Particle-in-a-box solution

These molecular-orbital functions are particle-in-a-box solutions, and not surprisingly, the molecular-orbital states satisfy the following condition under the operation of the inversion operator i,... 

Without performing an HMO calculation, sketch the MOs for the pentadienyl radical. Use the particle-in-a-box solutions and the pairing theorem as a guide. 

All solutions of the Schroedinger equation lead to a set of integers called quantum numbers. In the case of the particle in a box, the quantum numbers are n= 1,2,3,. The allowed (quantized) energies are related to the quantum numbers by the equation... 

To begin a more general approach to molecular orbital theory, we shall describe a variational solution of the prototypical problem found in most elementary physical chemistry textbooks the ground-state energy of a particle in a box (McQuanie, 1983) The particle in a one-dimensional box has an exact solution... 

Lx, Lr, and L , so that the volume of the box is LXLVL . The potential energy of the particle inside the box is zero but goes to infinity at the walls. The quantum mechanical solution in the. v direction to this particle in a box problem gives... 

The time-dependent Schrddinger equation (2.30) for the particle in a box has an infinite set of solutions tpn(x) given by equation (2.40). The first four wave functions tpn(x) for = 1, 2, 3, and 4 and their corresponding probability densities ip (x) are shown in Figure 2.2. The wave function ipiix) corresponding to the lowest energy level Ei is called the ground state. The other wave functions are called excited states. 

In the above treatment of the problem of the particle in a box, no consideration was given to its natural symmetry. As the potential function is symmetric with respect to the center of the box, it is intuitively obvious that this position should be chosen as the origin of the abscissa. In Fig. 4b, x =s 0 at the center of the box and the walls are symmetrically placed at x = 1/2. Clearly, the analysis must in this case lead to the same result as above, because the particle does not know what coordinate system has been chosen It is sufficient to replace x by x +1/2 in the solution given by Eq. (68). This operation is a simple translation of the abscissa, as explained in Section 1.2. The result is shown in Fig. 4b, where the wave function is now given by... 

The one-dimensional problem of the particle in a box was treateduxSectiQn S.4.1. Exact solutions were obtained, which were then restricted by the boundary conditions (0) — ty t) = 0. If the exact solutions were not known, the problem... 

The only solution for Equation (3.5) when V = 0 is ip = 0. So that if ip is to be singlevalued and continuous, it must be zero at the walls, that is, at x = 0 and x = I. Thus the potential energy walls impose what are called boundary conditions on the form of the wave function. Figure 3.3 shows (a) the particle-in-a-box potential, (b) a wave function, that satisfies the boundary conditions and, (c) one that does not. We see that only certain wave func-... 

Wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron around a nucleus and to take into account the wave nature of the electron. The mathematical expressions for atomic orbitals—which are obtained as solutions of the Schrodinger equation—are more complicated than the sine functions for the particle in a box, but their essential features are quite simple. Moreover, we must never lose sight of their interpretation, that the square of a wavefunction tells us the probability density of an electron at each point. To visualize this probability density, we can think of a cloud centered on the nucleus. The density of the cloud at each point represents the probability of finding an electron there. Denser regions of the cloud therefore represent locations where the electron is more likely to be found. 

We shall need to know how to evaluate these separated partition functions. The translational energy levels can be derived from the quantum mechanical solution for a particle in a box they are so closely spaced that the partition function can be evaluated in closed form by integration, and has the value... 

Inside the box, the general solution is just the same as that given in the previous section, eqn 2.31. Outside the box, the potential is infinity, and the only sensible value of iff is zero otherwise, it would immediately go to infinity, which we assume to be impossible. We make a further assumption, that iff must be continuous, i.e. it cannot suddenly jump from one value to another. We therefore have the following boundary conditions for the particle-in-a-box problem ... 

Equation 3.48 is of course the same equation as we have solved before, e.g. for the particle in a box. Its solutions are simple sine and cosine functions of angular variable, which repeats itself every 2n radians. The boundary conditions for the wavefunction are therefore different from those for the particle in a box. There is no requirement that iff must be zero anywhere instead, it must be single valued, which means for any 0,... 

The particle-in-a-box problem, which we considered qualitatively in Chapter 5, turns out to be one of the very few cases in which Schrodinger s equation can be exactly solved. For almost all realistic atomic and molecular potentials, chemists and physicists have to rely on approximate solutions of Equation 6.8 generated by complex computer programs. The known exact solutions are extremely valuable because of the insight... 

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