Eigenfunctions and Energy Levels

A nuclear spin system may be described by a steady-state wave function arising from the Hamiltonian operator satisfying the time-independent Schrodinger equation 

E gives the energy of the system, which turns out to be quantized into discrete energy levels. 

The quantum mechanical computations here are extremely simple and furnish an excellent example of such calculations. The simplicity arises from the facts that 

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Up to now, we have seen that many of the optical properties of active centers can be understood just by considering the optical ion and its local surrounding. However, even in such an approximation, the calculation of electronic energy levels and eigenfunctions is far from a simple task for the majority of centers. The calculation of transition rates and band intensities is even more complicated. Thus, in order to interpret the optical spectra of ions in crystals, a simple strategy becomes necessary. 

Solution of the secular equation for benzene gives the following energy levels and eigenfunctions ... 

Having accurate semi-empirical values of the energy levels and eigenfunctions one is in a position to calculate wavelengths and oscillator strengths of allowed and forbidden electronic transitions in the inter-... 

The energy levels and eigenfunctions, obtained in one or other semi-empirical approach, may be successfully used further on to find fairly accurate values of the oscillator strengths, electron transition probabilities, lifetimes of excited states, etc., of atoms and ions [18, 141-144]. 

Finding energy levels and eigenfunctions for the general three-dimensional mixed crystal is a formidable problem, not yet solved in closed form except in special limiting situations. In other cases numerical solutions are available,6 but analytical approximations are desirable and only simplified model systems can at present provide them. 

The probability of proton tunnelling across the barrier of a double-well potential strongly depends on its symmetry [74, 75]. Figure 9.6 shows a symmetrical and an asymmetrical double-well potentials, their energy levels and eigenfunctions. In the case of the symmetrical PES profile (degenerate systems). 

The final step is to use the B s (taking only a set with the same L, S, Ml, and Ms) in equation 9 21 to determine the approximate energy levels and eigenfunctions of H. Before going into this last step in detail, we shall note some of the properties of the B functions. 

Since our spin functions are eigenfunctions of S2, we can drop the last term in Eq. (70), because it contributes a constant term to all energy levels and hence drops out when we compute energy differences. Many workers prefer to add a term — 5(5+ 1) to the spin Hamiltonian to get a Hamiltonian which transforms readily under a coordinate rotation. We thus have forJt ... 

It is very important for semi-empirical calculations to have experimental data measured accurately and identified correctly. Otherwise, it is impossible to achieve good correspondence between calculated and measured quantities. To overcome these dangers one has to analyse the structure of the eigenfunctions, energy levels and oscillator strengths of electronic transitions along the isoelectronic sequences. As a rule, all these quantities vary fairly smoothly and, therefore, the occurrence of any inconsistency in their behaviour usually indicates that there are uncertainities or errors in the data utilized. 

For the particularly important case of / = l/2 there are just two energy levels and two wave functions, which are called a and j3. It will be worthwhile to explore the properties of the functions a and j8 in more detail, a and /3 are orthogonal and normalized eigenfunctions of the quantum mechanical operators Iz and I2 but not of 4 and Iy. Specifically, as shown in standard quantum mechanics texts, the effects of these operators may be expressed as... 

In either case the eigenvalues of the resulting hamlltonian matrix give the energy levels, and the eigenfunctions give the wavefunctions In terms of the appropriate basis functions. 

The size of the phonon is where the sound wave is, for example, a guitar string. The eigenfunctions are tones (or notes). A sound or a noise is a wave packet that can be written as a superposition of tones, called a Fourier series, involving different frequencies. The phonon is created as an excitation among the quantized energy levels and disappears in a deexcitation. Phonons play the major role in heat conduction, and the role of resistance in the case of electric conduction as we have previously seen. 

Figure 2-4 The eigenfunctions corresponding to = 1, 2, 3, plotted on the corresponding energy levels. The energy units of the ordinate do not refer to the wavefunctions Each wavefunction has a zero value wherever it intersects its own energy level, and a maximum value of...

The L and >S values of a B function (or an energy level whose eigenfunction of H is a combination of B functions) is denoted by a capital letter showing the L value, the letter being chosen from the same code which is used in denoting the I value of an atomic orbital. The value of 2>S 1, the multiplicity of the term or group of levels with the same L... 

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