Time Independent Schrodinger Equation

10 Key excerpts on "Time Independent Schrodinger Equation"

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  eBook - ePub

  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 I

  The Fundamentals

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

    (Publisher)

  As stated earlier the state of a quantum mechanical system is represented by a wave function ψ. | ψ | 2 is assumed to give the position probability density. From the solution ϕ (X) of time-independent Schrödinger Eq. (2.17), the solution of the time-dependent Schrödinger equation is written as ψ n (X, t) = ϕ n (X) e − i E n t / ℏ. (2.87) We note that | ψ n (X, t) | 2 is independent of time. Further, the time dependence of ψ n (X, t) is periodic with angular frequency ω = E n / ℏ. Such states are called stationary states. In quantum mechanics a stationary state does not mean that the system is strictly in a rest state or in a classical equilibrium point. In this state, expectation values of all dynamical variables 〈 A 〉 = ∫ − ∞ ∞ ψ n * (X, t) A ψ n (X, t) d τ = ∫ − ∞ ∞ ϕ n * (X) A ϕ n (X) d τ (2.88) are independent of time. It means that though ψ may vary with time, ψ * ψ at any point in the configuration space is independent of time. The solution ϕ n (X) of the time-independent Schrödinger Eq. (2.17) for a fixed value of E n is obviously time-independent and thus ϕ n * ϕ n is independent of time. Therefore, we call Eq. (2.17) the stationary state Schrödinger equation and its solutions as stationary state solutions or eigenfunctions. This is an energy eigenvalue equation, H ϕ = E ϕ. Consequently, when the state of the particle is the solution of Eq. (2.17) then the energy of it has a definite value given by the eigenvalue E. Suppose a particle is initially in a stationary state represented by a ψ n (X, 0) = ϕ n (X). Then from Eq. (2.87) we write ψ n (X, t) = ψ n (X, 0) e − i E n t / ℏ (2.89) and | ψ n (X, t) | 2 = | ψ n (X, 0) | 2 is independent of time. That is, if a system is initially in a stationary state then it remains in that state forever. Though ψ n (X, t) changes with time the energy and expectation values of observables remain the same. We note that the wave function given by Eq. (2.89) is a particular solution of the time-dependent Schrödinger equation

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  Mechanics

  Classical and Quantum

  • T. T. Taylor, D. Ter Haar(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R 7 THE TIME-INDEPENDENT SCHRÖDINGER EQUATION AND SOME OF ITS APPLICATIONS 7.01. Time-independent Potential Energy Functions and Stationary Quantum States The Schrödinger equation for a system consisting of one particle of mass m moving in three dimensions and having a potential energy V(r, t) was derived in Chapter 6. It is: --£-ν 2 Ψ+ VW =ih^~. (7.001) 2m dt The present chapter introduces the case in which the potential energy does not depend explicitly upon the time and can be written V(r). This case has wide applicability; it permits solutions of (7.001) in which Ψ can be written as a function of r only times a complex expo-nential factor involving a single frequency ω. Thus: ψ(τ, t) = v

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  Computational Modeling and Visualization of Physical Systems with Python

  • Jay Wang(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Unlike deterministic Newtonian mechanics, quantum mechanics is both deterministic and probabilistic. As a result, quantum mechanics suffers from what may be called the “unsolvability” problem, that is, meaningful time-dependent quantum systems that are analytically solvable are few and far in between. This is where numerical simulations are particularly valuable to reveal aspects of quan- tum mechanics that cannot be easily perceived. We will describe several methods suitable for simulating time-dependent quantum systems. We begin with a direct method that converts the Schrödinger equation into a system of ordinary differential equations (ODEs), and use it to study the quantum motion of a simple harmonic oscillator. This is the most direct and straightforward approach. Next, we discuss a split-operator method to deal with motion in open space such as free fall. We also introduce a two-state model to study transitions and to illustrate the concept of coherent states. Finally, we briefly discuss quantum waves in 2D and quantum revival. 8.1 TIME- DEPENDENT SCHRÖDINGER EQUATION In quantum mechanics the notion that a particle’s position and velocity can be simultaneously determined is abandoned. Instead, due to particle-wave duality, a particle is described by the wave function,  (x, t) [44]. The interpretation of the wave function is such that | | 2 dx is the probability of finding the particle in space from x to x + dx at time t. For a normalized wave function, the probability over all space is unity, ∫ ∞ −∞ | (x, t)| 2 dx = 1. (8.1) 272 8.1 Time-Dependent Schr  odinger Equation 273 The wave function evolves in time according to the time-dependent Schrödinger equation (TDSE) − ℏ 2 2m  2  (x, t) x 2 + V (x, t)  (x, t) = iℏ  (x, t) t . (8.2) Here, V (x, t) is the potential, ℏ the rationalized Planck constant, and m the mass of the parti- cle. The TDSE is fundamental to quantum mechanics as Newton’s second law is to classical mechanics.

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  Semiconductor Quantum Optics

  • Mackillo Kira, Stephan W. Koch(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 Central concepts in stationary quantum theory Whereas the time-dependent Schrödinger equation (3.1) uniquely determines the quantum dynamics of particles, it is often sufficient to use a much simpler static description. In particular, the stationary Schrödinger equation, ˆ H φ E (r) = E φ E (r), defines an eigenvalue problem whose solutions can be used to obtain a relatively simple interpretation of the pertinent quantum features. For example, if the system is initially in a particular eigenstate φ E (r), it stays there for all times such that all expectation values remain constant. Clearly, such a system does not have time-dependent currents that could radiate energy in the form of electromagnetic fields. Thus, quantum mechanics predicts that an atom is stable if it is in an eigenstate φ E (r). This stationarity explains why quantized atoms do not collapse despite the classical description’s complete failure to explain this elementary aspect. In this chapter, we summarize how stationary quantum-mechanical properties emerge from the dynamic Schrödinger equation. Especially, we formulate the stationary Schrödinger equation in terms of a generic Helmholtz equation. We develop a transfer- matrix technique that can be applied to calculate the stationary wave functions for arbitrary one-dimensional problems. These solutions show how interference effects, tunneling phenomena, and bound states with a discrete energy spectrum relate to the wave aspects of the particles. 4.1 Stationary Schrödinger equation Fourier transforming the Schrödinger Eq. (3.39) with respect to time yields the eigenvalue problem ¯ h ω ψ(r, ω) = H ( ˆ r, ˆ p) ψ(r, ω), ⇔ H ( ˆ r, ˆ p) φ E (r) = E φ E (r). (4.1) Associating ¯ h ω with the energy E defines the stationary Schrödinger equation with the eigenfunction ψ(r, ω) via ψ E (r). The energy eigenvalue E can be used to identify the state and thus plays the role of an energy quantum number.

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  A Student's Guide to the Schrödinger Equation

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

    (Publisher)

  This says that the acceleration of an object is directly proportional to the vector sum of the forces acting on it, and inversely proportional to the object’s mass. But in classical physics the acceler-ation does not depend on the energy, momentum, or velocity of the object. So if the Schr¨ odinger equation is to serve a purpose in quantum mechanics similar to that of Newton’s Second Law in classical mechanics, the time evolution of the wavefunction shouldn’t depend on the particle or system’s energy or momentum. Hence the time derivative cannot be of second order. So although the Schr¨ odinger equation can’t be derived from first principles, the form of the equation does make sense. More importantly, it gives results 78 3 The Schr¨ odinger Equation that predict and describe the behavior of quantum particles and systems over space and time. But one very useful form of the Schr¨ odinger equation is independent of time, and that version is the subject of the next section. 3.3 Time-Independent Schr¨ odinger Equation Separating out the time-dependent and space-dependent terms of the Schr¨ odinger equation is helpful in understanding why the quantum wavefunction behaves as it does. That separation can be accomplished, as for many differential equations, using the technique of separation of variables. This technique begins with the assumption that the solution ( ( x , t ) in this case) can be written as the product of two separate functions, one depending only on x and the other depending only on t . You may have encountered this technique in one of your physics or mathematics classes, and you may recall that there’s no a priori reason why this approach should work. But it often does work, and in any situation in which the potential energy varies only over space (and not over time), you can use separation of variables to solve the Schr¨ odinger equation.

