Quantum mechanics reduced mass

The wave functions for a particle in a box illustrate another important principle of quantum mechanics the correspondence principle. We have already stated earlier (and will often repeat) that all successful physical theories must reproduce the explanations and predictions of the theories that preceded them on the length and mass scales for which they were developed. Figure 4.25 shows the probability density for the n = 5, 10, and 20 states of the particle in a box. Notice how the probability becomes essentially uniform across the box, and at m = 20 there is little evidence of quantization. The correspondence principle requires that the results of quantum mechanics reduce to those of classical mechanics for large values of the quantum numbers, in this case, n. 

Consider an excited condensed-phase quantum oscillator Q, witli reduced mass p and nonnal coordinate q j. The batli exerts fluctuating forces on the oscillator. These fluctuating forces induce VER. The quantum mechanical Hamiltonian is [M, M]... 

Using Jacobi coordinates and reduced masses, the Hydrogen-Chlorine interaction is modeled quantum mechanically whereas the Ar-HCl interaction classically. The potentials used, initial data and additional computational parameters are listed in detail in [16]. 

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

The reason that a compound ion can be field dissociated can be easily understood from a potential energy diagram as shown in Fig. 2.23. When r is in the same direction as F, the potential energy curve with respect to the center of mass, V(rn) is reduced by the field. Thus the potential barrier width is now finite, and the vibrating particles can dissociate from one another by quantum mechanical tunneling effect. Rigorously speaking, it... 

That is, the simple classical picture we described applies to the quantum mechanical case provided the masses of solitons and breathers are changed. This remarkably simple result is due to the special characteristics of the sine-Gordon equation. The quantization reduces to factorization of the action to the classical action and a constant factor that is independent of the soliton velocity and the breather frequency. 

Structure Whereas the remaining e stays in the Is ground orbital, the captured p occupies a large-(n, l) state n no = JM /rne 38, where M is the reduced mass of the p-He system. The angular momentum l which is brought by the captured p can be as much as that of the circular state, l n — 1. As shown in Fig. 1, the p is orbiting in a classical trajectory, while the e is distributed quantum mechanically. 

In the non-relativistic quantum mechanics the nuclear recoil effect for a hydrogenlike atom is easily taken into account by using the reduced mass p = mM/(m + M) instead of the electron mass m (M is the nuclear mass). It means that to account for the nuclear recoil effect to first order in m/M we must simply replace the binding energy E by E(1 — m/M). 

Ia) is included in the electronic Hamiltonian since, as we shall see, its most important effects arise from interactions involving electronic motions. The interactions which arise from electron spin, 30(5 ), will be derived later from relativistic quantum mechanics for the moment electron spin is introduced in a purely phenomenological manner. The electron-electron and electron-nuclear potential energies are included in equation (2.36) and the purely nuclear electrostatic repulsion is in equation (2.37). The double prime superscripts have been dropped for the sake of simplicity. We remind ourselves that // in equation (2.37) is the reduced nuclear mass, M M2/(M + M2). 

Positronium, reduced mass, 109 Postulates, quantum mechanics, 51-52 Propanol, infrared spectroscopy, 222-223... 

With both types of vibrational spectroscopy, distinctive spectra and facility in interpretation are possible because only vibrational transitions corresponding to changes in the vibrational quantum number of+1 are allowed by the spectral selection rules. That is, An = 1, where n is the vibrational quantum number. Due to this, the frequencies observed are usually the fundamental frequencies. In addition, because of analogies between the mathematical descriptions of classical and quantum mechanical vibrating molecular systems, it is possible to rationalize many spectral observations by analogy with classical vibrating systems that possess characteristic force constants and reduced masses. This rationalization has become the basis for systematizing much of the structural and chemical information derived from vibrational spectra. 

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