Introduction to a Quantum Mechanical Harmonic Oscillator Using a Modified Particle-in-a-Box Problem Harvey F. Blanck
Austin Peay State University, Clarksville, TN 37044 The particle-in-a-one-dimensional-box (BOX)problem is often the first quantum mechanical problem solved in an introductory quantum mechanics course. The differential equation is solved easily and illustrates the fundamentals of quantum mechanics. The solution of the Schriidinger wave equation for a simple harmonic oscillator (HO)is significantly more complicated. Although some physical chemistry texts qualitatively relate the problems, no mathematical energy relationships are developed between them. However, it is possible to create a hybrid problem having the simplicity of the BOX problem and energy level spacing similar to those of the HO. The solution to the BOX problem for a particle of mass m in a box of length L results in energies given by En = n2h2/8m~2 (1) Energy level separation increases as the integer n increases ( n = 1, 2, 3, . . .) and decreases a s L increases
Figure 2. Relative energy levels El, E,, and E3for a hybrid BOWHO problem showing three boxes of different lengths.
where v = 0,1,2, . . . . Aproblem combining characteristics of both the HO and the BOX may be formulated by using eq 2 for a harmonic oscillator to determine the size of the box and the total energy but using infiiite potential energies a t the boundaries and zero potential energy inside as in the BOX problem. This results in a box whose size increases as its energy increases. The energy levels for this hybrid BOXIHO problem may be obtained by setting the box length, L, equal to in eq 2, which results in L2 22,, and replacing k and x, = 2E/x2mv2.Substituting for L2 in eq 1 and solving for E gives the energy for the hybrid BOWHO problem
Figure 1.Relativeenergy levels E,, E2.and E3for a particle in boxes of length L, and L2. X
(Fig. 1). For a harmonic oscillator, the potential energy is given by kx2/2 where x is the displacement and the force constant, k, is related to the frequency by k = 4x2mv2.Because the kinetic energy is zero at the extremities, the total energy, E, is given by
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Solution of the quantum mechanical HO problem results in energy levels with a constant separation above a value of 112 hv as expressed in the equation
Presented at the 100th meeting of the Tennessee Academy of Science at the University of Tennessee at Chattanooga. Chattanooga, TN, Nov. 1990.
98
Journal of Chemical Education
Figure 3. Relative energy levels E,, E2, and E3 associated with the circumferences C,, &, and C3,respectively, forthe circular orbits of the Bohr hydrogen atom model.
The energy level separation produced by eq 3 is uniform as in the true harmonic oscillator problem, although the separation is 0.8 hv rather than hv. Thus, using a harmonic oscillator equation to determine the energy of the particle and the width of a box results in a box whose size increases a t just the right rate to keep the energy level spacing constant (Fig. 2). The problem could have been posed in a slightly different fashion by asking what equation relating energy and size would be required to generate a relationship such that the increase in size is just sufficient to lower the next higher energy level to generate uniform spacing. The answer is E = ax2,where a is a constant. If the exponent ofx is greater than 2, the energy level separation increases a s n increases and the opposite is true if the exponent is less than 2. A somewhat similar situation to the latter arises for the hydrogen atom problem because the energy relationship to
the radius of a circular orbit is Er = -nstant. Usine this equation, a box may be simulated by drawing two hy&bolas. The box size. which is the circumference. increases so rapidly with en& that the energy level separation decreases rapidly with an increase in n (Fig. 3). While the similarities of the wave shapes for the BOX and HO are evident, this hybrid problem extends the similarity to include energy spacing. Although the quantum mechanical HO problem is still no easier to solve, development of this hybrid lrroblem encouraees the extension of into an unfamiliar known results'from ihe ROX situation and rewards those efforts by correctly predicting uniform energy separation in the HO. The hybrid BOXNO problem gencrakd from the user-friendly particle-in-a-box has onwen to be a useful bride0 bctwccn the BOX and HO problems in teaching introductory quantum mechanics in undergraduate physical chemistry.
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Volume 69 Number 2 February 1992
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