Quantum Harmonic Oscillator

11 Key excerpts on "Quantum Harmonic Oscillator"

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  Quantum Mechanics

  A Paradigms Approach

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

    (Publisher)

  275 C H A P T E R 9 Harmonic Oscillator In the last four chapters, you have learned the tools for analyzing the motion of particles in quantum mechanics. You applied these tools to three important problems: (1) a particle bound in an infinite square potential energy well in one dimension, (2) a free particle in one dimension, and (3) the hydro- gen atom in three dimensions. In this chapter we will solve another system with bound states in a one- dimensional potential energy well: the harmonic oscillator. This system resembles the infinite square well or particle-in-a-box system—the harmonic oscillator box just has a different shape. To solve the harmonic oscillator problem, we introduce a new method and some new tools in the process. Then we use the solutions to the harmonic oscillator problem as a means to review the fundamental tools and concepts of quantum mechanics. 9.1  CLASSICAL HARMONIC OSCILLATOR Let’s first review the classical harmonic oscillator before we study the quantum mechanical case. A prototypical classical harmonic oscillator system is a mass m connected to a spring that is fixed to a wall at its other end. The spring force is governed by Hooke’s law, which says that the force F is a restoring force and is proportional to the displacement x of the mass from equilibrium: F = - kx , (9.1) where k is the spring constant. This linear restoring force is derivable from the quadratic potential energy function V1x2 = 1 2 kx 2 . The beauty of the mass-on-a-spring system is that it is a model for many other systems in nature that behave as harmonic oscillators. To see why this is so, consider the generic potential energy curve shown in Fig. 9.1. We are typically interested in finding the motion in the ground state or other low energy states of the system.

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  Radiation Of Atoms In A Resonant Environment

  • V P Bykov(Author)
  • 1994(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 3 HARMONIC OSCILLATOR In this chapter we shall give a review of possible states of a Quantum Harmonic Oscillator and discuss their correspondence with the states of a classical harmonic oscillator. Now Quantum Harmonic Oscillator is mathematically studied very well (all the more, classical harmonic oscillator), however some problems of correspondence between the states of the quantum and classical oscillators till recently were not clear [1-3]-The harmonic oscillator plays an enormously important role in quantum electrody-namics. The reason is that the Maxwell's equations that describe the electromagnetic field in a vacuum are linear, and because of this linearity, the electromagnetic field in vacuum can be considered as a collection of linear, or harmonic oscillators, for example, plane waves. The electromagnetic field retains its linear properties up to extremely high field strength, where effects due to scattering of light by light, involv-ing the creation of virtual electron-positron pairs, become significant. Modern laser technology has shown that it is possible to excite one individual oscillator of the field, a single mode, in optical cavities. Because of the interaction of the field with the mirrors of the cavity the region of linearity of the oscillator is narrower than in free space, but nevertheless it is still very great. The range of applicability of the theory of the harmonic oscillator is thus very broad. We would remind basic properties of the classical and harmonic oscillators. 3.1 Classical harmonic oscillator The theory of a classical harmonic oscillator is based on the Hamilton function * -& + #> ( 1 ) where p and q are the oscillator momentum and coordinate correspondingly; m is the mass of the oscillating body and k is the elasticity coefficient of the string, which returns the body to the equilibrium position. To be sure, p and q can be generalized 39

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  Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë

  • Guillaume Merle, Oliver J. Harper, Philippe Ribiere(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  219 5 Solutions to the Exercises of Chapter V (Complement M V ). The One-Dimensional Harmonic Oscillator The harmonic oscillator is one of the most important models in classical mechan- ics. It is the easiest way of describing a system about an equilibrium: mass-spring systems, near-equilibrium pendula, floating, or bobbing masses in fluids, etc. Generally speaking, for any potential V , equivalent to the potential energy in classical mechanics, presenting a stable equilibrium denoted x eq , we can write: dV dx (x = x eq ) = 0 and d 2 V dx 2 (x = x eq ) = K > 0 The Taylor expansion to order 2 of V (x) therefore allows the local approximation of any potential by a harmonic potential: V (x) ≃ V (x eq ) + dV dx (x eq ) ⏟⏞ ⏟⏞ ⏟ 0 (x − x eq ) + 1 2 d 2 V dx 2 (x eq ) ⏟⏞⏞ ⏟⏞⏞ ⏟ K (x − x eq ) 2 + · · · Figure 5.1 illustrates how the Lennard–Jones potential, which typically models inter- molecular interactions, can be locally approximated by a harmonic potential about the equilibrium. Therefore, the vibration of an atom within a crystal or molecule about its equilibrium position can always be modeled as a harmonic oscillator, i.e. a mass- spring system, whatever the crystal or molecule to the second order, that is as long as the oscillations remain small. The natural angular frequency is the only parameter that must be set for each system. Chapter V highlights the importance of the harmonic oscillator in quantum mechanics. This set of exercises further explores the role of harmonic oscillators in the description of quantum systems. It is interesting to make comparisons with classical results and there are often similarities, but there are also fundamental differences. In particular, the fact that a particle is trapped in a harmonic potential well implies a discretization or quantization of possible states for the particle and the corresponding energies.

