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