Hermitian operators eigenvalues

Operators that correspond to physical observables are Hermitian operators. Eigenvalues and expectation values of Hermitian operators are real numbers. Eigenfunctions of a Hermitian operator are orthogonal functions. Some pairs of operators commute, and some do not. When a complete set of functions are simultaneously eigenfunctions of a set of operators, then every pair of operators in that set commutes. 

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... 

Aeeording to the rules of quantum meehanies as we have developed them, if F is the state funetion, and (jin are the eigenfunetions of a linear, Hermitian operator. A, with eigenvalues a , A( )n = an(l)n, then we ean expand F in terms of the eomplete set of... 

J-1,J-1> and J,J-1> are eigenfunetions of the Hermitian operator J2 eorresponding to different eigenvalues, they must be orthogonal). This same proeess is then used to generate J,J-2> J-l,J-2> and (by orthogonality eonstruetion) J-2,J-2>, and so on. 

Since e is the eigenvalue of a hermitian operator, it is real hence, upon taking the hermitian adjoint of Eqs. (9-208) and (9-209) we deduce that... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

The eigenvalues of a hermitian operator are real. To prove this statement, we consider the eigenvalue equation... 

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... 

The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator 4 of one variable... 

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 characteristic-value problem - more often referred to as the eigenvalue problem - is of extreme importance in many areas of physics. Not only is it the very basis of quantum mechanics, but it is employed in many other applications. Given a Hermitian operator a, if their exists a function (or functtofts) g such that... 

It is important to note that the eigenvalues of a Hermitian operator are real. If Eq. (20) is multiplied by g and the integration is carried out over all space, the result is... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

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 ... 

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

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

The linear Hermitian operators of quantum mechanics can be divided into two categories with respect to time reversal. In the first category are those operators A which correspond to dynamical variables that are either independent of t or depend on an even power of t. Let rjjk be an eigenfunction of A with (real) eigenvalue ak. Then Qipk is also an eigenfunction of A with the same eigenvalue,... 

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