Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

= mv, is substituted in quantum mechanics by the differential operator , defined by

(2.2)

operating (or being applied to) the wavefunction Ψ(x, t). In Eq. (2.2), i is the imaginary unit, defined by Equation (2.2) often is considered the central postulate of QM.

      The form of Eq. (2.2) can be made plausible from equations of classical wave mechanics, de Broglie's equation (Eq. [1.10]) and Planck's equation (Eq. [1.7]), but cannot be derived axiomatically. It was the genius of E. Schrödinger to realize that the substitution described in Eq. (2.2) yields differential equations that had long been known and had solutions that agreed with experiments. In the Schrödinger equations to be discussed explicitly in the next chapters (for the H atom, the vibrations and rotations of molecules, and molecular electronic energies), the classical kinetic energy T given by

(2.3)

      is, therefore, substituted by

      (2.4)

which is, of course, obtained by inserting Eq. (2.2) into Eq. (2.3). The total energy of a system is given as the sum of the potential energy V and the kinetic energy T:

      (2.5)

Postulate 3: All experimental results are referred to as observables that must be real (not imaginary or complex). An observable is associated with (or is the “eigenvalue” of) a quantum mechanical operator . This can be written as

      (2.6)

      where a are the eigenvalues and ϕ the corresponding eigenfunctions. The terms “operator,” “eigenvalues,” and “eigenfunctions” are terminology from linear algebra and will be further explained in Section 2.3 where the first real eigenvalue problem, the particle in a box, will be discussed. Notice that the eigenfunctions often are polynomials, and each of these eigenfunctions has its corresponding eigenvalue.

      In this book, following generally accepted notations, the total energy operator is generally identified by the symbol and referred to as the Hamilton operator, or the Hamiltonian, of the system. With the definition of the Hamiltonian above, it is customary to write the total energy equation of the system as

(2.7)

Equation (2.7) implies that the energy “eigenvalues” E are obtained by applying the operator on a set of (still unknown) eigenfunctions ψ that are here assumed to be time‐independent and a function of spatial coordinates x only, ψ(x). Solving the differential equations given by Eq. (2.7) yields the eigenfunctions ψ_i and their associated energy eigenvalues E_i.

      Postulate 4: The expectation value of an observable a, associated with an operator , for repeated measurements, is given by

(2.8)

      If the wavefunctions Ψ(x, t) are normalized, Eq. (2.7) simplifies to

(2.9)

since the denominator in Eq. (2.8) equals 1. This expectation value may be viewed as an expected average of many independent measurements and embodies the probabilistic nature of quantum mechanics.

      Postulate 5: The eigenfunctions ϕ_i, which are the solutions of the equation , form a complete orthogonal set of functions or, in other words, define a vector space. This, again, will be demonstrated in Section 2.3 for the particle‐in‐a‐box wavefunctions, which are all orthogonal to each other and therefore may be considered unit vectors in a vector