Is this operator Hermitian?

  • Thread starter sgoodrow
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  • #1
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1. Let G be an operator on H (Hilbert Space). Show that:
(a) H = 1/2 (G + G[tex]^{\dagger}[/tex]) is Hermitian.
(b) K = -1/2 (G - G[tex]^{\dagger}[/tex]) is Hermitian.
(c) G = H + iK
.

Homework Equations


...

3. The Attempt at a Solution :

(a) Since the adjoint of the sum of two operators does not change their position (addition of operators is commutative), it is very straight forward.

(b) This is where I run into trouble, and I think it is because the problem is wrong. The operator given appears to be anti-hermitian (skew-hermitian), however I would like confirmation. This result makes (c) very difficult, as it uses an incorrect answer. My suspicion is that the intended question includes an i (as in, H = -i/2 (...)):

K[tex]^{\dagger}[/tex] = -1/2 (G[tex]^{\dagger}[/tex] - G) = 1/2 (G - G[tex]^{\dagger}[/tex]) = -K.

(c) Well if I am correct about (b), (c) is wrong. I was just hoping someone could confirm I am right, or show me how K is Hermitian. I can take it from there.
 

Answers and Replies

  • #2
318
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If the problem should be as stated this will give a condition for G that allows to solve all equations.
If G is arbitrary and the equations should be satisfied for general G there has to be a change in definition.
K= -i/2(...)
 

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