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2: Some Essential Math

Last updated

Mar 5, 2022
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- 1.5: Electron in a Box
- 2.1: Fourier Series and Integrals, the Dirac Function

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Topic hierarchy

2.1: Fourier Series and Integrals, the Dirac Function
We begin with a brief review of Fourier series. Any periodic function of interest in physics can be expressed as a series in sines and cosines—we have already seen that the quantum wave function of a particle in a box is precisely of this form. The important question in practice is, for an arbitrary wave function, how good an approximation is given if we stop summing the series after N terms.
2.2: Linear Algebra
This linearity of the sets of possible solutions is true generally in quantum mechanics, as is the representation of physical variables by operators on the wave functions. The mathematical structure this describes, the linear set of possible states and sets of operators on those states, is in fact a linear algebra of operators acting on a vector space. From now on, this is the language we’ll be using most of the time. To clarify, we’ll give some definitions.
2.3: Function Spaces
The motivation for our review of linear algebra was the observation that the set of solutions to Schrödinger’s equation satisfies some of the basic requirements of a vector space, in that linear combinations of solutions give another solution to the equation. Furthermore, Schrödinger’s equation itself, as a differential operator acting on a function, suggests that the concept of a matrix operator acting on vectors in an n-dimensional vector space can be extended to more general operators.
2.4: Complex Variable, Stationary Path Integrals

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