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afrano
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Someone told me that any operator can be decomposed in a Hermitian and Antihermitian part. Is this true? How? By addition?
A Hermitian operator is a mathematical operator that is equal to its own adjoint. In other words, the operator is symmetric with respect to the inner product in the space in which it operates. This means that the operator is self-adjoint, meaning it is both Hermitian and anti-Hermitian at the same time.
The Hermitian part of an operator is defined as half of the sum of the operator and its adjoint, divided by two. This means that it is the average of the operator and its adjoint, which results in a Hermitian operator.
In quantum mechanics, Hermitian operators represent physical observables, such as position, momentum, and energy. This is because the eigenvalues of a Hermitian operator are always real, which corresponds to measurable quantities in the physical world.
The anti-Hermitian part of an operator is defined as half of the difference between the operator and its adjoint, divided by two. This means that the anti-Hermitian part is the imaginary component of the operator, while the Hermitian part is the real component. Together, they make up the full operator.
Yes, an operator can be both Hermitian and anti-Hermitian at the same time. This is known as a self-adjoint operator, and it is a special case in which the Hermitian and anti-Hermitian parts are equal to each other. Self-adjoint operators are important in quantum mechanics because they represent observables that are both real and imaginary at the same time.