Hermitian Operators

  • Thread starter yakattack
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  • #1
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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


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?
 

Answers and Replies

  • #2
dextercioby
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1. Use this TEX parse [tex] \hat{Q} [/tex].
2. For a vector [itex] \psi [/tex], the expectation value of the linear operator A is [itex] \langle \psi, A\psi [/itex]. If A is hermitean, can you show that the exp. value is real ?

The 3-rd point is a little bit involved.
 
  • #3
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Another question about Hermitians..

If A and B are Hermitian, then is AB also hermitian?

b
 
  • #4
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If A and B are Hermitian, then is AB also hermitian?

b

think of what the hermitian conjugate is for AB...
 

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