Particle-in-a-box problem

Applied to each of the independent coordinates of the two-dimensional particle in a box problem, this expression reads ... 

The particle-in-a-box problem provides an important model for several relevant chemical situations... 

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... 

Here, the momentum vector p contains the momenta along all coordinates of the system, and the coordinate vector q likewise contains the coordinates along all such degrees of freedom. For example, in the two-dimensional particle in a box problem considered above, q = (x, y) has two components as does p = (Px, py), and the action integral is ... 

The one-dimensional particle-in-a-box problem is that of a single particle subject to the following potential-energy function ... 

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 ... 

The three-dimensional particle-in-a-box problem is an obvious extension from two dimensions. For a cubic box with sides of length a, the allowed wavefunctions satisfying the boundary conditions are... 

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... 

Figure 1.5.1 summarizes the results of this particle-in-a-box problem. From this figure, it is seen that when the electron is in the ground state (n= 1), it is most likely found at the center of the box. On the other hand, if the electron is in the first excited state (n = 2), it is most likely found around x = a A or x = 3a/4. 

This model is popularly known as the particle-in-a-box problem. 

We can diagram the solutions to the particle-in-a-box problem conveniently by showing a plot of the wave function that corresponds to each energy level. The energy level, wave function, and probability distribution are shown in Fig. 12.14 for the first three levels. 

Consider a variant of the one-dimensional particle-in-a-box problem in which the x-axis is bent into a ring of radius R. We can write the same Schrbdinger equation... 

The exact wavefunction leads to the minimum energy and the more an approximate wavefunction approaches the exact one the closer the corresponding expectation value becomes to the true energy (for a recent application of the variation principle to the particle in a box problem, see ref. 81). 

Particle-in-a-box problems The energies of a particle in a one-dimensional box are given by E = and the normalized wave functions by... 

Before going on to something as complex as an atom, let s look at a model problem in some detail. The first one is the one-dimensional particle-in-a-box problem. This turns out to be an excellent conceptual model for conjugated dye molecules (see Chapter 21) and also a model for trapped charged particles. The problem and its solutions are similar to the vibrating string just discussed. The potential term is shown graphically and mathematically in Fig. 7.1. 

The confinement of electrons or holes in potential wells leads to the creation of discrete energy levels in the wells, compared with the continuum of states in hulk material quantisation also leads to a major change in the density of states. The energy levels can be calculated by solving the Schrodinger equation for the well-known particle in a box problem. Using the effective mass envelope function approximation (Bastard, 1981 and 1982 Altarelli, 1985 Bastard and Brum, 1986), the electron wavefunction % is then... 

Fig. 3.1. Ground state and first excited state wave functions associated with particle in a box problem. The wave functions are drawn such that their zeroes correspond to the energy of the corresponding state.

As yet, our quick tour of quantum mechanics has featured the key ideas needed to examine the properties of systems involving only a single particle. However, if we are to generalize to the case in which we are asked to examine the quantum mechanics of more than one particle at a time, there is an additional idea that must supplement those introduced above, namely, the Pauli exclusion principle. This principle is at the heart of the regularities present in the periodic table. Though there are a number of different ways of stating the exclusion principle, we state it in words as the edict that no two particles may occupy the same quantum state. This principle applies to the subclass of particles known as fermions and characterized by half-integer spin. In the context of our one-dimensional particle in a box problem presented above, what the Pauli principle tells us is that if we wish to... 

Fig. 3.4. Schematic illustration of the nodes used to discretize the particle in a box problem and the corresponding finite element shape functions.

From the standpoint of the Schrodinger equation, this is another example of a particle in a box problem which in this case is effectively two-dimensional. In particular, the potential within the corral region is assumed to be zero, while the... 

Fig. 2.5. A schematic diagram showing the formulation of the energy structure of butadiene as a particle in a box problem. Only the pi electrons are shown. In the free pi-electron model (which is a simplification) each state can be occupied by two carriers, each with opposite spin.

We solved an equation of this form in Section 8.5, for two alternative sets of boundary conditions. The preceding solution of the Helmholtz equation in Cartesian coordinates is applicable to the Schrddinger equation for the quantum-mechanical particle-in-a-box problem. 

There is no such a thing in nature as infinitely steep potential energy walls or infinite values of the potential energy (as in the particle-in-a-box problem). This means we should treat such idealized cases as limit cases of possible continuous potential energy functions. From the Schrddinger equation = Ei/r — V r, we see... 

After substitution for A and / into Equation (3.53), the acceptable solutions to the particle in a box problem are given in Equation (3.60). 

The acceptable solutions to the one-dimensional particle in a box problem are sketched in Figure 3.27(a) for the first several quantum numbers. The Born interpretation of the wave function states that the product y/ i// represents the probability density of finding the electron in a finite region of space. Because the Born interpretation of the wave function is this function is shown in Figure 3.27(b). 

The first of several solutions to the particle in a box problem showing (a) vr(x) and (b) v (x), along with their corresponding energies. [Copyright University Science Books, Mill Valley, CA. Used with permission. All rights reserved. McQuarrie, D. A. Simon, J. D. Physical Chemistry A Molecular Approach, 1997.]... 

Example 3-12. Using Postulate 4, show that the average position of the electron in the one-dimensional particle in a box problem lies exactly in the center of the box. 

The first several energy levels for the three-dimensional particle in a box problem (where a = b = c) are shown in Figure 3.29, where the energy axis has units of h /8 ma. Many of the energy levels are degenerate—having more than one acceptable set of quantum numbers at the same energy. For example, there are three ways to have the sum = 6 n, = 2, I, I 1,2, I and 1, 1, 2. As a... 

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