1.2: Wavefunctions

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The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts; these functions are called wavefunctions

In addition to operators corresponding to each physically measurable quantity, quantum mechanics describes the state of the system in terms of a wavefunction \(\Psi\) that is a function of the coordinates {q\(_j\)} and of time \(t\). The function |\(\Psi(q_j ,t)|^2 = \Psi^*\Psi\) gives the probability density for observing the coordinates at the values \(q_j\) at time t. For a many-particle system such as the \(H_2O\) molecule, the wavefunction depends on many coordinates. For the \(H_2O\) 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 \(\Psi\).

In classical mechanics, the coordinates qj and their corresponding momenta \(p_j\) are functions of time. The state of the system is then described by specifying \(q_j\) (t) and \(p_j\) (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 \(q_j\) at a particular value at a particular time t: \(|\Psi(q_j,t)|^2\). Knowledge of the corresponding momenta as functions of time is also relinquished in quantum mechanics; again, only knowledge of the probability density for finding \(p_j\) with any particular value at a particular time \(t\) remains.

Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry and Jeff A. Nichols (Oak Ridge National Laboratory)