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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

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