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  Essentials of Physical Chemistry

  • Don Shillady(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  12 The Quantized Harmonic Oscillator: Vibrational Spectroscopy INTRODUCTION Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more dif fi cult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was fi rst solved [1]. Then we proceed to application in the form of worked examples. We have tried to give suf fi cient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘ What does this mean? ’’ and absorb the conclusions for a future activity called ‘‘ thinking ’’ rather than just memorizing facts. The previous particle-in-a-box (PIB) and particle-on-a-ring (POR) problems both had V ¼ 0 and only dealt with the kinetic energy operator. The essence of the harmonic oscillator is a parabolic potential energy V ¼ kx 2 2 where k is the ‘‘ spring constant ’’ or ‘‘ restoring force constant. ’’ According to the idea that a force is the negative derivative of a potential, we have f ( x ) ¼ dV dx ¼ 2 kx 2 ¼ kx so k is the proportionality factor of the force of a spring stretched or compressed away from x ¼ 0. The classical case can be solved easily by equating two forces f ¼ ma ¼ m d 2 x dt 2 ¼ kx . The solution from sophomore physics is x ¼ A sin ffiffiffi k m r ! t # and this can be shown by direct substitution. dx dt ¼ A ffiffiffi k m r cos ffiffiffi k m r ! t # and d 2 x dt 2 ¼ A k m sin ffiffiffi k m r ! t # ¼ k m x .

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  How to Be a Quantum Mechanic

  • Charles G. Wohl(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  6. THE HARMONIC OSCILLATOR 6.1. The Classical Oscillator 6.2. The Quantum Oscillator: Series Solution 6.3. The Operator Solution 6.4. States as Vectors, Operators as Matrices Problems The restoring force acting on an object displaced from a position of stable equilibrium is often, for small enough displacements, proportional to the displacement x; hence the im- portance of Hooke’s law F (x) = −kx and the harmonic oscillator. Of course, since F (x) becomes infinite as x →±∞, it can only be valid over a limited range of x. Nevertheless, the oscillator approximation is one of the most important models in both classical and quantum physics. “Essentially, all models are wrong, but some are useful” (George E.P. Box). We solve the Schr¨ odinger equation for the oscillator potential energy, V (x) = 1 2 mω 2 x 2 , in two very different ways. The first is the series-solution method taught in courses on differential equations—the method that led to the special functions and polynomials carrying the names of such as Bessel, Legendre, Laguerre, and (relevant for this chapter) Hermite. The second method is more exotic: The eigenvalues and eigenfunctions are found, almost by magic, from the commutator relation [ˆ x, ˆ p] = i¯ h and the form of the Hamiltonian, which is quadratic in ˆ x and ˆ p. Each method of solution will be used again—the operator method for angular momentum, the series-solution method for the radial equation of hydrogen. Solving differential equations with the series method is not everyone’s favorite mathematics. However, if you slight Section 2, you still need to read Section 2(a), on “rescaling” the Schr¨ odinger equation. With the eigenvalues and eigenfunctions in hand, we write general states of an oscillator as vectors and the operators as matrices. Applications of the physics (most of them beyond the scope of this book) include molecular vibrations, specific heats of solids and gases, and quantization of the electromagnetic field. 111

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  Quantum Mechanics for Scientists and Engineers

  • David A. B. Miller(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  Chapter 15 Harmonic oscillators and photons Prerequisites: Chapters 2–5, 9, 10, 12, and 13. For additional background on vector calculus, electromagnetism and modes see Appendices C and D. In this section, we return to the harmonic oscillator and consider it in a mathematically more elegant way. This approach leads to the introduction of “raising” and “lowering” operators that take us from one harmonic oscillator state to another. The introduction of these operators allows us to rewrite the harmonic oscillator mathematics quite economically. We then show that the electromagnetic field for a given mode can also be described in a manner exactly analogous to a harmonic oscillator. In this case, we describe the states of this generalized harmonic oscillator in terms of the number of photons per mode, with that number corresponding exactly to the quantum number for the corresponding harmonic oscillator state. The raising and lowering operators are now physically interpreted as “creation” and “annihilation” operators for photons and are key operators for describing electromagnetic fields. By this process, we can describe the electromagnetic field quantum mechanically, rather than our previous semiclassical use of quantum mechanical electron behavior but with classical electric and magnetic fields. We say we have “quantized” the electromagnetic field. This quantization is then the basis for all of quantum optics. This approach also prepares us for discussions in subsequent chapters of fermion operators and of the full description of stimulated and spontaneous emission. 15.1 Harmonic oscillator and raising and lowering operators We remember the (time-independent) Schrödinger equation that we had constructed before for the harmonic oscillator (Eq. (2.75)).

