Harmonic oscillator quantum mechanical

By replacing variables in the classical Hamiltonian with corresponding quantum mechanical operators, develop the quantum mechanical Hamiltonian operator and express the Schrodinger equation specific to the problem. 

Find the wavefunction from the Schrodinger equation, which means finding the eigenfunction or eigenfunctions of the Hamiltonian operator. 

From the earlier analysis of the classical harmonic oscillator we have the Hamiltonian function. 

For convenience, let us now define the x-coordinate origin to be the location of the mass when the system is at equilibrium. This means choosing the origin such that Xg = 0. Thus, 

The quantum mechanical Hamiltonian is constructed formally by making x, p, and H operators. 

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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... 

FIG. 2.5. Temperature dependence of the mean-square (ms) displacement of a quantum-mechanical harmonic oscillator with a mass of 200 daltons for a number of frequencies. The linearity of the ms displacement in the high-temperature region is evident. 

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

Note that there is nothing wrong widi Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively. 

It is interesting to note that the vibrational model of the nucleus predicts that each nucleus will be continuously undergoing zero-point motion in all of its modes. This zero-point motion of a quantum mechanical harmonic oscillator is a formal consequence of the Heisenberg uncertainty principle and can also be seen in the fact that the lowest energy state, N = 0, has the finite energy of h to/2. 

For the vibrational partition function the molecule is regarded as a quantum mechanical harmonic oscillator, for which ... 

Just as London did, these frequencies are considered as frequencies of two quantum mechanical harmonic oscillators with lowest eigen values and where Expanding a + and ot one finds for the interaction energy ... 

For a classical harmonic oscillator, the particle cannot go beyond the points where the total energy equals the potential energy. Identify these points for a quantum-mechanical harmonic oscillator in its ground state. Write an integral giving the probability that the particle will go beyond these classically-allowed points. (You need not evaluate the integral.)... 

Moreover, when the probability flux correlation function decays on a time scale shorter than the time scale of the solvent effects on the R motion, the solvent effects on the R motion can be neglected. In this case, Cf,(t) can be approximated by the standard analytical expression for the time correlation function of an undamped quantum mechanical harmonic oscillator [53] ... 

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... 

Figure 21.5 The Bell tunnel model (a) Quantum mechanical harmonic oscillator with its ground state wavefunctions. (b) Inverted harmonic oscillator potential, (c) A stream of particles with a Boltzmann distribution of energies hits the barrier. Classical only those particles with W>Vq can pass the barrier.

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.]... 

From prior experience with the one-dimensional quantum mechanical harmonic oscillator [4], we know that its eigenfunctions and eigenvalues are... 

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. 

The manner in which the total energy is partitioned into average potential and kinetic parts is the same for classical and quantum-mechanical harmonic oscillators, namely, half and half. 

Light of wavelength 4.33 x 10 m excites a quantum-mechanical harmonic oscillator from its ground to its first excited state. Which one of the following wavelengths would accomplish this same transition if i) the force constant only was doubled ii) the mass only was doubled ... 

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

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