Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

alt="left-bracket ModifyingAbove p With Ì‚ Subscript x Baseline comma ModifyingAbove x With Ì‚ right-bracket"/> 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_p and σ_x that can be related to the uncertainties Δp_x and Δx.

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.

2.2 The Potential Energy and Potential Functions

      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 2.1 along 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 2.1 as 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