Field of Science

Smooth Operators

The formulation of quantum mechanics in terms of operators lends it dash of elegance compared to the stodgy differential equations of Newtonian physics. In 1932 John von Neumann brought operator algebra to bear on quantum mechanics. von Neumann was unusually social, for a mathematician, and his home in Princeton the venue for many parties. He was also gifted across a wide range of field in mathematics, doing fundamental work in both my field and that of my husband (and our work is not connected in any way!).

What's an operator? The basic definition is a rule that changes one function into another. A more sophisticated one is a mapping between two function spaces. A function space is a collection of functions, each point in the same corresponds to a function. The functions are collected according to a set of rules, different function spaces have different rules associated with them. For example, the set of all functions that are real-valued on the interval 0 to 1 and have continuos 2nd derivatives would constitute a function space. A famous function space for quantum mechanics is Hilbert space.

You can, of course, construct classical physics in terms of operators as well; and quantum mechanics in terms of differential equations! The pedagogical approach to quantum mechanics generally brings operators explicitly to the table very early one, while the introduction of classical physics is often done without recourse even to the calculus.

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