Continuous set of basis functions

A continuous set of basis functions refers to a collection of functions that can be used to represent or approximate other functions in a continuous manner over a specified domain. In mathematics and physics, basis functions are building blocks that can be combined to represent more complex functions through a linear combination.

In the context of mathematics and physics, the continuous set of basis functions is often used in fields like calculus, Fourier analysis, and quantum mechanics. Instead of using a fixed number of predefined functions (like the blocks with fixed shapes), a continuous set of basis functions allows you to work with an infinite number of functions that span a whole range of possibilities.

For example, in Fourier analysis, you can represent any periodic function as a sum (or integral) of sinusoidal functions with different frequencies and amplitudes. These sinusoidal functions form a continuous set of basis functions that can approximate any periodic function.

Having a continuous set of basis functions is powerful because it provides a flexible and versatile way to represent and analyze complex functions, and it allows us to work with functions that might not have a simple and predefined representation. It's like having an endless supply of blocks with different shapes to build and create all sorts of interesting structures!

As an example of continuous set of functions, we consider momentum eigen functions,

where p is the eigen value of momentum operator of free particle.

Such function Vp(x) is said to be normalized in Dirac sense. The norm of such function is infinite. The momentum eigenfunctions are characterized by their specific momentum values, and they form a continuous set because momentum can take on any real value (positive, negative, or zero). Unlike discrete basis functions, such as the energy eigenstates in a finite potential well, the momentum eigenfunctions cover an infinite range of possible momenta. These functions are complex-valued and describe the wave-like nature of particles in quantum mechanics. They are essential for describing the momentum state of a particle and for analyzing various physical phenomena, such as scattering, quantum tunneling, and wave-particle duality.

Norm = √ (V'p(x) Vp(x) = 𝛿(p - p') - > ∞

So such functions do not belong to function space. However, we can use such functions as basis in function space. We can expand any function

If we expand wavefunction 𝜓(x) as

𝜓(x) = Σ Ciui(x)

where ui(x); i = 1,2,3,....energy eigen function

then the coefficient

C1 = [u1, 𝜓1(x)]

C2 = [u2, 𝜓2(x)] and so on

represent energy representation of the wave function of energy representation of the state.

If we expand wave function 𝜓(x) as

Eigen functions of position operator of free particle are Delta functions

Thus delta functions are also normalized in dirac sense

𝜓(x) = ∫ 𝜓(Xo) 𝛿(Xo - x)dXo

Φ(x) = ∫ Φ(Xo) 𝛿(Xo - x)dXo

𝜓(Xo) represents the coordinate representation of the wavefunction.

In summary, a continuous set of basis functions for the momentum operator consists of the momentum eigenfunctions, which are complex-valued functions representing the momentum state of a particle and forming an infinite range of possible momentum values. They play a crucial role in the mathematical formalism and understanding of quantum mechanics.

This note is a part of the Physics Repository.