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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 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.