Waves quantum-mechanical tunneling

A model which takes into account the spin-rotation interaction has been found to satisfactorily explain the 0 rotation band of PHg. The millimetre-wave spectra of HCP and DCP have been compared with those of HCN and DCN. A method of estimating frequencies of bands in this region due to processes such as pseudorotation has been suggested. This new approach involves calculation of the rovibronic energy levels from the effects of quantum-mechanical tunnelling. ... 

Electron-transfer proteins have a mechanism that is quite different from the conduction of electrons through a metal electrode or wire. Whereas the metal uses a continuous conduction band for transferring electrons to the centre of catalysis, proteins employ a series of discrete electron-transferring centres, separated by distances of I.0-I.5nm. It has been shown that electrons can transfer rapidly over such distances from one centre to another, within proteins (Page et al. 1999). This is sometimes described as quantum-mechanical tunnelling, a process that depends on the overlap of wave functions for the two centres. Because electrons can tunnel out of proteins over these distances, a fairly thick insulating layer of protein is required, to prevent unwanted reduction of other cellular components. This is apparently the reason that the active sites of the hydrogenases are hidden away from the surface. 

Quantum mechanical tunneling. Tunneling is the phenomenon by which a particle transfers through a reaction barrier due to its wave-like property.Figure 1 graphically illustrates this for a carbon-hydrogen-carbon double-well system Hydrogen... 

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. 

Returning to the one-dimensional box of constant width, if the potential does not increase suddenly to infinity at one of the walls then the wave-function does not vanish there. Due to the continuity requirement of the wavefunction, it decreases exponentially to zero inside the wall of finite height. Therefore, there is a non-zero probability that the particle will penetrate the wall, although its kinetic energy is lower than the potential barrier (Fig. 2.6). This effect is called quantum-mechanical tunnelling. 

Quantum mechanical tunneling is a result of the finite potential barrier at the metal-vacuum interface. The electronic wave function ip and its first derivative dip/dz are continuous across this interface (or finite potential discontinuity), the electron wave function decaying exponentially, e in the forbidden region where the barrier exceeds the total electron energy. In this context, k is approximately related to the apparent work function or mean local tunneling barrier, = 2m((p — E)/ h/2nf-. Thus, the tunneling current. It, or transmission probability also decays with barrier width, z [6.10-6.13]... 

Fig. 6.2. Electron wave functions for two metal electrodes separated by a thin insulating layer showing quantum mechanical tunneling with exponential decay of the wave function in the barrier (a) the wave function of electrode 1 extending smoothly across the entire tunnel junction structure (b) separate wave functions for electrodes 1 and 2, xjj and Jp2 respectively [modified from reference 6.10] The wave functions have been offset from the Fermi level for clarity.

Experimental and theoretical interest in USCSs has existed since the early days of quantum mechanics. For example, a textbook picture of such an unstable state is that of the one-dimensional potential with a local minimum and a finite barrier that is used to explain, in terms of quantum mechanical tunneling, the instability of a nucleus, the concomitant emission of an alpha particle, and ifs energy. Another textbook example of basic importance is the formal construction of a wave packet from a superposition of a complete set of stationary states and the determination, at least for simple one-dimensional motion, of its time evolution. Finally, another example often presented in books is the appearance of structures ("peaks") in the energy-dependent transition rates (cross sections) over the smoothly varying continuum characterizing a physico-chemical process, which are normally called resonances and which are associated with the transient formation of USCSs. 

The first quantum-mechanical consideration of ET is due to Levich and Dogonadze [7]. According to their theory, the ET system consists of two electronic states, that is, electron donor and acceptor, and the two states are coupled by the electron exchange matrix element, V, determined in the simplest case by the overlap between the electronic wave functions localized on different redox sites. Electron transfer occurs by quantum mechanical tunneling but this tunneling requires suitable bath fluctuations that bring reactant and product energy levels into resonance. In other words, ET has... 

The physical significance of quantum-mechanical tunneling was recognized very early In the development of wave mechanics, and there are many examples of physical phenomena In which tunneling Is Important. Here Is a very Incomplete list of examples, chosen principally on the basis of historical Interest ... 

Quantum Mechanical Tunneling If the material of interest is an insulator, classical physics predicts that electrons incident on the metal/insulator interface will be reflected if the energy of an electron is less than the interfacial potential barrier of the interface. The electron cannot penetrate the barrier, and its passage from one electrode to the other is inhibited. However, quantum mechanics predicts that an electron will overcome this barrier due to the wave nature of the electrons. The electron wavefunction decays rapidly with depth of penetration from the electrode/insulator interface, and for barriers of microscopic thickness, the wavefunction is essentially zero at the opposite interface [18]. This means that the probability of finding the electron is essentially zero at the other electrode. However, if the barrier is very thin (<5 nm), the wavefunction has a nonzero value at the other electrode, allowing... 

Quantum-mechanical tunneling For barrier thickness of 1-4 nm, the wave-nature of the electrons enables penetration through thin barriers. This is especially true for heavily doped semiconductors. 

Resonance charge transfer describes the transfer of electrons between electronic states of the same energy, one of which must be vacant so as to accept an electron. This transfer will occur via quantum mechanical tunneling of electrons between atoms/ions making up the substrate surface and the atom/ion in close vicinity ( 1 nm) to the surface. As the wave function of electronic states closest to the Fermi edge Ef) extend out the greatest distance from the respective atom or ion s nucleus, only these levels are considered within this process (these are the first and last to come into contact with each other). 

Tunneling may be conceptualized in several ways. For example, the wave-particle duality of quantum mechanics indicates that particles of mass m and velocity v have an associated de Broglie wavelength given in Eq. 10.6, where h is Planck s constant."... 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

All this material is described in introductory textbooks of physics and chemistry. However, it is interesting to recall the headlines here because the veiy first application to a chemical theme of the ideas of waves in quantum mechanics was to explain how electrons were emitted from, or accepted by, electrodes. This was the achievement of Ronald Gurney,1 the first physical electrochemist, and much of this chapter is based on developments that sprang from his seminal paper of 1931. In this paper, he related electric currents across the electrode solution interface to the tunneling of electrons through energy barriers formed between the electrode and the ions or molecules in the first layer next to the electrode (possessing electronic states ). 

The last line, Table 5.1, reports the purely classical moments. The zeroth classical moment is a little smaller than the zeroth quantum moment, because of the wave mechanical tunneling of the collisional pair into the classically forbidden region which enhances the intensities. All odd moments of classical profiles are, of course, zero. The second and fourth moments are significantly smaller than the quantum moments, because... 

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