Quantum Mechanical Foundations of Molecular Spectroscopy. Max DiemЧитать онлайн книгу.

= mv, is substituted in quantum mechanics by the differential operator ModifyingAbove p With Ì‚

, defined by

(2.2) ModifyingAbove p With Ì‚ equals StartFraction normal h with stroke Over i EndFraction StartFraction d Over normal d x EndFraction equals minus i normal h with stroke StartFraction d Over normal d x EndFraction

operating (or being applied to) the wavefunction Ψ(x, t). In Eq. (2.2), i is the imaginary unit, defined by StartRoot negative 1 EndRoot equals i period 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) upper T equals one half m v squared equals p squared slash 2 m

is, therefore, substituted by

(2.4) ModifyingAbove upper T With Ì‚ equals minus StartFraction normal h with stroke squared Over 2 m EndFraction StartFraction partial-differential squared Over partial-differential x squared EndFraction

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) upper E Subscript total Baseline equals upper T plus upper V equals minus StartFraction normal h with stroke squared Over 2 m EndFraction StartFraction partial-differential squared Over partial-differential x squared EndFraction plus upper V

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 ModifyingAbove upper A With Ì‚ . This can be written as

(2.6) ModifyingAbove upper A With Ì‚ phi equals a phi

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 ModifyingAbove upper H With Ì‚ 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) ModifyingAbove upper H With Ì‚ normal psi left-parenthesis x right-parenthesis equals left-brace minus StartFraction normal h with stroke squared Over 2 m EndFraction StartFraction partial-differential squared Over partial-differential x squared EndFraction plus upper V right-brace normal psi left-parenthesis x right-parenthesis equals upper E normal psi left-parenthesis x right-parenthesis

Equation (2.7) implies that the energy “eigenvalues” E are obtained by applying the operator ModifyingAbove upper H With Ì‚ 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 ModifyingAbove upper A With Ì‚ , for repeated measurements, is given by

(2.8)

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

(2.9) left pointing angle a right pointing angle equals integral normal upper Psi Superscript asterisk Baseline left-parenthesis x comma t right-parenthesis ModifyingAbove upper A With Ì‚ normal upper Psi left-parenthesis x comma t right-parenthesis normal d x

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 ModifyingAbove upper A With Ì‚ phi equals a phi , 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

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