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.