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  Lectures on Quantum Mechanics

  A Primer for Mathematicians

  • Philip L. Bowers(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  1.1 The Classical Harmonic Oscillator 3 1.1 The Classical Harmonic Oscillator Consider a particle moving in a one-dimensional quadratic potential well V (x) = 1 2 kx 2 , where the spring constant k is positive and x rep- resents position along a line. The force experienced by the particle is F = −∇V = −kx, and serves to restore the particle to its equilibrium position x = 0. The force on the particle diminishes as it moves toward equilibrium but, since the particle has positive speed as it reaches the equilibrium point, it overshoots past equilibrium and begins to feel the restorative force again. Eventually the force slows the particle to rest at some maximum displacement from equilibrium, at which point the particle begins its journey back toward the equilibrium position, over- shooting again, and reaching a maximum displacement opposite to that of its previous maximum displacement. And so it goes ad infinitum. This is the classical harmonic oscillator, the quintessential physical system of classical mechanics. The newtonian analysis of the harmonic oscillator is straightforward. From Newton’s second law, the equation of motion of the harmonic oscillator is m¨ x + kx = 0, where m is the positive mass of the particle and x = x(t) is position at time t. Here we are using Newton’s notation, where the derivative of a function of the time variable is indicated by a dot above the function. The solution to the equation of motion, equally transparent, is x(t) = a cos ωt + b sin ωt, where the natural or characteristic frequency is ω =  k m and the initial conditions are x(0) = a and ˙ x(0) = ωb. For simplicity, take x(0) = X > 0 and ˙ x(0) = 0, so that x(t) = X cos ωt. Of course X is the magnitude of the maximum displacement. The kinetic energy of the system is K = 1 2 m ˙ x 2 = 1 2 mω 2 X 2 sin 2 ωt and the potential energy is V = 1 2 kx 2 = 1 2 kX 2 cos 2 ωt, so that, using k = mω 2 , the total energy E = K + V is a constant of the motion: E = 1 2 kX 2 = 1 2 mω 2 X 2 .

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  Quantum Mechanical Foundations of Molecular Spectroscopy

  • Max Diem(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  HCl has a characteristic vibrational frequency, but this leaves no room for the concept that electromagnetic radiation causes a transition to a more highly excited state. In a classical system, the energy can increase in infinitesimally small increments by increasing the amplitude of the vibration, whereas in the quantum mechanical and experimentally verified situation, the energy can only increase in certain quantized increments, leading to the absorption and annihilation of a photon. This aspect will be discussed next.

  4.2 The Harmonic Oscillator Schrödinger Equation, Energy Eigenvalues, and Wavefunctions

  Assuming that the chemical bond in a diatomic molecule obeys Hook's law, the vibrational Schrödinger equation for a harmonic oscillator with one degree of freedom (x) is then

  (4.9)

  This follows from Eqs. (2.5) and (4.2) . (In Eq. [4.9] and the following discussion, the subscript “R” in m

  R

  for the reduced mass has been dropped to simplify the notation.) This differential equation is known as “Hermite's” differential equation, in which the wavefunctions ψ(x) are the time‐independent (stationary state) vibrational wavefunctions and E denotes the energy of the vibrational states. Equation (4.9) is a typical operator–eigenvalue equation notation commonly found in linear algebra. This formalism is an instruction to operate with an operator, here, the vibrational Hamiltonian

  (4.10)

  on a set of (yet unknown) functions to obtain the eigenvalues. Substituting the eigenvalues into the trial solution and considering the boundary conditions yield the eigenfunctions ψ(x). Detailed methods for solving the vibrational Schrödinger Eq. (4.9) can be found in many books on vibrational spectroscopy or quantum chemistry textbooks [1 ]. Here, the approach to solve Eq. (4.9) is only outlined to demonstrate how involved such a solution is.

  Equation (4.9) is reformatted to read

  (4.11)

  Here, the results from Eq. (4.8)

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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)

  Sec. 40c ).

