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 [itex] \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 ?

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...