The Particle in a Box Problem

Ultimately, our goal is to solve the Schrodinger equation in three-dimensions for the electron in the hydrogen atom. This electron is subject to a potential energy term that involves a Coulombic attraction toward the nucleus. However, the solution to this differential equation is not a trivial one. We therefore choose a somewhat similar, but simpler, problem—the particle in a box—to demonstrate the procedure and to illustrate some of the principles of quantum mechanics. We then extrapolate those results to the hydrogen atom in a later chapter. 

The wave function is zero everywhere outside the box. Inside the box, the time-independent Schrodinger equation in one-dimension reduces to Equation (3.57), which has the trigonometric solution given by Equation (3.53), where p = 

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). 

Example 3-10. Use the normalization procedure discussed in the previous section to prove that the value of 6 in Equation (3.60) is (2/ j).  

Solution. In the process of normalization, the integral of i//p dr over all space is determined to be N. By substitution of Equation (3.60) for i//, the following integral must be evaluated  

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

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 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... 

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. 

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). 

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

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... 

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.]... 

The first formula says that Vi(x) has a constant value in [O, L] (this eorresponds to the particle-in-a-box problem). The bottom of this box is at energy — beeause only then the ground state energy will equal 0 (as it was assumed). Its SUSY partner eorresponds to V2(x) = [2ctg x + l]. As it follows from Eq. (4.71), the wave functions corresponding... 

SQE is modeled similarly to the particle in a box problem , in which the smaller the box , the larger is the lower energy eigen value. The correlation for energy states between bulk material and corresponding size-quantized particle is schematically depicted in Figure 9.7. 

In the particle-in-a-box problem, the wavefunction must be zero-valued at the edges and outside the confines of the box. Thus, the allowed solutions within the box. Equation 8.20, are acceptable as wavefunctions only if they become zero-valued at the edges of the box. This is expressed as boundary conditions ... 

Hypothetical step potential for a one-dimensional particle. To understand this system, it is helpful to break it into regions. In the first region (/), there is a potential well that is almost like the potential of the particle-in-a-box problem. The second region (II) has a constant but not infinite potential. The third region (HI) has a flat potential, and this continues to infinity. 

Also, as in the particle-in-a-box problem, the wavefimction must vanish at x = c because the potential is infinite at that point ... 

If the lowest energy state of the particle-in-a-box problem shown in Figure 8.10 were at an energy slightly greater than Vg, in what ways would the wavefunctions of the lowest and first excited states differ from the wavefunctions of the same system but in the absence of the barrier step potential Use a sketch to show qualitative features. 

Find the first-order energy corrections for the first four states of the particle-in-a-box problem if there is a perturbing potential of the form V (x) = O.l-sJl/I sin(jrx//). 

The solution to the particle-in-a-box problem connects the wavenumber kx to a quantum number through the equation... 

This exercise requires calculus.) In this exercise, use ideas from this chapter to develop the solution to the particle-in-a-box problem. We begin by writing the Schrodinger equation for a particle of mass m moving in one dimension ... 

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