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Relaxation of a quantum harmonic oscillator

We next consider another example of quantum-mechanical relaxation. In this example an isolated harmonic mode, which is regarded as our system, is weakly coupled to an infinite bath of other harmonic modes. This example is most easily analyzed using the boson operator formalism (Section 2.9.2), with the Hamiltonian [Pg.322]

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations  [Pg.322]

The initial conditions at Z = 0 are the corresponding Schrodinger operators. This model is seen to be particularly simple All operators in Eq. (9.46) commute with each other, therefore this set of equations can be solved as if these operators are scalars. [Pg.323]

Note that Eqs (9.46) are completely identical to the set of equations (9.6) and (9.7). The problem of a single oscillator coupled linearly to a set of other oscillators that are otherwise independent is found to be isomorphic, in the rotating wave approximation, to the problem of a quantum level coupled to a manifold of other levels. There is one important difference between these problems though. Equations (9.6) and (9.7) were solved for the initial conditions Co(Z = 0) = 1, Cz(Z = 0) = 0, while here a t = 0) and h/(Z = 0) are the Schrodinger representation counterparts of fl(Z) and Still, Eqs (9.46) can be solved by Laplace transform following the route used to solve (9.6) and (9.7). [Pg.323]

In what follows we take a different route (that can be also applied to (9.6) and (9.7)) that sheds more light on the nature of the model assumptions involved. We start by writing the solution of Eq. (9.46b) in the form [Pg.323]

The Heisenberg equations of motion,. = ilh)[H,A for the Heisenberg-representation operators a t) and b t) are derived using these commutations [Pg.322]


Z is proportional to the gas pressure, and, since Z1>0, the collision number for energy transfer, is constant for a particular transition, the actual value of fi is inversely proportional to the pressure. For convenience relaxation times are usually referred to a pressure of 1 atm. Equation (1) is an approximation, and requires modification to take into account the reversibility between quantum states 0 and 1. For example, the correct equation for vibrational relaxation of a simple harmonic oscillator of fundamental frequency, v, is... [Pg.184]

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. [Pg.278]

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. [Pg.509]

In this example the master equation formalism is applied to the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator. [Pg.278]

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum harmonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T -> 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. [Pg.467]

A more interesting point was made by Bader and Berne, who noted that the vibrational relaxation rate of a classical oscillator in a classical harmonic bath is identical to that of a quantum oscillator in a quantum harmonic bath [71]. On the other hand, when the relaxation of the quantum system is calculated using the corrected correlation function of the classical bath [Eq. (31)], the predicted rate is slower by a factor of j/3h(i) coth(/3h(o/2), which can be quite substantial for high-frequency solutes. The conclusions of a number of recent studies were shown to be strongly affected by this inconsistency [42,43,72]. Quantizing the solvent by mapping the classical correlation functions onto a quantum harmonic bath corrects the discrepancy. [Pg.93]

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. [Pg.30]

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... [Pg.204]

Equation 3.5, where v is the vibrational quantum number, means that only transitions between nearest vibrational states can directly occur in the case of the harmonic oscillator. This means that the 1R spectrum is generally mostly constituted hy fundamental transitions, that is, those associated with excitation from the fundamental state to the first excited state. This condition, however, is relaxed in the case of anharmonic oscillators, so that not only fundamental transitions but also overtone and combination modes (also called the harmonics, i.e. modes associated with the excitation from the fundamental state to a second or third excited state) can be sometimes observed, although they are usually weak. [Pg.99]

The Fourier component of interaction force F(t) on the transition frequency (2-184) characterizes the level of resonance. Matrix elements for harmonic oscillators m y n) are non-zero only for one-quantum transitions, n = m . The W relaxation probability of the one-quantum exchange as a function of translational temperature To can be found by averaging the probability over Maxwelhan distribution ... [Pg.73]

The validity of Eq. (15.11) even at the limit Na Nm means in fact that for harmonic oscillators relaxing in a heat bath there exists a closed equation for the mean energy. This is intimately connected with the linear dependence of transition probabilities on the vibrational quantum number (see Eq. (14.1)) and implies that the energy relaxation rate is independent of the initial distribution of harmonic oscillators over vibrational states. Still another peculiarity of this system is known the initial Boltzmann distribution corresponding to the vibrational temperature Tq =t= T relaxes to the equilibrium distribution via the set of the Boltzmann distributions with time-dependent temperatures. If (E ) is explicitly expressed by a time-dependent temperature, this process is again described by Eq. (15.11). [Pg.88]


See other pages where Relaxation of a quantum harmonic oscillator is mentioned: [Pg.322]    [Pg.323]    [Pg.325]    [Pg.327]    [Pg.322]    [Pg.323]    [Pg.325]    [Pg.327]    [Pg.322]    [Pg.323]    [Pg.325]    [Pg.327]    [Pg.322]    [Pg.323]    [Pg.325]    [Pg.327]    [Pg.512]    [Pg.512]    [Pg.251]    [Pg.202]    [Pg.7]    [Pg.79]    [Pg.67]    [Pg.242]    [Pg.307]    [Pg.137]    [Pg.646]    [Pg.20]    [Pg.321]    [Pg.498]    [Pg.307]    [Pg.103]    [Pg.47]    [Pg.499]    [Pg.321]    [Pg.468]   


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