Properties of Hermitian Operators: Show Real Expectation Value & Commutativity

[yakattack]

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 \hat{}Q and \hat{}R are Hermitian, \hat{}Q\hat{}R is only hermitian if [\hat{}Q,\hat{}R]=0

Homework Equations

The Attempt at a Solution

1) Expectation Value <\hat{}Q>= \int\Psi*\hat{}Q\Psi and for a Hermitian Operator \hat{}Q*=\hat{}Q
Therefore does
1) Expectation Value <\hat{}Q>= \int\Psi*\hat{}Q\Psi=(\int\Psi*\hat{}Q*\Psi )* 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?

 

[dextercioby]

1. Use this TEX parse \hat{Q}.
2. For a vector \psi [/tex], the expectation value of the linear operator A is \langle \psi, A\psi. If A is hermitean, can you show that the exp. value is real ?<br /> <br /> The 3-rd point is a little bit involved.

 

[beerchop]

Another question about Hermitians..

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

b

 

[Brad Barker]

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

b

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