The Quantum-Mechanical Harmonic Oscillator

Quantum mechanically, a wavefunction for a one-dimensional harmonic oscillator can be determined using the (time-independent) Schrodinger equation 

This differential equation does have an analytic solution. The method we use here is one general technique for solving differential equations We define the wave-function as a power series. What we will ultimately find is that in order to solve the Schrodinger equation, the power series must have a special form. 

the Schrodinger equation 11.4 will be rewritten using equation 11.3 to substitute for k. Rearranging equation 11.3, one finds that the force constant k is 

Second, we divide both sides of the equation by the term — fi l2m. Third, we bring all terms over to one side of the equation so that we have an expression equaling zero. The Schrodinger equation becomes 

We now assume that the form of the wavefunction T that satisfies this Schrodinger equation has the form of a power series in the variable x. That is, the wavefunction is some function/(x) that has some term containing xP (which is simply 1), some term containing some term containing ad infinitum, all added together. Each power of x has some constant called a coejficient multiplying it, so the form of/(x) (recognizing that x° = 1) is  

We have already seen [Eq. (3-8)] that the potential energy of a harmonic oscillator is 

The detailed solution of this differential equation is taken up in the next section. Here we show that we can understand a great deal about the nature of the solutions to this equation by analogy with the systems studied in Chapter 2. 

In Fig. 3-2a are shown the potential, some eigenvalues, and some eigenfunctions for the harmonic oscillator. The potential function is a parabola [Eq. (3-14)] centered at x = 0 and having a value of zero at its lowest point. For comparison, similar information 

The wavefunctions for the harmonic oscillator are either symmetric or antisymmetric under reflection through x = 0. This is necessary because the hamiltonian is invariant 

The harmonic oscillator has finite zero-point energy. (The evidence for this in Fig. 3-2a is the observation that the line for the lowest (n = 0) energy level lies above the lowest point of the parabola, where V = 0.) This is expected since the change from square well to parabolic well does not remove the restrictions on particle position it merely changes them. 

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For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

In the classical high-temperature limit, kBT hv, where kB is the Boltzmann constant, and hv is the spacing of the quantum-mechanical harmonic oscillator energy levels. If this condition is fulfilled, the energy levels may be considered as continuous, and Boltzmann statistics apply. The corresponding distribution is... 

Modeling Stretching Modes of Common Organic Molecules with the Quantum Mechanical Harmonic Oscillator 162... 

As soon as bound states are considered there are only discrete energy levels. Nevertheless it was shown by Bell [77] that it is possible to employ approximately a continuum of energy levels for the calculations of the tunnel rates, which is adequate for the description of many experimental systems. In the simplest form (see Fig. 21.5) of the Bell model, the potential barrier is an inverted parabola. This allows the use of the known solution of the quantum mechanical harmonic oscillator for the calculation of the transition probability through the barrier. The corresponding Schrodinger equation is... 

The quantum-mechanical harmonic oscillator satisfies the Schrodinger equation ... 

There are several very important features to note in Figure 3.32. First of all, according to classical mechanics, the mass on the end of the spring can have any energy, while the quantum mechanical harmonic oscillator must have the discrete energy levels given by Equation (3.78). Second, the potential energy of the mass will... 

The probability function for the harmonic oscillator with v= 10, showing how the potential energy for the quantum mechanical harmonic oscillator approaches that of the classical harmonic oscillator for very large values of v. [ Michael D Payer, Elements of Quantum Mechanics, 2001, by permission of Oxford University Press, USA.]... 

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. 

A8-1. Use the methods outlined in this appendix to show that F = r for any stationary state of the quantum mechanical harmonic oscillator. 

The energy levels of the quantum mechanical harmonic oscillator depicted as horizontal lines intersecting the potential energy function for the oscillator. 

Probability density functions of low-lying levels of the quantum mechanical harmonic oscillator. These functions are the squares of the wavefunctions. They are drawn so that the baseline, or point of zero probability density, is the energy level line as in Figure 7.6. Notice that as n increases, the probability density at = 0 diminishes. Furthermore, at levels such as w = 10 the points of maximum probability density are close to the classical turning points, unlike the situation for the lower states. 

Oscillators the size of molecules obey the laws of quantum mechanics. The vibrational energy of the quantum mechanical harmonic oscillator is not continuously vari-... 

To better understand the quantum mechanical harmonic oscillator, the results of the quantum mechanical system can be compared to those for the classical mechanical system (described in Section 1.3). The classical turning point for the mass, , occurs when the energy of a given state is equal... 

Another interesting feature of the quantum mechanical harmonic oscillator is that the energy difference between subsequent levels is the same -E, =h(0. This feature of uniform energy levels is a result of... 

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