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Linear operators - operator representation (Read 3069 times)
Gerrit-Jan Linker
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Linear operators - operator representation
17.01.08 at 08:20:47
Linear operators - operator representation in a basis
A linear vector or operator is defined by the way that for each vector a of V a vector a is added. This has to happen in a linear way:
( b + u ) = b + u  
where b and v are vectors and and are scalars.
Because the vector ei is also in V we can write
ei = ∑j Ejiej ,in Dirac notation: |i> = ∑jEji|j>  
This means that operator is completely defined by the matrix Eij and the basis e1, e2, ...
In other words, matrix E is the operator representation for in that basis.
All the matrix elements Eij can be found by left multiplying by the ket vector <k|:
<k||i> = ∑jEji<k|j> = ∑jEjideltakj = Eki
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« Last Edit: 21.01.08 at 19:04:51 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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