Harmonic oscillator probability density

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

Figure E.4 Low-lying levels and associated probability densities for the harmonic oscillator.

The probability is calculated according to classical statistical mechanics (Appendix A.2). According to Eq. (A.49), the density of states (number of states per unit energy) for s uncoupled harmonic oscillators with frequencies z/j is... 

The assumption about a uniform probability for any distribution of the energy between the harmonic oscillators may now be used to determine the probability Pet >e (E). It can be expressed as the ratio between the density of states corresponding to the situation where the energy exceeds the threshold energy in the reaction coordinate and the total density of states at energy E, that is, N(E) of Eq. (7.36). 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared with those of a harmonic oscillator. Figure 6.6 shows some wave functions and probability density functions (i//, i// )2 for an anharmonic oscillator. The asymmetry in ifjv and (i// i//,)2. compared with the harmonic oscillator wave functions in Figure 1.13, increases their magnitude on the shallow side of the potential curve compared with the steep side. 

The wave functions for u = 0 to 4 are plotted in figure 6.20 the point where the function crosses through zero is called a node, and we note that the wave function for level v has v nodes. The probability density distribution for each vibrational level is shown in figure 6.21, and the difference between quantum and classical behaviour is a notable feature of this diagram. For example, in the v = 0 level the probability is a maximum at y = 0, whereas for a classical harmonic oscillator it would be a minimum at v = 0, with maxima at the classical turning points. Furthermore the probability density is small but finite outside the classical region, a phenomenon known as quantum mechanical tunnelling. 

Figure 6.21. Probability density distribution for the v = 0 to 4 vibrational levels of the harmonic oscillator.

Fig. 2.26. The harmonic oscillator energy levels and wave functions. The potential is bounded by the curve V = /ikiArf (heavy solid line). The quantum-mechanical probability density functions 4> M are shown as light solid lines for each energy level, while the corresponding classical probabilities are shown as dashed lines (after McMillan, 1985 reproduced with the publisher s permission).

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. 

We wish to compare the quantum probability distributions with those obtained from the classical treatment of the harmonic oscillator at the same energies. The classical probability density P y) as a function of the reduced distance y(—l y l)is given by equation (4.10) and is shown in Figure... 

Figure Al.1.2. Probability density ( i/ vi/) for the n = 29 state of the harmonic oscillator. The vertical state is chosen as in Figure Al.l.L so that the locations of the turning points coincide with the superimposed potential function.

Figure Bl.2.4. Lowest five harmonic oscillator wavefimctions v]/ and probability densities vi/. ...

The maximum of the ground-state probability density for the harmonic oscillator corresponds indeed just to the equilibrium geometry. This is why the selection rules work at all (although in an approximate way). 

The transformation of Fokker-Planck equation into Schrodinger equation also serves in finding the FP solution for the transition probability density, here for the harmonic oscillator. To this aim, one firstly considers... 

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. 

In contrast to the harmonic oscillator of our classical model for absorption, real atoms or molecules have many energy levels and therefore also many possible transitions from lower thermally populated levels. This means that there are many absorption lines at different frequencies coi. The magnitude of the absorption coefficient a depends on the population density of the absorbing level and on the transition probabilities for the different transitions. These transition probabilities can be only calculated by using quantum mechanical methods, but there are several experimental techniques for their determination (see next section). 

Fig. 9. 39 (a) The wavefunctions and (b) the probability densities of the first three states of a harmonic oscillator. Note how the probability of finding the oscillator at large displacements increases as the state of excitation increases. The wavefunctions and displacements are expressed in terms of the parameter a =... 

Fl 9.40 A schematic illustration of the probability density for finding a harmonic oscillator at a given displacement. Classically, the oscillator cannot be foimd at displacements at which its total energy is less than its potential energy (because the kinetic energy cannot be negative). 

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