Hermitian Operator: Definition & Overview

In summary, a Hermitian operator is a matrix representation in quantum mechanics that has a diagonal with real numbers and all other elements are the complex conjugates of each other. This property allows the transpose and conjugate of the operator to be equal to itself. It represents a measurable quantity in quantum mechanics and is used in the measurement procedure of a quantum state. A key feature is that its eigenvalues are always real numbers.
  • #1
sm09
9
0
what is it?
 
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  • #2
an operator that is hermitian!

in a matrix representation, it means that the diagonal Mii is real, and any Mij is the complex conjugate of Mji. this gives the hermitian conjugate of M (transpose and conjugate) is itself.or... google Hermitian

it represents a measurable quantity in QM.

ps.
am i right?
 
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  • #3
This question is well enough described in a any QM textbook. You do not make much effort to look there. In simple, Hermitian operator is the observable represetation, or if not rigorously speaking, it reflects the measurement procedure in a some quantum state. For example, let we have an spin-up directed electron state [itex]\left | + \right>[/itex]. Measurement of the z-directed spin by [itex]\hat S_z[/itex] in this state is reflected in the equality [itex]\hat S_z \left | + \right> = +\hbar/2 \left | + \right> [/itex]. This mean the result you will get is [itex]+\hbar/2[/itex]. A key feature of a Hermitian operator is real numbers of their eigenvalues.
 
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FAQ: Hermitian Operator: Definition & Overview

1. What is a Hermitian operator?

A Hermitian operator is an operator in quantum mechanics that has the property of being self-adjoint, meaning that the operator is equal to its own adjoint. This means that the operator is symmetric with respect to a certain inner product.

2. What is the importance of Hermitian operators in quantum mechanics?

Hermitian operators play a critical role in quantum mechanics because they represent physical observables, such as energy, momentum, and angular momentum. These operators have real eigenvalues, making them essential for making predictions about the behavior of quantum systems.

3. How is a Hermitian operator related to a quantum state?

A Hermitian operator acts on a quantum state to produce a new quantum state. The eigenvalues of the operator correspond to the possible outcomes of a measurement of that observable on the quantum state.

4. Can a non-Hermitian operator be used in quantum mechanics?

Yes, non-Hermitian operators can be used in quantum mechanics, but they do not represent physical observables. Instead, they are used to describe non-unitary processes, such as decay, in quantum systems.

5. How is the Hermitian property of an operator related to its eigenvalues?

The Hermitian property of an operator ensures that its eigenvalues are real. This is important because the eigenvalues of a Hermitian operator correspond to the outcomes of a measurement, and real eigenvalues represent physically meaningful results.

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