Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Linear Spaces of Functions

In our work in class so far we have used vector spaces to describe the position of objects in space. We have built up larger spaces by describing the motion of one particle in many dimensions or many particles (so far) in one dimension. As a result of these two different examples, we have created two mathematical abstractions -- a linear (vector) space and an inner product space -- that obey certain properties. Once we have these properties, we can apply these ideas to other examples, effectively finding cases in which the analogy to motion of a particle in many dimensions (or of many particles in one dimension) is so good that we can use the same math. One example that will turn out to be of great importance, both in wave theory in elastic media and in quantum physics, is the case of functions. We'll begin by working out a simple example.

(a) Consider the set of functions consisting of the sum of the two functions sin θ and cos θ:

f(θ) = a sin θ + b cos θ.
.

where a and b are arbitrary complex numbers. We can associate this function with a vector

Prove that this set of functions forms a linear space according to the rules given in class on Monday.

(b) From the usual definition of inner product in our vector space describing the motion of a single particle in one dimension, we might come up with one or more definitions of an inner product in this space. Construct one and show that it doesn't work -- that it doesn't give an inner product according to the definition given in class on Monday. Discuss why this is the case.

(c) Show that the definition

satisfies the conditions for an inner product.

(d) What is the dimension, n, of the inner product space constructed in parts (a) and (c)? Construct a set of n orthonormal vectors in this space that can serve as a basis.

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Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120
Email: redish@umd.edu

Last revision 24. October, 2004.