Basis functions

Consider a denumerable set of possible physical amplitudes, or wave functions, u_i, where i=1,..N and N could be any integer.

If they have the property, $(u_{i},u_{j})=\delta_{ij}$ then they are said to be an orthonormal set. Because of the linear property, we can ask if it is possible for any, physical, amplitude $\Psi$ to be expressed as a linear combination of the u_i basis functions. We want,

$$\Psi = \sum_{i} c_{i} u_{i}$$ (1.14)

where c_i are complex numbers, they are called expansion coefficients. We take the inner, or scalar, product of this equation with any one member of the set, lets say u_j. Or

$$(u_{j},\Psi)=c_{j}$$ (1.15)

where we have used the orthonormal property of the basis. Inserting the value of c_j back into the previous equation we get,

$$\Psi({\bf r}) = \sum_{i} (u_{i},\Psi) u_{i}({\bf r}) = \sum_{... ...3}{\bf r'} u^{*}_{i}({\bf r'}) u_{i}({\bf r}) \Psi({\bf r'}).$$ (1.16)

The identity holds if

$$\sum u^{*}_{i}({\bf r'}) u_{i}({\bf r}) = \delta^{3}({\bf r}-{\bf r'}).$$ (1.17)

This important relation, if it holds for the set u_j, is called the closure, or completeness, property of the basis functions. We can summarize the above results in the form of a table pt

basis functions, or vectors	uj
Orthonormality relation	$(u_{i},u_{j})=\delta_{ij}$
State expansion in basis	$\Psi = \sum_{i} c_{i} u_{i}$
Expansion coefficients	$c_{j}=(u_{j},\Psi)$
Closure	$\sum_{i} u^{*}_{i}({\bf r'}) u_{i}({\bf r})= \delta^{3}({\bf r'}-{\bf r})$