The One-Dimensional Harmonic Oscillator

In Chapter 2 we examined several systems with discontinuous potential energies. In this chapter we consider the simple harmonic oscillator—a system with a continuously varying potential. There are several reasons for studying this problem in detail. First, the quantum-mechanical harmonic oscillator plays an essential role in our understanding of molecular vibrations, their spectra, and their influence on thermodynamic properties. Second, the qualitative results of the problem exemplify the concepts we have presented in Chapters 1 and 2. Finally, the problem provides a good demonstration of mathematical techniques that are important in quantum chemistry. Since many chemists are not overly familiar with some of these mathematical concepts, we shall deal with them in detail in the context of this problem. 

In this section we will increase our quantum-mechanical repertoire by solving the SchrOdinger equation for the one-dimensional harmonic oscillator. This system is important as a model for molecular vibrations. 

Classical-Mechanical D eatment. Before looking at the wave mechanics of the harmonic oscillator, we review the classical treatment. We have a single particle of mass m attracted toward the origin by a force proportional to the particle s displacement from the origin  

The proportionality constant k is called the force constant. is the x component of the force on the particle. This is also the total force in this one-dimensional problem. Equation (4.22) is obeyed by a particle attached to a spring, provided the spring is not stretched greatly from its equilibrium position. 

Since the sine function has maximum and minimum values of 1 and -1, respectively, jc in (4.24) oscillates between A and —A. The sine function repeats itself every 2tt radi- 

Now consider the energy. The potential energy V is related to the components of force in the three-dimensional case by 

Since the sine function has maximum and minimum values of 1 and —1, respectively, X in (4.22) oscillates between A and —A. The sine function repeats itself every 2tt radians, and the time needed for one complete oscillation (called the period) is the time it takes for the argument of the sine function to increase by 2tt. At time t + /v, the argument of the sine function is 2Trv t + /v) + b = 27rvt + 2tt + b, which is 2tt greater than the argument at time t, so the period is l/v. The reciprocal of the period is the number of vibrations per unit time (the vibrational frequency), and so the frequency is v. 

In Chapter 2 we considered a particle confined to a square potential well with infinitely high sides. In that model there was no force acting on the moving particle until it reached one of the walls, where it underwent an elastic reflection. In this chapter a different type of one-dimensional potential well will be considered in which the particle experiences a force which is proportional to its displacement from the midpoint of the well. This model is particularly suited to an examination of the vibrations of molecules. 

a minus sign is required because the work done is positive when the movement is in the opposite direction to the action of the force. The potential energy of the diatomic molecule, V, is equal to the total work done in extending the spring and is therefore given by the equation  

Situation after bond has been extended by a distance x 

Application of Newton s second law of motion (force = mass x acceleration) to the vibration leads to the equation  

The fundamental frequency of the vibration, co, is equal to 1/t. After rearrangement, equation (4.10) becomes  

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Problem 8-17. To show that the general conclusions suggested in the previous example are false, consider the case of the one-dimensional harmonic oscillator, for which H = T + V = T + kx. The energy levels of the one-dimensional harmonic oscillator with frequency v are ... 

For example, consider the one-dimensional harmonic-oscillator states. These states are either even or odd, depending on whether the quantum... 

The 3/V —6 one-dimensional Schrodinger equations (6.50) are easily solved. The one-dimensional harmonic-oscillator Hamiltonian is... 

The polyad quantum number is defined as the sum of the number of nodes of the one-electron orbitals in the leading configuration of the Cl wave function [19]. The name polyad originates from molecular vibrational spectroscopy, where such a quantum number is used to characterize a group of vibrational states for which the individual states cannot be assigned by a set of normal-mode quantum numbers due to a mixing of different vibrational modes [19]. In the present case of quasi-one-dimensional quantum dots, the polyad quantum number can be defined as the sum of the one-dimensional harmonic-oscillator quantum numbers for all electrons. 

Next, let us compute, for the one-dimensional harmonic oscillator, the "first moment" integral n x n). First, remember that the eigenfunctions of the harmonic oscillator are orthonormal ... 

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative 8t/8p, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the TF statistical theory. This latter theory gives for the density in d dimensions... 

Figure 22 Potential energy of the one-dimensional harmonic oscillator as function of interatomic displacement r...

The problem considered is the one-dimensional harmonic oscillator perturbed by cubic and quartic potential terms. Thus, the unperturbed Hamiltonian operator is... 

The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Poschl-Teller potential in quantum mechanics. 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

For the case of the one-dimensional harmonic oscillator the potential energy is given by V (x) = kx, which is parameterized in terms of a stiffness constant k. After some manipulation, the Schrodinger equation associated with this potential may be written as... 

In order to illustrate this approach, we shall consider two simple quantum systems the rigid rotator and the one-dimensional harmonic oscillator. Since the rotational and vibrational motions are incorporated in a natural way in the quantum behavior of the 3-dimensional harmonic oscillator, these examples will provide a good basis for understanding the properties of more general systems. 

In both cases, in the limit q -> 1 (r -> 0) one gets the standard expression for the energy of the one-dimensional harmonic oscillator ... 

The one-dimensional harmonic oscillator, restricted to -motion along the x axis in accordance with the potential function V = %kx2 = 2ir2mv xi, is seen to carry out harmonic oscillations... 

Equation 15-9 is the same as the wave equation 11-2 for the one-dimensional harmonic oscillator which was solved in Section 11. Referring to that section, we find that X(x) is given by the expression... 

Let us now consider a system whose Schrodinger time functions corresponding to the stationary states of the system are k0, i, , kn, . Suppose that we carry out an experiment (the measurement of the values of some dynamical quantities) such as to determine the wave function uniquely. Such an experiment is called a maximal measurement. A maximal measurement for a system with one degree of freedom, such as the one-dimensional harmonic oscillator, might consist in the accurate measurement of the energy the result of the measurement would be one of the characteristic energy values W and the corresponding wave function would then represent the... 

Fig. 5 2 In a potential well with a single minimum, the wave function for the ground state has a single maximum The wells above are for (a) the one-dimensional box, (b) the one-dimensional harmonic oscillator, and (c) an atom, in cross-section...

Much of the previous discussion of surface properties has treated the surface as the termination of a rigid periodic lattice. In reality, at temperatures above 0 K, surface atoms are vibrating about their equilibrium position. A useful model of this process is the one-dimensional harmonic oscillator. In this model, atoms are treated as a mass attached through a spring to a fixed surface. As depicted in Figure 21, restoring forces are exerted on the atom as it moves from its equilibrium position r. This force is linearly proportional to the displacement x and a proportionality constant k, called the force constant. The force is given by the equation... 

The one-dimensional harmonic oscillator has two dimensions (p and q), so that its phase space volume for H < E is given by... 

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