Hermitian operators completeness

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... 

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... 

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 ... 

The functions (2.50) are called basis functions The matrices F, G,. .. are called matrix representatives of the operators F, G,. .. in the

specific form of the matrix representation of a set of operators depends on the basis chosen. Equation (2.53) shows that the effect of the operator G on the basis functions is determined by the matrix elements GkJ. Since an arbitrary well-behaved function can be expanded using the complete set (2.50), knowledge of the matrix G allows one to determine the effect of the operator G on an arbitrary function. Thus, knowledge of the square matrix G is fully equivalent to knowledge of the corresponding operator G. Since G is a Hermitian operator, its matrix elements satisfy Gij = (GJi). Hence the matrix G representing G is a Hermitian matrix (Section 2.1). 

Let us assume that zero-order Hamiltonian Ho is a Hermitian operator having a complete set of eigenfunctions... 

Because the eigenfunctions of any linear Hermitian operator form a complete set, in the sense that any arbitrary function that satisfies appropriate boundary conditions can be expressed as a linear superposition of this set, eq. (3) holds also for such arbitrary functions. Therefore,... 

The prime on the delta function indicates differentiation with respect to the variable given in the subscript. The prime on the coordinate is just another coordinate value, different from the coordinate without a prime. This prime should not be confused with the prime on the delta function. The operator corresponding to a dependent variable ui(q,p) is given by a Hermitian operator Cl(q,p) = u> q —> q,p —> p). At the end of this section the complete expression for the relations with all coordinates is given. For brevity of notation, we usually only include the coordinate of interest, as in Eqs(F.8) and (F.9). 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

This is simply the completeness relation for the eigenvectors of an Hermitian operator H. We introduce formally the resolvent of the operator H as a function of a complex variable z ... 

The operator serves to extract information from the state vectors and may correspond to any physical observable such as position, momentum, angular momentum (spin or orbital) or energy. The state vector itself is not observable. In most systems the number of eigenfunctions is infinite and it is an axiom of quantum mechanics that the set of all eigenfunctions of a Hermitian operator forms a complete set. These eigenfunctions define a Hilbert space on which the operator acts. 

Postulate 4. If G is any linear Hermitian operator that represents a physical observable, then the eigenfunctions 4>, of the eigenvalue equation above form a complete set. 

For example, suppose that after a full Cl calculation with a complete N-electron set, we have obtained the exact wavefunction, F. The formal rela-fions regarding energy are given by Eqs. (3-7). Now consider a one-electron Hermitian operator, representing a property, say 0(0, where i are electron... 

Postulate 1 Correspondence Postulate. Some linear Hermitian operators on Hilbert space which have complete orthonormal sets of eigenvectors (eigenfunctions) correspond to physical observables of a system. If operator P corresponds to observable P, then operator F(P), where F is a function, corresponds to observable F(P). 

The Hermitian operator L acting on this Hilbert space of square integrable functions possesses a complete set of eigendistributions pa(p,q) with associated eigenvalue Aa. For an N degree of freedom quasiperiodic system with Hamiltonian H(I), action-angle variables (1,0) and associated frequencies oj(I) = 8H/81, the eigenvalue problem Lpx = Xpx becomes... 

Mathematiceilly, these solutions are the eigenvalues (tj) and eigenfunctions Hermitian operator (h) and, as such, they have several important properties. One of these properties is that they are complete, any function of ordinary three-space (the coordinates of a single electron) with sufficiently similar boundary conditions can be expanded as a linear combination of these functions. That is, any function /(f) can be written exactly as... 

We now postulate that the set of eigenfunctions of any Hermitian operator that represents a physical quantity forms a complete set. (Completeness of the eigenfunctions can be proved in the one-dimensional case and in certain multidimensional cases, but must be postulated for some multidimensional systems.) Thus, any well-behaved function that satisfies the same boundary conditions as the set of eigenfunctions can be expanded according to (7.39). Equation (7.29) is an example of (7.39). 

How about using the hydrogen-atom bound-state wave functions to expand an arbitrary function /(r, 6,4>)1 The answer is that these functions do not form a complete set, and we cannot expand / using them. To have a complete set, we must use all the eigenfunctions of a particular Hermitian operator. In addition to the bound-state... 

The contents of Sections 7.2 and 7.3 can be summarized by the statement that the eigenfunctions of a Hermitian operator form a complete, orthonormal set, and the eigenvalues are real... 

THEOREM 5. If the Hermitian operators A and B commute, we can select a common complete set of eigenfunctions for them. 

Extension of the above proofs to the case of more than two operators shows that for a set of Hermitian operators A,B,C,... there exists a common complete set of eigenfunctions if and only if every operator commutes with every other operator. 

We postulated in Section 7.3 that the eigenfunctions of any Hermitian operator that represents a physically observable property form a complete set. Since the g/s form a complete set, we can expand the state function as... 

This postulate is more a mathematical than a physical postulate. Since there is no mathematical proof (except in various special cases) of the completeness of the eigenfunctions of a linear Hermitian operator, we must assume completeness. Postulate 4 allows us to expand the wave function for any state as a superposition of the orthonormal eigenfunctions of any quantum-mechanical operator ... 

To find we expand it in terms of the complete, orthonormal set of unperturbed eigenfunctions of the Hermitian operator H° ... 

The eigenfunctions a and p of the Hermitian operator form a complete, orthonormal set, and any one-electron spin function can be written as Ci -I- C2P. We saw in Section 7.10 that functions can be represented by column vectors and operators by square matrices. For the representation that uses a and p as the basis functions, (a) write down the column vectors that correspond to the functions a, 8, and c a -I- C2 (b) use the results of Section 10.10 to show... 

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