Wave equation harmonic oscillator

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

You might remember from your physics that this is the differential equation that describes a harmonic oscillator. The solution is a sine wave with a frequency of l/ip. We will discuss these kinds of functions in detail in Part V when we begin our Chinese" lessons covering the frequency domain. 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

These manipulations have brought us to a familiar equation we recognize (4.23) as the Schrodinger equation (1.132) for a one-dimensional harmonic oscillator with force constant ke. Before we can conclude that (4.23) and (1.132) have the same solutions, we must verify that the boundary conditions are the same. For quadratic integrability, we require that S(q) vanish for q = oo. Also, since the radial factor F(R) in the nuclear wave function is... 

The electromagnetic field is quantized as a set of harmonic oscillators. Maxwell s equations, and the resulting wave equations, are described by partial differential equations that formally have an infinite number of degrees of freedom. Physically this means that the electromagnetic held is described by an infinite number of harmonic oscillators, where one sits at every point in space. The modes of the electromagnetic held are then completely described by this ensemble of harmonic oscillators. 

The eigenvalues and eigenfunctions of the simple harmonic oscillator are well known. A detailed account of the solution of the wave equation in (2.157) is given by Pauling and Wilson [11], The solution of equation (2.163) using creation and annihilation operators is described in the book by Bunker and Jensen [12]. The energy levels of the harmonic oscillator are given by... 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

The interaction of a light wave and electrons in atoms in a solid was first analysed by H. A. Lorentz using a classical model of a damped harmonic oscillator subject to a force determined by the local electric field in the medium, see Equation (2.28). Since an atom is small compared with the wavelength of the radiation, the electric field can be regarded as constant across the atom, when the equation of motion becomes ... 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

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... 

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