Hermitian operators eigenfunctions

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

However, in order to give an unambiguous answer to the question of how one is to calculate the probability of finding a Klein-Gordon particle at some point x at time t, we must first find a hermitian operator that can properly be called a position operator, and secondly find its eigenfunctions. It is somewhat easier to determine the latter since these should correspond to states wherein the particle is localized at a given point in space at a given time. Now the natural requirements to impose on localized states are ... 

If Ip I and ip2 ( fe eigenfunctions of a hermitian operator A with different eigenvalues a and ai, then ipi and p2 are orthogonal. To prove this theorem, we begin with the integral... 

Consider a set of orthonormal eigenfunctions 0, of a hermitian operator. Any arbitrary function / of the same variables as 0, defined over the same range of these variables may be expanded in terms of the members of set 0,-... 

Suppose the members of a complete set of functions tpi are simultaneously eigenfunctions of two hermitian operators A and B with eigenvalues a,- and j3i, respectively... 

We have shown in Section 3.5 that commuting hermitian operators have simultaneous eigenfunctions and, therefore, that the physical quantities associated with those operators can be observed simultaneously. On the other hand, if the hermitian operators A and B do not commute, then the physical observables A and B cannot both be precisely determined at the same time. We begin by demonstrating this conclusion. 

A hermitian operator A has only three normalized eigenfunctions tp, ip2,... 

Because of these properties of Hermitian functions it is accepted as a basic postulate of wave mechanics that operators which represent physical quantities or observables must be Hermitian. The normalized eigenfunctions of a Hermitian operator constitute an orthonormal set, i.e. 

This statement follows immediately from the elementary theorem that the eigenfunctions of a Hermitian operator are (or can be taken to be) mutually orthogonal. 

Because He is a Hermitian operator in r-space, its eigenfunctions (rlR) obey... 

P and D functions because S, 3P, and D are eigenfunctions of the hermitian operator L2 having different eigenvalues. The state that is normalized and is a combination of poCCpoPl, Ip.iapiPl, and lpi0cp i[3l is given as follows ... 

IJ-1,J-1> and IJ,J-1> are eigenfunctions of the Hermitian operator J2 corresponding to different eigenvalues, they must be orthogonal). This same process is then used to generate IJ,J-2> IJ-l,J-2> and (by orthogonality construction) IJ-2,J-2>, and so on. 

Since (°) is a Hermitian operator and 4 is an eigenfunction of it, the first and third integrals are equal and cancel, leaving an expression for the first-order correction to the energy,... 

The eigenfunctions Hermitian operator form a complete set. By this we mean that any well-behaved function / that satisfies the same boundary conditions as the (jp/s can be expanded as... 

Let a system be in the state T at some instant of time. Let orthonormal eigenfunctions of the Hermitian operator G that corresponds to the physical property G ... 

Prove that eigenfunctions of a Hermitian operator that correspond to different eigenvalues are orthogonal. 

When the integral in brackets in (7.6) vanishes, the electronic transition is forbidden. For example, since and )//, are both eigenfunctions of the Hermitian operator 5 (provided spin-orbit interaction is small), and since S2 commutes with del, we conclude [Equation (1.51)1 that electronic transitions with a change in S are forbidden (just as in atoms) ... 

Mj — + i and - respectively. (The same symbols were used previously to designate electronic spin functions with ms = the context will indicate whether a and ft mean electronic or nuclear spin functions.) Since the operator (8.42) is Hermitian, its eigenfunctions (8.43) form a complete set of well-behaved functions for the problem of two nuclei with spin Moreover, since there are only a finite number of functions in this complete set, the secular determinant in (2.77) is of finite order and easy to deal with. The complete set is then... 

The following properties of Hermitian operators follow from the definition (1.23). The eigenvalues of a Hermitian operator are real. Two eigenfunctions of a Hermitian operator that correspond to different eigenvalues... 

The eigenfunctions Y have been normalized moreover, being eigenfunctions of the Hermitian operators L2 and Lz, they are orthogonal hence... 

The electronic wave functions i//, and are eigenfunctions of the Hermitian operator H x with different eigenvalues hence they are orthog-... 

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