Is the following operator hermitian? C|Phi> = |Phi>*

In summary: However, the operator in question is defined differently and it is not clear what the eigen values are. They mention that for a real Phi, the only eigen value is one with infinite degeneracy. Hurkyl suggests writing down the complete definition of a hermitian operator and trying to prove that the operator in question satisfies each of the properties. They also give a hint that C is not linear. In summary, Gabriel is asking if the operator C, which takes a state function and gives its complex conjugate, is hermitian and is seeking help in determining its eigen values
  • #1
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-hey everyone,

this one might be a little too math based for this forum, but I ran across it studying for one of my quantum exams and it seemed like an interesting problem. Haven't figured it out completely.
We all know hermitian operators play a central role in quantum and so being able to tell if an operator is hermitian or not is important. Is the following operator hermitian?

C|Phi> = |Phi>*

(Takes a state function and gives the complex conjugate)

All the methods to show that an operator is hermitian (that I've seen) rely on the eigen-value equation

A|Phi> = a|Phi>

But the way this operator is defined, I'm not sure what the eigen values are (Or if it is an eigen value problem since the original function is not returned just its conjugate).

All I got was that for a Phi that is real, the only eigen value is one and it has infinite degeneracy. Not as exciting as faster than light travel, but dig in if you'd like...
 
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  • #2
Have you tried writing down the complete definition of a hermitian operator and try to prove that C satisfies each of the properties.
 
Last edited:
  • #3
Below is another hint. (don't peek until you've tried the above!)















C is not linear.
 
  • #4
Thanks

Thanks Hurkyl,

Appreciate the push in the right direction.

Gabriel
 

1. What is a Hermitian operator?

A Hermitian operator is a linear operator that satisfies the condition C|Phi> = |Phi>*, where C is the operator, |Phi> is a quantum state, and * denotes the complex conjugate. This condition is also known as the Hermitian adjoint or Hermitian conjugate.

2. What does it mean for an operator to be Hermitian?

A Hermitian operator has the property that its eigenvalues (possible outcomes of a measurement of the operator) are real numbers, and its eigenvectors (states where the operator yields a definite value) are orthogonal (perpendicular) to each other. In other words, a Hermitian operator represents an observable physical quantity in quantum mechanics.

3. How do you determine if an operator is Hermitian?

To determine if an operator is Hermitian, you can apply the Hermitian condition C|Phi> = |Phi>* to the operator and its adjoint. If the result is equal, the operator is Hermitian. Another way is to check if the operator satisfies the Hermiticity property C^dag = C, where C^dag is the Hermitian adjoint of the operator.

4. What are some examples of Hermitian operators?

Some examples of Hermitian operators include the position operator, momentum operator, and energy operator in quantum mechanics. These operators represent physical observables and have real eigenvalues and orthogonal eigenvectors.

5. What are the implications of an operator being Hermitian?

The Hermiticity of an operator has important implications in quantum mechanics. It guarantees that the operator has real eigenvalues, which correspond to the possible outcomes of a measurement. It also ensures that the operator is unitary, meaning it preserves the normalization of quantum states. This property is crucial in the study of quantum systems and their behavior.

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