Mueller M.R. - Fundamentals of Quantum Chemistry[c] Molecular Spectroscopy and Modern Electronic Structure Computations (Kluwer, 2001)

The potential everywhere else in the box is zero. Figure 4-1 shows a picture of this system. Calculate the energy of the system up to the first-order energy correction for the (a) ground-state, (b) first excited state, and then (c) for any level n.

Solution: As can be seen by Figure 4-1, the system is broken down into three different regions. The perturbation is isolated to Region II.

Within Region II of the box:

(a) For the ground-state

(b) For the first-excited state

(c) For any level n:

Note that the perturbation in Example 4-2 results in a greater first-order energy correction in the ground-state than in the first excited state. In fact, it can be seen by the general solution in part (c) that all odd values of n will result in a larger first-order correction than even values of n. In order to understand the reason for this behavior, the wavefunctions for the 1- dimensional Particle-in-a-Box problem, shown in Figure 2-2 must be compared to the perturbing potential as shown in Figure 4-1. The perturbing potential is limited to the center 10% of the box. All states with an even value of n have a node in the wavefunction at the center of the box in the region of the perturbing potential. As a result, the effect of the perturbing potential is minimal in even valued n states. However, in states with an odd value of n, the effect of the perturbation is the greatest because these wavefunctions have an antinode at the center of the box. The physical interpretation can be made by recalling that the square of the wavefunction for a given state n (functions are real in this case) corresponds to the probability density of the particle. States with odd values of n have minimal probability densities in the center 10% of the box whereas states with even values of n have high probability densities in this region resulting in a larger effect to the energy of the particle. The general solution in part (c) also predicts that as n increases, the first-order energy correction becomes smaller. This is because the kinetic energy of the particle increases with increasing value of n and the effect of the potential becomes less significant.

If the energy of the model or known system changes as a result of a perturbation, the wavefunction for the system also changes from its unperturbed form. The first-order correction to the wavefunction for a quantum level n for the perturbed system, can now be obtained. The first-order correction of the wavefunction can be expressed as a sum over the unperturbed wavefunctions.

The sum is over all of the unperturbed wavefunctions (the basis set). The terms labeled are coefficients reflecting the contribution of each of the unperturbed wavefunctions to the sum. The coefficients can now be determined. Equation 4-14 is substituted into Equation 4-11b.

This result is then multiplied by

and integrated overall space.

When the same result is obtained as in Equation 4-13 due to the orthonormality of When all of the terms vanish except when

and

Solution: The perturbed wavefunction with a first-order correction summed to the unperturbed wavefunction is as follows (N is the normalization constant):

Each coefficient must be solved for using Equation 4-15.

decreases with higher-order wavefunctions. The wavefunction for the perturbed system becomes:

The wavefunction can now be normalized.

The normalized wavefunction becomes as follows:

For a positive potential the effect of the first-order correction to the ground-state wavefunction, is to reduce the probability density of the particle in the center of the box. This is shown in Figure 4-2 for the case where The probability density has diminished for the particle in the center 10% of the box where the positive potential blip exists.

The second-order correction to the state energy of the perturbed system, can now be developed. The approach is similar to obtaining the first-order energy correction. Equation 4-11c is multiplied by the complex conjugate of the unperturbed state wavefunction, and integrated over the perturbed system.

Upon taking advantage of the orthonormality of the wavefunctions and the hermiticity of the following expression results for the second-order correction for the n state of the perturbed system.

In terms of Dirac notation, Equation 4-18 may be expressed as follows:

The expressions in Equations 4-18 and 4-19 are valid only for nondegenerate systems Also note that it is possible to have a non-zero second-order energy correction (and even higher orders) even if the perturbed system has only a first-order perturbing Hamiltonian. While the first-order correction to the energy represents an “average” of the perturbation to a given unperturbed state, the second-order correction to the energy represents the “mixing” between unperturbed states as a result of the perturbation.

Example 4-5

Problem: Consider the same system as in Example 4-2, a Particle-in-a-Box with a sinusoidal potential inside,

Calculate the second-order energy correction to the ground-state of the perturbed system including up to the unperturbed wavefunction.

Solution: Since the perturbed system only has a first-order perturbation, Equation 4-18 summed to is as follows: