Hermitian Operators and Eigenvalues

In summary, C is an operator that changes a function to its complex conjugate. It is not hermitian, as shown by the fact that the integrals (psi*)(psi*) and (psi)(psi) are not equal. However, eigenvalues can still be found for non-hermitian operators, with a modulus of 1. The completeness and orthogonality of eigenfunctions in this case is unclear, as it depends on the form of psi and the chosen alpha constant.
  • #1
FlagellumDei
7
0

Homework Statement


C is an operator that changes a function to its complex conjugate
a) Determine whether C is hermitian or not
b) Find the eigenvalues of C
c) Determine if eigenfunctions form a complete set and have orthogonality.
d) Why is the expected value of a squared hermitian operator always positive?


Homework Equations


If C is hermitian, then <C(psi1)\(psi2)>=<(psi1)\C(psi2)>
For eigenvalues: C(psi)=a(psi), where a is a constant

The Attempt at a Solution


I don't know even if I'm doing wrong but using the condition for hermiticity described above I get the integrals for the products (psi*)(psi*) and (psi)(psi) are equal. (Being the terms with "*" the complex conjugate)
For d), if the operator is squared, then the constant is squared too, but how do I know "a" is not a complex constant?

Ok, I guess I was a little desperate ad didn't check my results as I had to, from the beggining.
Simply substituting C(psi) for (psi*) and (psi*) for C(psi) makes evident C is hermitian.
Then, there's a theorem stating eigenvalues of hermitian operators are real, because average values are always real numbers. Squaring the constant makes for a positive number. Yet I'm not sure how to get the eigenvalues of b). How "far" can I go with this information?
 
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  • #2
can you run through the proof of why C is hermitian please?
 
  • #3
I'm sorry, it's not hermitian. Got confused using bra-kets to prove it.

C(psi)=(psi*)

If it were hermitian then the integral (psi*)C(psi) would be equal to that of (psi)(C(psi*)), but substituting C(psi) in the integrals yields (psi*)(psi*) is equal to (psi)(psi). I guess it would be an hermitian operator in case (psi) is a real function, but that would be a specific case. So, in general it's not hermitian.

Sorry again for spreading my own confusion.
 
  • #4
yes. i also found it to be non-hermitian. but I am pretty sure its meant to be because otherwise the rest of the question doesn't make any sense as its all based around the properties of hermitian operators
 
  • #5
Well, I think you can still get eigenvalues for non-hermitian operators. If you calculate the modulus of the eigenvalue and the eigenfunction (multiplying by its complex conjugate) you'll find you need the eigenvalue's modulus to be 1. So you can propose it to be exp(-i*alpha). It should make sense, because the eigenvalues of non-hermitian operators are not necessarily real numbers, and its magnitude is 1. So this alpha constant depends on the form of psi, and it should be chosen so that the operator turns psi into psi*.

I can't figure out the answer about the complete set and their orthogonality. If C were hermitian there's no problem, the functions of different states are orthogonal with each other. But what can you tell for this case?
 
  • #6
/Cpsi>=Ipsi*>
multiply by <psi/

<psi/Cpsi>=<psi/psi*> ------ 1
<psi/Cpsi>=<psi/psi>* ------2

can we say from 1 & 2 that it is Hermitain
 

Related to Hermitian Operators and Eigenvalues

1. What is a Hermitian operator?

A Hermitian operator is a linear operator on a complex vector space that is equal to its own conjugate transpose. In other words, if A is a Hermitian operator, then A* = A, where A* is the conjugate transpose of A.

2. What are the properties of Hermitian operators?

Some key properties of Hermitian operators include the fact that their eigenvalues are always real numbers, they have orthogonal eigenvectors, and they can be diagonalized by a unitary matrix.

3. How do you find the eigenvalues of a Hermitian operator?

The eigenvalues of a Hermitian operator can be found by solving the characteristic equation det(A-λI) = 0, where A is the matrix representation of the operator and λ is the eigenvalue. This will result in a polynomial equation that can be solved for the eigenvalues.

4. What is the significance of Hermitian operators in quantum mechanics?

Hermitian operators play a crucial role in quantum mechanics because they represent observables in the physical world. In quantum mechanics, the eigenvalues of Hermitian operators correspond to the possible outcomes of a measurement of the observable represented by the operator.

5. Can a non-square matrix be a Hermitian operator?

No, a non-square matrix cannot be a Hermitian operator. This is because the definition of a Hermitian operator requires it to be equal to its own conjugate transpose, which can only be satisfied for square matrices.

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