Problem 1: Operation in Hilbert space (a) Show the following properties of Hermitian operators, where a E C (A +aB) = A + a*B (AB)t = Bt At (ABCD)t = DtCtBt At (b) Show that the inner product of any two vectors ), o) € 7 is invariant under unitary transformations UU UtU = 1. =

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Problem 1: Operation in Hilbert space
(a) Show the following properties of Hermitian operators, where a € C
(A+aB) = A + a*B
(AB)† = B¹At
(ABCD)† = DİC† Bt At
(b) Show that the inner product of any two vectors ), |ø) € H is invariant under unitary transformations UU†
UtU = 1.
=
Transcribed Image Text:Problem 1: Operation in Hilbert space (a) Show the following properties of Hermitian operators, where a € C (A+aB) = A + a*B (AB)† = B¹At (ABCD)† = DİC† Bt At (b) Show that the inner product of any two vectors ), |ø) € H is invariant under unitary transformations UU† UtU = 1. =
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