Resonance in Quantum Mechanics

Resonance in Quantum Mechanics.—In order to illustrate the resonance phenomenon in quantum mechanics, let us continue to discuss the system of interacting harmonic oscillators.1 Using the potential function of Equation 41-1, the wave equation can be at once separated in the coordinates and 17 and solved in terms of the Hermite functions. The energy levels are given by the expression 

This treatment, like the classical treatment using the coordinates and ij, makes no direct reference to resonance. Let us 

1 This example was used by Heisenberg in his first papers on the resonance phenomenon. Z.f. Phya. 38, 411 (1926) 41. 239 (1927). 

The perturbation energy for the non-degenerate level n = 0 is zero. For the level n = 1 the secular equation is found to be (Sec. 24) 

The correct zeroth-order wave functions for the two levels with n = 1 are found to be 

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At the end of this section, it is worthwhile to point out that resonances in quantum mechanics are intimately related to the existence of trapped classical trajectories. The smaller the classical forces between r and 7 on one hand and R on the other, the longer is the lifetime and vice versa. In this sense it might be helpful for understanding the complex quantum dynamics by imagining the trajectories of a classical billiard ball moving on multidimensional potential-energy surfaces (see, e.g., Chapter 5 of Ref. 4). 

Heisenberg, W. (1927) The problem of several bodies and resonance in quantum mechanics. II, Z. Phys. 41, 239. 

How can other classical features, e.g. resonances, be seen in quantum systems Perhaps the most direct and familiar consequence of a classical resonance, if it occupies a sufficient amount of phase space, is to lead to resonantly mixed quantum states (or metastable states). Typically, vdW stretching freqencies are 30 cm" and vdW bending frequencies are on the order of 15 cm as seen, for example, in a recent study of Ar-glyoxal complexes by Dai and co-workers. This was the case for our NeQ2 study and led, for example, to the importance of the (l,0,-2) resonance. Such 1 2 stretch-bend resonances, in quantum mechanics, are simply Fermi resonances. Indeed, examples of such Fermi resonances have been noted in quantum studies of the metastable states of NeQ2 and ArCl2. vdW stretch-bend interactions have recently been invoked to explain spectral patterns observed in substituted benzene-Ar complexes. 

In quantum mechanics, as we have already seen, one can approximately describe the hydrogen molecular ion as consisting of Ha+ and Hb, or Hb+ and Ha. Some combination of wave functions representing these two configurations is needed as an approximation of the actual state of affairs. The state of H2+ can then be thought of as a resonance hybrid of the two. 

By means of this combination of the cross section for an ellipsoid with the Drude dielectric function we arrive at resonance absorption where there is no comparable structure in the bulk metal absorption. The absorption cross section is a maximum at co = ojs and falls to approximately one-half its maximum value at the frequencies = us y/2 (provided that v2 ). That is, the surface mode frequency is us or, in quantum-mechanical language, the surface plasmon energy is hcos. We have assumed that the dielectric function of the surrounding medium is constant or weakly dependent on frequency. 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

The word resonance is a very widespread term in the scientific world. Common uses range from being in a or on resonance to resonance poles and peaks. As with many such ubiquitous terms, they evolve with time and tend to take a life of their own acquiring new meaning and connotations as time goes by. This can lead to some confusion and ambiguity when different definitions are evoked. Here, we wish to explore the meaning of this term attributed to unstable states in quantum mechanics. 

As the name suggests, shape-type resonances result from the shape of the potential at hand. But, what attributes must a potential have in order to trap the particle for a finite time and thus form a metastable state The wave nature of particles in quantum mechanics provides two typical ways for a... 

N. Hatano, K. Sasada, H. Nakamura, T. Petrosky, Some properties of the resonant state in quantum mechanics and its computation, Prog. Theo. Phys. 119 (2008) 187. 

The determination of accurate molecular structure from molecular rotational resonance (MRR) spectra has always been a great challenge to this branch of spectroscopy [/]. There are three basic facts which make this task feasible (1) the free rotation of a rigid body is described in classical as well as in quantum mechanics by only three parameters, the principal inertial moments of the body, Ig, g = x, v, z ... 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

Resonances in reactive collisions were first observed in quantum mechanical scattering calculations for the colllnear H + H2 reaction (1-9 for a review of early calculations on this system see reference 22. recent review of the quantum mechanical... 

Despite the fact that we are not required to introduce it, the concept of resonance in classical mechanical systems has been found to be very useful in the description of the motion of systems which are for some reason or other conveniently described as containing interacting harmonic oscillators. It is found that a similar state of affairs exists in quantum mechanics. Quantum-mechanical systems which are conveniently considered to show resonance occur much more often, however, than resonating classical systems, and the resonance phenomenon has come to play an especially important part in the applications of quantum mechanics to chemistry. 

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