2. Mathematical Prerequisites


Abstract

Quantum mechanics is based on a number of advanced mathematical ideas that are described in this chapter.

First the normal real numbers will be generalized to complex numbers. A number such as ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$ is an invalid real number, but it is considered to be a valid complex one. The mathematics of quantum mechanics is most easily described in terms of complex numbers.

Classical physics tends to deal with numbers such as the position, velocity, and acceleration of particles. However, quantum mechanics deals primarily with functions rather than with numbers. To facilitate manipulating functions, they will be modeled as vectors in infinitely many dimensions. Dot products, lengths, and orthogonality can then all be used to manipulate functions. Dot products will however be renamed to be “inner products” and lengths to be norms.

Operators will be defined that turn functions into other functions. Particularly important for quantum mechanics are eigenvalue cases, in which an operator turns a function into a simple multiple of itself.

A special class of operators, Hermitian operators will be defined. These will eventually turn out to be very important, because quantum mechanics associates physical quantities like position, momentum, and energy with corresponding Hermitian operators and their eigenvalues.




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