Max Diem - Quantum Mechanical Foundations of Molecular Spectroscopy

or abbreviated as . If the operators commutate and can be determined simultaneously; if the commutator is not zero, then the eigenvalues cannot be determined simultaneously. This case will be demonstrated in Example 2.2.

Example 2.2Determine the commutator of the momentum operator and the position operator when applied to a function f ( x ), i.e. determine

(E2.2.1)

(E2.2.2)

Answer:

The derivative of the product needs to be evaluated using the product rule of differentiation. Thus,

(E2.2.3)

(E2.2.4)

Thus, the commutator

(E2.2.5)

which predicts that the position and momentum of a moving particle cannot be determined simultaneously. This was stated earlier in Eq. (2.1)as the Heisenberg uncertainty principle as

(2.1)

To show the equivalency of Eqs. (E2.2.5)and (2.1), one has to determine the standard deviations in momentum and position σ pand σ xthat can be related to the uncertainties Δ p xand Δ x .

Figure 2.1 Potential energy functions and analytical expressions for (a) molecular vibrations and (b) an electron in the field of a nucleus. Here, f is a force constant, k is the Coulombic constant, and e is the electron charge.

In Postulate 2, the kinetic energy T was substituted by the operator

(2.4)

but the potential energy V was left unchanged, since it does not include the momentum of a moving particle. The potential energy, however, depends on the particular interactions describing the problem, for example, the potential energy an electron experiences in the field of a nucleus or the potential energy exerted by a chemical bond between two vibrating nuclei. The shape of these potential energy curves are shown in Figure 2.1along with the potential energy equations.

When these potential energy expressions are substituted into the Schrödinger equation

(2.7)

one obtains a differential equation:

(2.15)

for the harmonic oscillation of a diatomic molecule and

(2.16)

for the electron in a hydrogen atom. In Eqs. (2.15)and (2.16), f and k are constants that will be introduced later, and e is the electronic charge, e = 1.602 × 10 −19[C]. Equation (2.16)is not strictly correct since the potential energy is a spherical function in the distance r from the nucleus, but is presented here and in Figure 2.1as a one‐dimensional quantity. Also, the mass in the denominator of the kinetic energy operator needs to be substituted by the reduced mass to be introduced later.

Due to the difficulties in solving equations such as 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.

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 2.1and 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.

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.