**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(a

^{2}+b

^{2}) = |c

^{*}c|

|A|

^{2} = |A

^{†}A|

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 = ½(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 + i½(O + O

^{†}) <=>

B= -iO + i½O + i½O

^{†} <=>

B= -i½O + i½O

^{†} = -i½(O - O

^{†}) = -i½(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

^{†}) - O

^{†}O <=>

[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