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Transformation of operator in another basis set (Read 4405 times)
Gerrit-Jan Linker
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Transformation of operator in another basis set
24.01.08 at 21:43:04
 
Transformation of operator in another basis set
 
If we have an operator in the basis {|α>}N, Eαβ= <α||β>
Eαβ is a matrix element of the matrix E.  
Matrix E is said to be the matrix representation of the operator in the basis {|α>}N.
 
If we would like to get the representation of in another basis set, say {|i>}N we can find a matrix U that brings E into the new basis set: A=UEU where U is the adjoint of U.
 
Derivation:
Eαβ= <α||β> = <α| 1 1 |β>
 
We will use the closure relation: Σi|i><i| = 1
 
=> <α| 1 1 |β> = <α| Σi|i><i| Σj|j><j|β>
= Σi<α|i>Σj<i||j><j|β>
= ΣiΣj<α|i><i||j><j|β>
= ΣiΣjUαiEijU
= ΣiΣjU*EijU = UEU = A
 
So, we have transformed the matrix representation E of in the basis {|α>}N into another matrix representation A in the basis {|i>}N using the transformation matrix U.
 
The last formula ΣiΣjU*EijU = UEU=A should be read as follows. Use ΣiΣjU*EijU to compute the matrix element Aαβ
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« Last Edit: 24.01.08 at 22:05:17 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
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Posts: 75
Re: Transformation of operator in another basis se
Reply #1 - 26.01.08 at 11:09:53
 
Reversing the transformation
 
Transforming A into a new basis set gives a new representation B with B=UAU .  
 
To reverse the process, to obtain the matrix representation in the original basis set you need to reverse the operation:
 
B=UAU , multiply from the left by U and from the right by U<=>
UBU=UUAUU
 
Since UU = UU = 1 we can write:
 
UBU=1A1 => A = UBU
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« Last Edit: 26.01.08 at 11:17:30 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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