Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

Eqs. (2.15) and (2.16), a much simpler potential energy function will be used for the initial example of a quantum mechanical system, namely, a rectangular box function. The ensuing particle in a box is an artificial example but is pedagogically extremely useful and presents simple differential equations while offering real physical applications; see Section 2.5.

2.3 Demonstration of Quantum Mechanical Principles for a Simple, One‐Dimensional, One‐Electron Model System: The Particle in a Box

      Real quantum mechanical systems have the tendency to become mathematically quite complicated due to the complexity of the differential equations introduced in the previous section. Thus, a simple model system will be presented here to illustrate the principles of quantum mechanics introduced in Sections 2.1 and 2.2. The model system to be presented is the so‐called particle in a box (henceforth referred to as “PiB”) in which the potential energy expression is simplified but still has with wide‐ranging analogies to real systems. This model is very instructive, since it shows in detail how the quantum mechanical formalism works in a situation that is sufficiently simple to carry out the calculations step by step while providing results that much resemble the results in a more realistic model. This is exemplified by the overall similarity such as the symmetry (parity) of the PiB wavefunctions when compared with that of the harmonic oscillator wavefunctions discussed in Chapter 4.

      2.3.1 Definition of the Model System

The PiB model assumes that a particle, such as an electron, is placed into a potential energy well or confinement shown in Figure 2.2. This confinement (the “box”) has zero potential energy for 0 ≤ x ≤ L, where L is the length of the box. Outside the box, i.e. for x < 0 and for x > L, the potential energy is assumed to be infinite. Thus, once the electron is placed inside the box, it has no chance to escape, and one knows for certain that the electron is in the box.

      As discussed earlier, the total energy is written as the sum of the kinetic and potential energies, T and V, respectively:

      (2.17)

      As before, the kinetic energy of the particle is given by

      (2.3)

      where m is the mass of the electron. Substituting the quantum mechanical momentum operator,

      (2.4)

      into Eq. (2.3), the kinetic energy operator can be written as

      (2.5)

Figure 2.2 Panel (a): Wavefunctions

for n = 1, 2, 3, 4, and 5 drawn at their appropriate energy levels. Energy given in units of h²/8mL². Panel (b): Plot of the square of the wavefunctions shown in (a).

The potential energy inside the box is zero; thus, the total energy of the particle inside the box is

      (2.18)

      Since the potential energy outside the box is infinitely high, the electron cannot be there, and the discussion henceforth will deal with the inside of the box. Thus, one may write the total Hamiltonian of the system as

(2.19)

      In the notation of linear algebra, this operator/eigenvector/eigenvalue problem is written as

(2.20)

Equation (2.20) instructs to apply the Hamiltonian of Eq. (2.19) to a set of yet unknown eigenfunctions to obtain the desired energy eigenvalues. The eigenfunctions typically form an n‐dimensional vector space in which the eigenvalues appear along the diagonal. Thus, Eq. (2.20) implies

(2.21)

      that is, the Hamiltonian operating on a set of eigenfunctions such that

; ;