  FIG . 11–1.—Energy levels for the harmonic oscillator according to wave mechanics (see Fig. 6–1 ).

  11b. The Wave Functions for the Harmonic Oscillator and Their Physical Interpretation. —For each of the characteristic values

  Wn

  of the energy, a satisfactory solution of the wave equation 11–1 can be constructed by the use of the recursion formula 11–7. Energy levels such as these, to each of which there corresponds only one independent wave function, are said to be non-degenerate to distinguish them from degenerate energy levels (examples of which we shall consider later), to which several independent wave functions correspond. The solutions of 11–1 may be written in the form

  in which .

  Hn

  (ξ) is a polynomial of the n th degree in ξ, and

  Nn

  is a constant which is adjusted so that is normalized, i.e., so that satisfies the relation

  in which , the complex conjugate of , is in this case equal to . In the next section we shall discuss the nature and properties of these solutions in great detail. The first of them, which corresponds to the state of lowest energy for the system, is

  FIG . 11–2.—The wave function for the normal state of the harmonic oscillator (left), and the corresponding probability distribution function (right). The classical distribution function for an oscillator with the same total energy is shown by the dashed curve.

  Figure 11–2 shows this function. From the postulate discussed in Section 10a , , which is also plotted in Figure 11–2

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  Classical Mechanics

  • Tai L. Chow(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  199 © 2010 Taylor & Francis Group, LLC Harmonic Oscillator Harmonic oscillators, along with damped and driven oscillators, will be treated in considerable detail in this chapter not merely because harmonic motion is a good approximation of many physi-cal processes but also because a thorough understanding of this process aids comprehension of the other types of oscillations. By Fourier analysis, complicated oscillations often may be regarded as consisting of a number of simple harmonic oscillations. Simple harmonic motion (SHM) arises whenever a system vibrates around an equilibrium posi-tion. It is caused by a force that is directed toward the equilibrium position and that is proportional to the displacement of the particle from the equilibrium position, which causes its motion. Examples of simple harmonic motions are found in the motion of a weight on the end of a perfect elastic spring, the bob of a simple pendulum swinging through a very small arc, atoms in a crystal lattice, the nuclei of atoms in molecules, and so on. 7.1 SIMPLE HARMONIC OSCILLATOR We first consider two examples of simple harmonic motion. 7.1.1 M OTION OF M ASS M ON THE E ND OF A S PRING The spring has a natural length b and spring constant k when the system is in equilibrium (i.e., when the particle hangs motionless); the weight of the particle is exactly balanced by the restoring force of the spring where d is the extension of the spring (Figure 7.1): mg = kd . (7.1) Suppose the particle has been set into vertical vibration at the instant when the displacement of the particle is x ; the extension of the spring is then d + x .

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  Mathematical Methods for Oscillations and Waves

  • Joel Franklin(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  But the model’s utility in physics has little to do with the coiled metal itself. Instead, the “harmonic” oscillator behavior is really the dominant response of a particle moving near the equilibrium of a potential energy function. To review, a conservative force F comes from the derivative of a potential energy U via: F(x) = − dU(x) dx . (1.18) If we have a potential energy U(x) (from whatever physical configuration), then a point of equilibrium is defined to be one for which the force vanishes. For x e a point of equilibrium, F(x e ) = 0 = − dU(x) dx     x=xe ≡ −U  (x e ). (1.19) 6 Harmonic Oscillator Now if we expand the potential energy function U(x) about the point x e using Taylor expansion: U(x) = U((x − x e )    ≡Δx +x e ) = U(x e ) + ΔxU  (x e ) + 1 2 Δx 2 U  (x e ) + · · · , (1.20) then the first term is just a constant, and that will not contribute to the force in the vicinity of x e (since we take a derivative with respect to x sitting inside Δx to get the force). The second term vanishes by the assumption that x e is a point of equilibrium, and the first term that informs the dynamics of a particle moving in the vicinity of x e is the third term ∼ (1/2)U  (x e )(x − x e ) 2 , leading to a force, near x e : F(x) = − dU(x) dx ≈ −U  (x e )(x − x e ) + · · · (1.21) The effective force in the vicinity of the equilibrium is just a linear restoring force with “spring constant” k ∼ U  (x e ) (assuming U  (x e ) > 0 so that the equilibrium represents a local minimum) and equilibrium spacing x e . A picture of a local minimum in the potential energy and the associated force is shown in Figure 1.3. Near x e , the potential is approximately quadratic, and the force is a linear restoring force of the sort we have been studying. There is also an equilibrium point at the maximum of U(x) in that picture, but the associated force tends to drive motion away from this second equilibrium.

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