plus-electron subject to the boundary condition that the solution in a preselected direction of electron space goes over asymptotically into a plane wave with exactly this direction of propagation (the arriving electron). In a thus selected solution we are further interested principally in a behavior of the "scattered" wave at infinity, for it describes the behavior of the system after the collision. We spell this out a little further. Let ιt/*¾¾)* · · · be the eigenfunc- tions of the unperturbed atom (we assume that there is only a discrete spectrum). The unperturbed electron, in straight-line motion, corresponds to eigenfunctions sin (2π/λ)(αχ + βγ + yz + δ), a continuous manifold of plane waves. Their wave- length, according to de Broglie, is connected with the energy of translation τ by the relation τ = Ρ/(2μλ 2 ). The eigenfunction of the unperturbed state in which the electron arrives from the + ζ direction, is thus ^nAq k , Z) = Ά°(<Ζ)ί) sin (2πζ/λ).

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