Understanding Hermitian Operators: Exploring Their Properties and Applications

In summary, the conversation discusses the use of Hermitian Operators and the question of whether (AB+BA)+ is equal to (AB)++(BA)+. The participants also question the hermiticity of A-B and the properties of the zero matrix. The justification for (AB+BA)+ is explained using the involution identity property and the fact that the zero matrix is its own Hermitian conjugate.
  • #1
danmel413
12
0
Basically I've seen some expressions involving Hermitian Operators that I can't seem to justify, that others on the internet throw around like axiomatic starting points.

(AB+BA)+ = (AB)++(BA)+? Why does this work?

Assuming A&B are hermitian, I get why we can assume A+B is hermitian, but does it follow that A-B is hermitian? Because AB is only hermitian if AB=BA which means AB-BA=0, and I'm fairly sure 0 cannot be a hermitian operator.

Thanks
 
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  • #2
About the simplest way to see it is to work out (AB)+ + (BA)+ to get B+A+ + A+B+ then involution identity property to further simplify. Also, the zero matrix is its own Hermitian conjugate; has real eigenvalues, all zero.
 

What is a Hermitian operator?

A Hermitian operator is a mathematical operator that satisfies the property of being self-adjoint, meaning that it is equal to its own adjoint. In other words, the operator and its adjoint have the same eigenvalues and eigenvectors.

Why are Hermitian operators important?

Hermitian operators have several important properties that make them useful in physics and mathematics. For example, they have real eigenvalues, which correspond to physically observable quantities. They also play a key role in quantum mechanics, where they represent observables such as position, momentum, and energy.

How do you know if an operator is Hermitian?

To determine if an operator is Hermitian, you can check if it satisfies the Hermiticity condition: A† = A, where A† is the adjoint of A. This can be done by calculating the adjoint of the operator and comparing it to the original operator.

Can a non-Hermitian operator have real eigenvalues?

No, a non-Hermitian operator cannot have real eigenvalues. This is because real eigenvalues are a result of the Hermiticity property, which is only satisfied by Hermitian operators. Non-Hermitian operators may have complex eigenvalues, which can still be physically meaningful in certain contexts.

How are Hermitian operators used in quantum mechanics?

In quantum mechanics, Hermitian operators represent observables such as position, momentum, and energy. The eigenvalues and eigenvectors of these operators correspond to the possible outcomes and states of a quantum system, respectively. In addition, Hermitian operators play a key role in the formulation of the fundamental equations of quantum mechanics, such as the Schrödinger equation.

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