Wave functions and energy levels

The quantum description and space of phases 4.3.1. Wave functions and energy levels 

Now let us take the example where our collection of particles requires, in order to be described, the use of quantum mechanics and a stationary state that satisfies Schrodinger s equation of the system. We know that to solve this equation, we must determine the energy values E, called eigenvalues, that lead to acceptable wave functions which are the solution to the previous equation, called eigenfunctions. In fact, the energy values found can be quite close and numerous with the need to consider an energy interval 

1 The term acceptable refers to the fact that wave functions must have certain properties to be hermitic, of an integrable square, etc. 

Let us cortsider a collection of N iderrtical qrrantum objects each with three degrees of freedom (e.g. one punctual gas molectrle). To define the movemerrt of these objects, we must first define the three spatial coordinates and the three qrrantities of movement (or three velocity componerrls), i.e. a total of six coordinates in a space with six dimensions. In this space, called the quarrtum space, an object is defined by a poirrt. For N objects, we rrse an hyperspace with 6N dimensions, called the Gibbs space of phase. In this hyperspace, it is a system state which is represented by a poirrt Each ensemble defined constitutes a complexion of the system. 

Of course, if the objects concerned are not quantirm objects, Heisenberg s principle no longer applies and the evolutions can be followed from one state to the next the objects are supposedly discernible. We will find an example of this in our study of a canonical ensemble. 

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Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

Wood, J. H., and Pratt, G. W., Phys. Rev. 107, 995, "Wave functions and energy levels for Fe as found by the unrestricted Hartree-Fock method."... 

Born-Oppenheimer approximation (physchem) The approximation, used in the Born-Oppenheimer method, that the electronic wave functions and energy levels at any instant depend only on the positions of the nuclei at that instant and not on the motions of the nuclei. Also known as adiabatic approximation. born ap an.hT-mar 3,prak s3,ma shan J... 

Fig. 2. Wave functions and energy levels for the solvated electron in (a) methylamine (MeA) and (b) hexamethylphosphoramide (HMPA). The potential V(r) and wavefunction are based upon the model of Jortner (101) and computed using values of the optical and static dielectric constants of the two solvents. The optical absorption responsible for the characteristic blue color is marked by h v and represents transitions between the Is and 2p states. The radius of the cavity is 3 A in MeA, and —4.5 A in HMPA. 

Fassaert et al. (68) simulated H adsorption on a Cu surface by adding an additional electron per metal atom to the system. This approximation relies on the fact that atomic wave functions and energy levels are not too different for Ni and Cu and that their principal difference lies in the number of valence electrons. In the case of adsorption to Cu substrate, which has no unfilled d orbitals, the metal d orbitals do not participate in the bonding to H. All bonding takes place using the metal 4s orbitals. The calculated covalent bond energy is comparable on the Ni and Cu substrate models, so that from the results a distinction between the catalytic properties of the two metals cannot be made. 

Figure 4.2 Wave functions and energy levels for a particle in a harmonic potential well. The outline of the potential energy is indicated by shading.

Figure 4.19. Wave functions and energy levels of a perfect biradical (center), constructed from the most localized orbitals x nnd Xi> tind from the most delocalized orbitals < > and (right) (adapted from BonaCiC-Kouteck et al., 1987).

The helium ion He is a one-electron system whose wave functions and energy levels are obtained from those for H by changing the atomic number to Z = 2. Calculate the average distance of the electron from the nucleus in the 2s orbital and in the 2p orbital. Compare your results with those in Problem 9 and explain the difference. 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

The stationary-state wave functions and energy levels of a one-particle, one-dimensional system are found by solving the time-independent Schrodinger equation (1.19). In this chapter, we sdlve the time-independent Schrodinger equation for a very simple system, a particle in a one-dimensional box (Section 2.2). Because the Schrodinger equation is a differential equation, we first review the mathematics of differential equations (Section 2.1). 

For the particle in a one-dimensional box of length /, we could have put the coordinate origin at the center of the box. Find the wave functions and energy levels for this choice of... 

The stationary-state wave functions and energy levels of a particle in a three-dimensional rectangular box were found by the use of separation of variables. 

The approximate analytical solutions for wave functions and energy levels had shown the following. P-type contribution is dominant at the defect distance zo from the surface 0 10 ag. Here is effective Bohr radius that depends strongly on effective mass p, and dielectric permittivity 82 S2/p- (see Fig. 4.13). It is seen from the Fig. 4.13... 

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