The Particle in a Box Revisited 6. D. El-lssa University of Kuwait, P.O. Box 5969, Kuwait
We propose in this article to introduce some of the important concepts of quantum chemistry by dwelling on the "particle-in-a-box" model. We hope that the concepts introduced in this article will act as a basis for the introduction of an experiment in the basic physical chemistry laboratory courses. I t will probably make it easier for the student to understand orthogonality relationships, expansion of an orbital in a complete set of basis functions, perturbation theory, and transition probabilities with some attention to selection rules. The computer programs were developed in our laboratory on the HP-9845 desk-top calculator. We shall assume throughout this article that we are dealing with a particle whose mass, m, is 9.11 X 10-3' kg. The length of the box will he assumed to be 1.00 nm. As usual, an infinite potential is invoked on the two extremities of the box, i.e., x = 0 and x = 1. The set of normalized eigenfunctions satisfying the basic axioms of quantum theory are $, =
flsin(nsr1
(1)
Figure 1. The shapes of the first six exact eigenfunctionsof the particle-in-abox model. (The fourth wave function is actuailv shown in Fia. 21. The number
where n = 1,2,3,. . . and the superscript on $, indicates that the solutions are exact. This suggests that the Hamiltonian inside the box is exactly known. In fact, the only contrihution to the Hamiltonian will, a t this stage, be assumed to be the kinetic energy and is therefore given by
where h i s Planck's constant. This set of functions is the set of eigenfunctions satisfying the eigenvalue equation HO*Z = Ei*,
(3)
where E, represents the exact energy of the nth level. The eigenfunctions of the above equation form a complete set of orthonormal basis functions since (*;I*;)
Figure 2. The shape of the fourth exact eigenfunctionof the particle-in-a-box model. The number of nodes is equal to 3.
= 8"m
where the Dirac notation for the above integral was used and where the Kroenecker delta is given by:
We shall throughout our study use the symbol (n0lm")to represent the Dirac bra-ket notation; i.e., (n"lmo)= ($I$:)
(6)
Theorthogonality I $ thr sigenfuncriunscan acfuallv he vrrified hy elemcnrrl integration t~rhniaues.Hwvever. urrhoeuuality can be easity by studying graphical p k ,., Figure 1shows the shape of six funcof the functions $$ tions orthogonal to $., Figure 2 shows the shape of $, whereas Figure 3 shows the shapes of the functions $&" (n = 1,2,3,5,. . . I ) . The fact that the area under these curves is zero can be easily verified. We shall now study perturbation theory. The idea is simply to introduce an additional term to the exact Hamiltonian. This term will be assumed to be time-independent and will be identified as the perturbation H'. We seek to find
Figure 3. Shapes of the functions $;1,°. (n = 1.2.3.5,. . . 7). Orthogonality is apparent since the area under these functions is equal to 0.If each of these functionsis normalized,then the set of functions$; is saidto form acompleta orthonormal set. Any function that satisfiesthe boundary conditions of the problem can be written as a linear combination of this orthonormal basis set. Volume 63 Number9 September 7986
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appropriate eigenfunctions $, and eigenvalues E, satisfying the eigenvalue equation If. however. H' >> H'. then an acceotable solution to the above equation is to have both the eigenfunctions and the eigenvalues written in terms of their exact counterparts, i.e.,
and
where in eq 8 the summation represents the deviation of the nth eigenfunction from its exact counterpart and where E, in eq 9 represents an energy correction to the nth level. The following results ensue directly from first-order time-independent perturbation theory ( I )
..
.
.
where Ns is the normalization factor for the kth level. However, because of the orthogonality of the exact solutions, one obtains
Table 2 shows the unperturbed exact energy, the perturbation enerpv, -. and the normalization constants for the first five states. The student is now asked to consult any standard text (24) on the time evolution of a state of a particle (i.e., electron) from an initial state i to a final state f. Time-dependent perturbation theory reveals that the time average of the maximum transition probability is given by P,,,(f-i)
=
