- #1
yakattack
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I have some questions about the properties of a Hermitian Operators.
1) Show that the expectaion value of a Hermitian Operator is real.
2) Show that even though [tex]\hat{}Q[/tex] and [tex]\hat{}R[/tex] are Hermitian, [tex]\hat{}Q[/tex][tex]\hat{}R[/tex] is only hermitian if [[tex]\hat{}Q[/tex],[tex]\hat{}R[/tex]]=0
1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex] and for a Hermitian Operator [tex]\hat{}Q[/tex]*=[tex]\hat{}Q[/tex]
Therefore does
1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex]=([tex]\int[/tex][tex]\Psi[/tex]*[tex]\hat{}Q*[/tex][tex]\Psi[/tex] )* prove that the expectaion value is real as the complex conjugate = the normal value?
attempt at 2)
AB*=(AB)transpose=BtransposeAtranspose=BA
now if A, B are hermitian this is only true if AB is also hermitian?
1) Show that the expectaion value of a Hermitian Operator is real.
2) Show that even though [tex]\hat{}Q[/tex] and [tex]\hat{}R[/tex] are Hermitian, [tex]\hat{}Q[/tex][tex]\hat{}R[/tex] is only hermitian if [[tex]\hat{}Q[/tex],[tex]\hat{}R[/tex]]=0
Homework Equations
The Attempt at a Solution
1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex] and for a Hermitian Operator [tex]\hat{}Q[/tex]*=[tex]\hat{}Q[/tex]
Therefore does
1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex]=([tex]\int[/tex][tex]\Psi[/tex]*[tex]\hat{}Q*[/tex][tex]\Psi[/tex] )* prove that the expectaion value is real as the complex conjugate = the normal value?
attempt at 2)
AB*=(AB)transpose=BtransposeAtranspose=BA
now if A, B are hermitian this is only true if AB is also hermitian?