GerritJan Linker

Eigen functions and eigen vectors Usually when an operator Ω acts on a function f the result is another function. In certain cases the outcome of Ωf is wf; the outcome is the same function f multiplied by a constant w. Functions of this kind are called eigen functions. They satisfy the relation Ωf = wf which is called an eigen value equation. The constant w in the eigen value equation is called the eigen value of operator Ω. All the eigen functions of Ω are called a complete set of functions. Any general function g can be written as a linear combination of the complete set: g = Σ_{n}c_{n}f_{n}. The benefit of doing this is that it allows us to deduce the effect of the operator on a function that is not its own eigen functions: Ωg = ΩΣ_{n}c_{n}f_{n} = Σ_{n}c_{n}Ωf_{n} = Σ_{n}c_{n}w_{n}f_{n} From a basis of n basis functions it is possible to create n linear independent combinations. A set of functions f_{1}, f_{2}, ..., f_{n} is linearly independent if we cannot find a set of constants c_{1}, c_{2}, ..., c_{n} for which Σ_{i}c_{i}g_{i} = 0 See also: Molecular Quantum Mechanics, 3rd edition, Atkins and Friedman, p 910
