The Particle in a Box

The Schrodinger wave equation, Hip = Eip, lies at the heart of the quantum mechanical description of atoms. Recall from the preceding discussion that H represents an operator (the Hamiltonian) that extracts the total energy E (the sum of the potential and kinetic energies) from the wave function. The wave function ip depends on the x, y, and z coordinates of the electron s position in space. 

Although the detailed solution of the Schrodinger equation for the hydrogen atom is not appropriate in this text, we will illustrate some of the properties of wave mechanics and wave functions by using the wave equation to describe a very simple, hypothetical system commonly called the particle in a box, a situation in which a particle is trapped in a one-dimensional box that has infinitely high sides. It is important to recognize that this situation 

Consider a particle with mass m that is free to move back and forth along one dimension (we arbitrarily choose x) between the values x = 0 and x = L (that is, we are considering a one-dimensional box of size L meters). We will assume that the potential energy V(x) of the particle is zero at all points along its path, except at the endpoints x = 0 and x = L, where V(x) is infinitely large. In effect, we have a repulsive barrier of infinite strength at each end of the box. Thus the particle is trapped in a one-dimensional box with impenetrable walls (see Fig. 12.13). 

As we mentioned before, the Schrodinger equation contains the energy operator H. In this case, since the potential energy is zero inside the box, the only energy possible is the kinetic energy of the particle as it moves back and forth along the x axis. The operator for this kinetic energy is 

A schematic repiesentation of a particle in a one-dimensional box with infinitely high potential walls. 

The total probability of an electron being somewhere in space = 1. This is called normalizing the wave function.  

Sine and cosine functions have the properties we associate with waves— well-defined wavelength and amplitude—and we may therefore propose that the wave characteristics of our particle may be described by a combination of sine and cosine functions. A general solution to describe the possible waves in the box would then be 

Because P must be continuous and must equal zero at x 0 and x a (because the particle is confined to the box), must go to zero at x = 0 and x = a. Because cos ix = 1 for X = 0, P can equal zero in the general solution above only if B = 0. This reduces the expression for to 

At X = a, must also equal zero therefore, sin ra = 0, which is possible only if ra is an integral multiple of tt  

These are the energy levels predicted by the particle-in-a-box model for any particle in a one-dimensional box of length a. The energy levels are quantized according to quantum numbers n = 1,2,3,. .. 

Recalling that p = mv and assuming that the electron mass is constant (ignoring any relativistic corrections), we have 

if we know the electron s position with a minimum uncertainty of 5 X 10 m, the uncertainty in the electron s velocity is at least 1 X 10 m/s. This is a very large number in fact, it is the same magnitude as the speed of light (3 X 10 m/s). At this level of uncertainty we have virtually no idea of the velocity of the electron. 

This means there is a very small (undetectable) uncertainty in our measurements of the speed of a ball. Note that this uncertainty is not caused by the limitations of measuring instruments At/ is an inherent uncertainty. 

Thus the uncertainty principle is negligible in the world of macroscopic objects but is very important for objects with small masses, such as the electron. 

FIGURE 2-4 Wave Functions and Their Squares for the Particle in a Box with n = 1, 2, and 3. 

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I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

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... 

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

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. 

Most students are introduced to quantum mechanics with the study of the famous problem of the particle in a box. While this problem is introduced primarily for pedagogical reasons, it has nevertheless some important applications. In particular, it is the basis for the derivation of the translational partition function for a gas (Section 10.8.1) and is employed as a model for certain problems in solid-state physics. 

Fig. 4 Wavefunctions for the particle in a box (a) without symmetry considerations (b) (be symmetric box.

Although the variational theorem was expressed in Eq. (111) with respfect to the ground state of the system, it is possible to apply it to higher, so-called excited, states. As an example, consider again the particle in a box. In Section 5.4.2 a change in coordinate was made in order to apply symmetry considerations. Thus, fire DOtential function was written as... 

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... 

Although the net results obtained above for the particle in a box are physically the same, the mathematieal consequences are quite different. From Fig. 4b it can be seen that the wavefunction is either even or odd, depending on the parity of n. Specifically, ifrtt(x) = iM— ) where the plus sign is appropriate when n is odd and the minus sign when n is even. As Eq. (70) contains the sine of the sum of two terms, it can be rewritten in die form... 

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-... 

The theory assumes that the nuclei stay fixed on their lattice sites surrounded by the inner or core electrons whilst the outer or valence electrons travel freely through the solid. If we ignore the cores then the quantum mechanical description of the outer electrons becomes very simple. Taking just one of these electrons the problem becomes the well-known one of the particle in a box. We start by considering an electron in a one-dimensional solid. 

The underlying assumption in several of these cards was that a numerical example would overcome all of the troubles the student was having. The critical feature in comments that reflected the first of the students expectations was that examples were only valid if they were numerical. When instructors presented non-numerical examples, the students generally viewed the information as more theory. Consider Craig s reaction to the question of whether discussion of an example based on the particle in a box was what he wanted to see. 

I . .. people kept saying Oh, I want to see examples. We want to [see] examples. And finally he showed you an example of the particle-in-a-box. And. .. did you like that example of a particle in a box Doesn t that make you happy as an example ... 

Craig made several interesting points in this quote from an interview at the end of the semester. The first was that the particle-in-a-box was not a good example because it wasn t useful as Craig pointed out, it is unlikely that he would ever deal with the particular case of a one-dimensional particle in a square well. Second, and even more intriguing, Craig commented that although he understood that the particle-in-a-box was an example, it was not the one he needed to see. 

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