Why hermitian operator

But because is Hermitian, the left side vanishes. If eigenvalues and are not degenerate, then , so the eigenfunctions are orthogonal. If the eigenvalues are degenerate, the eigenfunctions are not necessarily orthogonal.

Now take. The integral cannot vanish unless , so we have and the eigenvalues are real. For a Hermitian operator ,. Given Hermitian operators and ,.

Because, for a Hermitian operator with eigenvalue ,. Therefore, either or. But iff , so. This means that , namely that Hermitian operators produce real expectation values. Every observable must therefore have a corresponding Hermitian operator.

For i. Define the adjoint operator also called the Hermitian conjugate operator by. Furthermore, given two Hermitian operators and ,. Given two Hermitian operators and ,. Given an arbitrary operator ,. Arfken, G. Orlando, FL: Academic Press, pp. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end.

Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions. What other properties do Hermitian matrices have, which for example triangular matrices lack, that makes them desirable for this purpose? See also this Phys. SE post. If you want to see something different, there are actually a few articles by Carl Bender developing quantum mechanics formulated with parity-time symmetric operators.

He shows that some Hamiltonians are not Hermitian, yet they have real eigenvalues and seem to represent valid physical systems. If you think about it, the requirement that your operator is parity-time symmetric makes more sense physically than hermiticity.

In a later article, his quantum mechanics approach was proven to be equivalent to the standard one where operators are hermitian. To give an answer that is a little more general than what you're asking I can think of three reasons for having hermitian operators in quantum theory:. And unitary Lie group representations come with a lie algebra of hermitian operators.

This structure if efficiently represented by a hermitian operator that comes with an eigenstructure that matches these requirements precisely. For ensembles this can be seen from the construction as a convex sum of projectors, which are necessarily hermitian. For subsystem states it comes out of tracing a projector over tensor factor spaces. This is related to point 2 because processes like decoherence connect measurement outcomes with density operators.

In particular, this means:. Where the summation is taken over all the eigenstates of an operator. That shows that the matrices must be normal. That they are chosen to be Hermitian is non-essential, but useful, as has already been discussed.

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