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# 1.2: Wavefunctions

• • Contributed by Jack Simons
• Professor Emeritus and Henry Eyring Scientist (Chemistry) at University of Utah

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.