Harmonic oscillator quantum

You have already encountered the classical version of the harmonic oscillator in our discussion of blackbody radiation in Section 4.2. The equations for the force and the potential energy of the harmonic oscillator are  

To obtain the quantum version, we substitute the potential energy function for the harmonic oscillator into the Schrodinger equation to get 

TABLE 4.2 The First Four Flarmonic Oscillator Wave Functions 

Note that, unlike the particle in a box, m = 0 is perfectly acceptable because the 1 

Potential energy diagrams for diatomic molecules were introduced in Section 3.5, and you can see that they are not parabolic over the entire region 0 r 00 (for example, see Fig. 3.9). Near the equilibrium internuclear separation the potential appears to be well approximated by a parabola. This similarity suggests that the harmonic oscillator should be a good model to describe the vibrations of diatomic molecules. The dependence of the vibrational frequency v on the force constant k and the mass has the same form as Equation 4.44, but now the mass is the reduced mass /t of the two nuclei 

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Mavri, J., Berendsen, H.J.C. Dynamical simulation of a quantum harmonic oscillator in a noble-gas bath by density matrix evolution. Phys. Rev. E 50 (1994) 198-204. 

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. 

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

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). 

The ground-state effective Hamiltonian is diagonal with eigenvalues ha n + 5], whereas the excited state one is that of a driven quantum harmonic oscillator that must lead to coherent states. 

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

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 effective Hamiltonian /7 °f, related to the ground states 0 ) and [0]) of the fast and bending modes, is the Hamiltonian of a quantum harmonic oscillator characterizing the slow mode ... 

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

Figure 5.11 The shape of the lxo(2)l and x2o(Q) functions for a quantum harmonic oscillator.

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... 

Boltzmann Density Operators in Both Representations The Evolution Operator of a Driven Quantum Harmonic Oscillator [59]... 

Thermal Average of the Translation Operator The Driven Damped Quantum Harmonic Oscillator Quantum Representation II ... 

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

There are well-known temperature effects, particularly dealing with the two first moments of the spectra that evoke those of the thermal average appearing in the statistical mechanics of quantum harmonic oscillator coordinates. 

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... 

H° and HFree are, respectively, the Hamiltonians of the fast and slow modes viewed as quantum harmonic oscillators, whereas Hint is the anharmonic coupling between the two modes, which are given by Eqs. (15), (21), and (22). Besides, He is the Hamiltonian of the thermal bath, while Hint is the Hamiltonian of the interaction of the H-bond bridge with the thermal bath. 

InEq. (113), I)] (tfv is the IP time-evolution operator of the driven quantum harmonic oscillator interacting with the thermal bath,... 

Another possibility is to extract the reduced time evolution operator from the analytical solution obtained by Louisell and Walker for the reduced time-dependent density operator of a driven damped quantum harmonic oscillator. 

Now, perform the trace over the eigenstates of the slow mode quantum harmonic oscillator involved in the ACF (114). This leads, after neglecting the zero-point energy of H-bond bridge oscillator, to... 

Appendix D shows that the IP time evolution operator of a driven quantum harmonic oscillator is given by Eq. (D.23), that is,... 

Alternatively, according to Eqs. (147) and (148), the classical limit of the mean number occupation of a quantum harmonic oscillator is... 

From this viewpoint, which is the most fundamental, the line shape as a whole is the sum of the diagonal matrix elements of the time evolution operator of the driven damped quantum harmonic oscillator in the IP representation with respect to the diagonal part of the Hamiltonian of this oscillator. According to Eq. (120), each diagonal element is a sum of time-dependent terms... 

Figure 16. The effects of the parity operator C2 on the ground and the first excited states of the symmetrized g and u eigenfunctions of the g and u quantum harmonic oscillators involved in the centrosymmetric cyclic dimer. (The subscripts 1 and 2 refer, respectively, to the a and b moieties of the centrosymmetric cyclic dimer).

It may be observed that the two Hamiltonians (269) are those of quantum harmonic oscillators, whereas Hamiltonian (270) is that of a driven damped quantum harmonic oscillator, and Hamiltonians (271) are those of driven undamped quantum harmonic oscillators perturbed by the Davydov coupling... 

Consider the effective Hamiltonian (47) of a driven quantum harmonic oscillator. Since it is not diagonal, it may be suitable to diagonalize it with the aid of a canonical transformation that will affect it or its equivalent form (50), but not that of (46) or its equivalent expression (49), which is yet to be diagonal [13]. 

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