## Abstract

Differential equations that arise in seeking mathematical descriptions of physical systems frequently take the form as we saw in Chapter 7. Here the parameter λ is constant and the operator Ω involves differentiation with respect to Functions with this property are called

$$\Omega u(x)\, = \,\lambda u(x)$$

*x*. For different values of λ, the equation has different solutions*u*(*x*) We also saw in Chapter 7 that physical constraints may restrict λ to only certain values. We denote these “allowed” values by an index*n*. To each λ_{ n }there corresponds a solution to*u*_{ n }(*x*) this equation. Typically, these functions obey the relation$$\int\limits_a^b {u_m ^ * (x)u_n (x)\,dx\, = \,0} \,\,\,\,\text{for}\,m \ne \,n.$$

(8.1)

*orthogonal functions*. They are very important and have wide application in applied mathematics. To illustrate their usefulness, we look at some specific applications, first in quantum mechanics and laer in classical physics. In Section 8.6, we consider a more general treatment of orthogonal functions.## Keywords

Fourier Series Harmonic Oscillator Classical Physic Arbitrary State Fourier Amplitude
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York, Inc. 2002