1.) Given A and B are both hermitian operators,
a.) Prove the product will be hermitian only if and commute
b.) Prove is hermitian.
2.) Through the use of dirac notation, prove the eigenstates of a hermitian operator that belong to 2 different eigenvalues are orthogonal
A Hermitian operator is defined as an operator C such that .
So let's take any two Hermitian operators A and B and any two arbitrary kets . (Please note that are not being taken as eigenkets of A or B.)
(since [A, B] = 0)
If you don't require quite this amount of formality, then
will do just fine.
Part b) of this is easy. Just let C = A + B. Since A and B are Hermitian, so is C. Obviously C commutes with itself, so we know that is Hermitian by the theorem in 1a). Now, and C obviously commute, thus etc.
-Dan
So assume we are given a Hermitian operator A and two normalized eigenkets such that a and b are not equal.
First we show that the eigenvalues of A are real. (If you have already proven this, then just skip to the next part.)
We have
but
and
(since is real.)
Thus for all c.
So to prove the theorem consider
Again:
But the eigenvalues of a Hermitian operator are real, thus
Thus
So either b - a = 0, which is not possible according to assumption, or .
-Dan
VERY thorough and helpful as usual. I'm grateful that this site has someone good at both physics and math to help with these! The answers will very clear. Thanks.
I have an exam next week, and I plan to study this weekend, so I'll probably have more questions if you have time. Thanks again.