Linker IT Software
Google
Web www.oraxcel.com
menubar-top-links menubar-top-rechts
Home Help Search Login
Welcome, Guest. Please Login.
SQL*XL: Database to Excel bridge litLIB: Excel power functions pack ExcelLock: Locking and securing your valuable Excel spreadsheets encOffice: Protect your Excel file easy and safe encOffice: Protect your Excel file easy and safe
Pages: 1
Hermitian, normal and complex operators (Read 4002 times)
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Hermitian, normal and complex operators
31.01.08 at 20:03:36
 
Hermitian, antihermitian and normal operators
 
An operator is Hermitian if O = O
An operator is antihermitian if O = -O
An operator is normal if it commutes with its adjoint operator:[ O,O] = 0
 
A complex operator can be written in a real part and a complex part. For a complex number c we can write c = a + ib where a is a and b are real numbers. The complex conjugate of c is a - ib.
 
Similarly a complex operator O can be written as O = A + iB where A and B are Hermitian operators. The complex conjugate of operator O is O = A - iB.
 
Also:  
|c|2 = sqrt(a2+b2) = |c*c|
|A|2 = |AA|
 
Now the real operators A and B can be expressed in terms of O and O:
 
O = A+iB and O = A-iB
 
Taking the last equation we can write B = -(O-A)/i = i(O-A)  
 
Entering this defintion of B into the first equation yields:
 
O = A+iB = A + i ( i(O-A) ) = A - O + A = 2A - O
<=> A = (O + O)
 
B is obtained by using the expression for A in the first equation O=A+iB
B= (O-A)/i = -iO + iA <=>
B= (O-A)/i = -iO + i(O + O) <=>
B= -iO + iO + iO <=>
B= -iO + iO = -i(O - O) = -i(O - O) <=>
B= (O - O)/2i
 
Using these expressions for A and B we can state the commutator [O,O] in terms of the commutator [A.B]:
[O,O] = OO) - OO <=>
[O,O] = (A+iB)(A-iB)
 
Now using the expressions we obtained for A and B we can write
[O,O] =-2iAB + 2iBA = -2i[A,B] = 2i[B,A]
 
We arrive at this equation:
[O,O] = 2i[B,A]
 
From this equation we derive that if O is normal and [O,O] = 0 A and B must commute.
 
See also:
Complex number
http://en.wikipedia.org/wiki/Complex_number
Back to top
 
« Last Edit: 07.02.08 at 13:13:02 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Re: Hermitian, normal and complex operators
Reply #1 - 04.02.08 at 15:00:10
 
Theorem:
If operator is normal |α> is an eigenfunction with ωα as eigenvalue : |α> = ωα|α>
then |α> is an eigenfunction with ω*α as eigenvalue : |α> = ω*α|α>
 
Theorem:
If two Hermitian operators A and B commute then at least one complete set of functions exists which are simultaneous eigenfunctions of both operators.
 
Theorem:
If O is a normal operator that commutes with operator A and |α> and |> are two non-degenerate eigenfunctions of O the following matrix element is zero:
<α|A|> = 0
Back to top
 
« Last Edit: 04.02.08 at 15:48:50 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Pages: 1