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  Foundations for Nanoscience and Nanotechnology

  • Nils O. Petersen(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  C H A P T E R 4 Solving the time-independent Schrödinger equation The solutions to the Time-Independent Schrödinger Equation (TISE) can be found as long as we can define the Hamiltonian, which in the simplest cases, means defining the potential in which the particle of interest is moving. In the following sections, we will consider a single particle of mass, m p , moving in a potential that is time-independent, but can vary in space. We will follow four steps: • Define the Hamiltonian by defining the Potential Energy Operator since  H = −  2 2mp  ∇ 2 +  V (r ). • Solve the TISE to get the wave functions ψ(x,y,z ) and then the total stationary state wave functions Ψ(x,y,z ; t) = ψ(x,y,z )e −iωt . • Determine the energies of the various states of the system. • Examine the implications of the solutions to the Schrödinger equations. We will progress from the simplest systems to more complex systems, start- ing with one-dimensional problems and moving towards three-dimensional sys- tems of various geometries. As we progress, the mathematical treatment will become less rigorous, but we will still be able to examine the key trends of introducing confinement on the movement of the particle. 4.1 THE FREE PARTICLE Let us consider first a particle of mass, m p , moving in space where the potential energy is zero everywhere, which means that V (r ) = 0 for all values of r . 27 28 ■ Foundations for Nanoscience and Nanotechnology Further, let us initially chose a coordinate system in which the particle moves along the x-axis so that it becomes a 1-dimensional problem. Then the TISE becomes −  2 2m p  ∇ 2 ψ(x) = −  2 2m p ∂ 2 ∂ 2 x ψ(x) = Eψ(x). (4.1) This can be re-written as ∂ 2 ∂ 2 x ψ(x) + 2m p E  2 ψ(x) = ∂ 2 ∂ 2 x ψ(x) + k 2 ψ(x) = 0. (4.2) This differential equation has a general solution of the form ψ(x) = ψ + (x) + ψ − (x) =Ae ikx + Be −ikx =(A + B)cos(kx) + i(A − B)sin(kx).

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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 Quantum Mechanics

  • J. J. Sakurai, Jim Napolitano(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Quantum Dynamics So far we have not discussed how physical systems change with time. This chapter is devoted exclusively to the dynamic development of state kets and/or observables. In other words, we are concerned here with the quantum-mechanical analogue of Newton’s (or Lagrange’s or Hamilton’s) equations of motion. 2.1 Time Evolution and the Schrödinger Equation The first important point we should keep in mind is that time is just a parameter in quantum mechanics, not an operator. In particular, time is not an observable in the language of the previous chapter. It is nonsensical to talk about the time operator in the same sense as we talk about the position operator. Ironically, in the historical development of wave mechanics both L. de Broglie and E. Schr¨ odinger were guided by a kind of covariant analogy between energy and time on the one hand and momentum and position (spatial coordinate) on the other. Yet when we now look at quantum mechanics in its finished form, there is no trace of a symmetrical treatment between time and space. The relativistic quantum theory of fields does treat the time and space coordinates on the same footing, but it does so only at the expense of demoting position from the status of being an observable to that of being just a parameter. 2.1.1 Time-Evolution Operator Our basic concern in this section is, How does a state ket change with time? Suppose we have a physical system whose state ket at t 0 is represented by | α. At later times, we do not, in general, expect the system to remain in the same state | α. Let us denote the ket corresponding to the state at some later time by | α, t 0 ; t (t > t 0 ), (2.1) where we have written α, t 0 to remind ourselves that the system used to be in state | α at some earlier reference time t 0 . Because time is assumed to be a continuous parameter, we expect lim t→t 0 | α, t 0 ; t = | α (2.2) 62

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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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