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Eigen functions and eigen vectors (Read 2454 times)
Gerrit-Jan Linker
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Eigen functions and eigen vectors
19.01.08 at 11:44:34
 
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 = Σncnfn. 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 = ΩΣncnfn = ΣncnΩfn = Σncnwnfn
 
From a basis of n basis functions it is possible to create n linear independent combinations. A set of functions f1, f2, ..., fn is linearly independent if we cannot find a set of constants c1, c2, ..., cn for which Σicigi = 0
 
See also:
Molecular Quantum Mechanics, 3rd edition, Atkins and Friedman, p 9-10
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« Last Edit: 19.01.08 at 12:11:54 by Gerrit-Jan Linker »  

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