self-adjoint operator
Hermitian operator

See the article on self-adjoint operators for a full treatment.

The class of self-adjoint operators is especially important in mathematical physics.

Self-adjoint operators are used in functional analysis and quantum mechanics.

A is given by a densely defined self-adjoint operator on H.

This is a positive self-adjoint operator so its principal minors do not vanish.

In general, spectral theorem for self-adjoint operators may take several equivalent forms.

In other words, a compact self-adjoint operator can be unitarily diagonalized.

There is also a spectral theorem for self-adjoint operators that applies in these cases.

In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function.

Mathematically, is a self-adjoint operator on a Hilbert space.

This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian.

The proofs use the following results about self-adjoint operators:

There is a close relation between the symmetric forms in a Hilbert space and self-adjoint operators.

This reduction uses the Cayley transform for self-adjoint operators which is defined in the next section.

Moreover, L gives rise to a self-adjoint operator.

As we know in the loop representation a self-adjoint operator generating spatial diffeomorphims.

The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space.

In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators.

Every self-adjoint operator is densely defined, closed and symmetric.

We would like there to be a unique, positive, self-adjoint operator whose matrix elements reproduce .

It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator.

However, for self-adjoint operators, all the above definitions for the essential spectrum coincide.

See self-adjoint operator for a detailed discussion.

Self-adjoint operators. (3 lectures) Partial differential equations: separation of variables.

Further analysis shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded.

When two Hermitian operators commute, a common set of eigenfunctions exists.

In general the eigenstates and of two different Hermitian operators and are not the same.

But because this is the square of a Hermitian operator, the right hand side coefficient must be positive for all .

Since H is a Hermitian operator, the energy is always a real number.

This implies that infinitesimal transformations are transformed with a Hermitian operator.

For the Hermitian operator we define the repeated commutator by:

The real functions correspond to the Hermitian operators.

Mathematically, it is represented by a Hermitian operator.

Hermitian operators then follow for infinitesimal transformations of a classical polarization state.

The definition of a Hermitian operator is:

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.

Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.

The momentum operator is always a Hermitian operator when it acts on physical (in particular, normalizable) quantum states.

Important properties of Hermitian operators include:

Hermitian operators' eigenvalues are real.

Creation and Annihilation operators are not Hermitian operators.

Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.

Number operators are Hermitian operators.

Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state in time naturally emerge.

The antisymmetrizer commutes with any observable (Hermitian operator corresponding to a physical-observable-quantity)

All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state obtaining the completeness relation:

'P' is both Hermitian operator and Unitary operator.

An arbitrary Hermitian operator on the Hilbert space of the system need not correspond to something which is a physically measurable observable.

In quantum mechanics, two eigenstates of a Hermitian operator, and , are orthogonal if they correspond to different eigenvalues.

Because of the sign-change under rotations by 2π, Hermitian operators transforming as spin 1/2, 3/2 etc., cannot be observables.