The Quantum Harmonic Oscillator

In Chapter 14, we solved the classical equation of motion for a harmonic oscillator. We now solve the time-independent Schrodinger equation for this model system. The Hamiltonian operator must contain the potential energy expression from 

The Hermite equation Is named for Charles Hermite, 1822-1901, a great French mathematician who made many contributions to mathematics, Including the proof that e (2.71828...) Is a transcendental irrational number. 

If we set the constant o equal to zero, the time-independent Schrodinger 

This differential equation is the same as a famous equation known as the Hermite equation (see Appendix F). Hermite solved this equation by assuming that the solution was of the form 

Assume that the foUowing equation is valid for any value of x  

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The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

Using MMd. calculate A H and. V leading to ATT and t his reaction has been the subject of computational studies (Kar, Len/ and Vaughan, 1994) and experimental studies by Akimoto et al, (Akimoto, Sprung, and Pitts. 1972) and by Kapej n et al, (Kapeijn, van der Steen, and Mol, 198.V), Quantum mechanical systems, including the quantum harmonic oscillator, will be treated in more detail in later chapters. 

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). 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

The solvent will be treated classically (1) to avoid the quantum harmonic oscillator treatment of the polar solvent which is... 

These operators allow for the description of the quantum harmonic oscillator that is very parsimonious. The quantum harmonic oscillator has evenly spaced eigenstates, and the state of the system may be changed according to... 

Some Properties of Coherent States Expansion of the Coherent State on the Eigenvectors of the Quantum Harmonic Oscillator Hamiltonian... 

In this expression, according to the theory of the quantum harmonic oscillator, the operator q appearing on the right-hand side, may couple two successive eigenstates /c ) of the Hamiltonian of the harmonic oscillator. Consequently, by ignoring the scalar term p(0,0), which does not couple these states, we may write the dipole moment operator according to... 

The eigenvalue equations of the quantum harmonic oscillators Hamiltonians Hpree and H° given by Eqs. (21) and (15) are, respectively,... 

Again, insert a closeness relation on the eigenstates of the quantum harmonic oscillator involved in Eq. (23) and then, perform the trace over the basis involving these eigenstates. That gives... 

It is shown in Appendix N that the action of the translation operator on the ground state (0) ) of the Hamiltonian of the quantum harmonic oscillator gives a coherent state a ) ... 

Here, H is the Hamiltonian of the quantum harmonic oscillator coupled to the thermal bath, which is that given by Eq. (1.1). 

N.l EXPANSION OF THE COHERENT STATE ON THE EIGENVECTORS OF THE QUANTUM HARMONIC OSCILLATOR HAMILTONIAN... 

By definition, a coherent state a) is the eigenvector of the non-Hermitean lowering operator a of the quantum harmonic oscillator. Thus, the basic equation and its conjugate are, respectively,... 

Now, we may insert in front of a coherent state, the closeness relation on the eigenstates of the number operator at a of the quantum harmonic oscillator (with [a, at] = 1), in the following way ... 

As a consequence, the expansion (N.2) of the coherent state on the eigenkets of the quantum harmonic oscillator Hamiltonian, gives... 

Then, owing to the orthonormality (n m) = 8 m of the eigenstates of the quantum harmonic oscillator Hamiltonian, the above expression reduces to... 

Again, note that when the absolute temperature is vanishing, the Boltzmann density operator reduces to that of the ground state of the Hamiltonian of the quantum harmonic oscillator, that is,... 

Bosons of the quantum harmonic oscillator describing the H-bond bridge with... 

We simply list the solutions, which you can verify by substituting them into the Schrodinger equation. The first four wave functions for the quantum harmonic oscillator are listed in Table 4.2 and plotted in Figure 4.31. The energy levels of the harmonic oscillator are given by... 

First, in contrast with the classical harmonic oscillator, the quantum harmonic oscillator in its ground state is most likely to be found at its equilibrium position. A classical harmonic oscillator spends most of its time at the classical turning points, the positions where it slows down, stops, and reverses directions. (It might... 

FIGURE 4.31 Solutions for the quantum harmonic oscillator, (a) The first four wave functions, (b) Probability densities corresponding to the first four wave functions. 

Particle-in-a-box models and the qnantnm harmonic oscillator illustrate a number of important features of quantum mechanics. The energy level structure depends on the natnre of the potential in the particle in a box, E n, whereas for the harmonic oscillator, E n. The probability distributions in both cases are different than for the classical analogs. The most probable location for the particle-in-a-box model in its gronnd state is the center of the box, rather than uniform over the box as predicted by classical mechanics. The most probable position for the quantum harmonic oscillator in the ground state is at its equilibrium position, whereas the classical harmonic oscillator is most likely to be fonnd at the two classical turning points. Normalization ensures that the probabilities of finding the particle or the oscillator at all positions add np to one. Finally, for large values of n, the probability distribution looks mnch more classical, in accordance with the correspondence principle. 

The quantum harmonic oscillator is a model system that describes chemical and physical systems in which the restoring force is proportional to the displacement from the equilibrium position xq and is opposite to the direction of the displacement. The restoring force is given by F(x) = —k(x — Xq) and the... 

The Schrodinger equation for the quantum harmonic oscillator can be solved... 

Section 4.7 introduces the quantum harmonic oscillator and provides the groundwork for subsequent discussions of vibrational spectroscopy. This section is completely new. 

Back to the quantum harmonic oscillator, the Hermite polynomials helps in writing its eigen-functions as ... 

This result allows the determination of the normalization constant above for the wave-function of the quantum harmonic oscillator. 

FIGURE 3.8 The quantum harmonic oscillator eigen-function probabilities (density) representation (thick continuous curves) for ground state ( = 0), and few excited vibronic states ( = 2, 5, and 10) for the working case of HI molecule (respecting the coordinated centered on its mass center) the classical potential is as well illustrated (by the dashed curve in each instant) for facihtating the correspondence principle discussion. 

Moreover, the search for the best approximation of effective-classical partition function (4.524) will be conducted as such the quantum fluctuations be not dependent on the classical displacement (4.522), abstracted from the free motion, but being driven by the quantum harmonic oscillations - through they constitute a generalization of the free motion itself, see for instance the equivalence of classical paths or propagators of free with harmonic motion in the zero-frequency limit, see (Putz, 2009). 

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