Basis Sets

## Question:

In words, how would you compare the idea of a basis set from
quantum to vector space?
## Answer:

A basis set of a quantum mechanical observable is a complete set
of orthonormal eigenfunctions built up from the eigenfunctions of
that observable. If two observables share a common basis set they
commute.
A basis set of a finite dimensional vector space is the a set
of linearly independent vectors that span the space.

It is not obvious that the two uses of the words "basis
set" are related but fundamentally they convey the same
idea. Any element of a vector space may be expressed in terms of
the basis set. Any element of a commuting set of observables may
have its eigenfunctions expressed as a well defined function of
eigenfunctions of the basis set.

As I re-read this it strikes me that this is a real stretch,
but it is the best I can do short of too much research. I am
curious how you got to this question from the courses you signed
up for. Is this possibly an exam for the instructor?:).

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JDJ