Quantum Mechanics in Chemistry (Simons and Nichols)

1: The Basic Tools of Quantum Mechanics

Quantum Mechanics Describes Matter in Terms of Wavefunctions and Energy Levels and physical Measurements are Described in Terms of Operators Acting on Wavefunctions

1.1: Operators Each physically measurable

The mapping from F to F is straightforward only in terms of cartesian coordinates. To map a classical function F, given in terms of curvilinear coordinates (even if they are orthogonal), into its quantum operator is not at all straightforward. Interested readers are referred to Kemble’s text on quantum mechanics which deals with this matter in detail. The mapping can always be done in terms of cartesian coordinates after which a transformation of the resulting coordinates and differential operators to a curvilinear system can be performed. The corresponding transformation of the kinetic energy operator to spherical coordinates is treated in detail in Appendix A. The text by EWK also covers this topic in considerable detail. The relationship of these quantum mechanical operators to experimental measurement will be made clear later in this chapter. For now, suffice it to say that these operators define equations whose solutions determine the values of the corresponding physical property that can be observed when a measurement is carried out; only the values so determined can be observed. This should suggest the origins of quantum mechanics’ prediction that some measurements will produce discrete or quantized values of certain variables (e.g., energy, angular momentum, etc.).

In addition to operators corresponding to each physically measurable quantity, quantum mechanics describes the state of the system in terms of a wavefunction that is a function of the coordinates {q } and of time . The function | gives the probability density for observing the coordinates at the values at time t. For a many-particle system such as the molecule, the wavefunction depends on many coordinates. For the example, it depends on the x, y, and z (or r,q, and f) coordinates of the ten electrons and the x, y, and z (or r,q, and f) coordinates of the oxygen nucleus and of the two protons; a total of thirty-nine coordinates appear in .

In classical mechanics, the coordinates qj and their corresponding momenta are functions of time. The state of the system is then described by specifying (t) and (t). In quantum mechanics, the concept that qj is known as a function of time is replaced by the concept of the probability density for finding at a particular value at a particular time t: . Knowledge of the corresponding momenta as functions of time is also relinquished in quantum mechanics; again, only knowledge of the probability density for finding with any particular value at a particular time remains.