Operators Hermitian

Two or more properties F,G, J whose eorresponding Hermitian operators F, G, J eommute... 

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... 

In quantum meehanies, physieally measurable quantities are represented by hermitian operators. Sueh operators R have matrix representations, in any basis spanning the spaee of funetions on whieh the R aet, that are hermitian ... 

The equality of the first and third terms expresses the so-ealled turn-over rule hermitian operators ean aet on the funetion to their right or, equivalently, on the funetion to their left. 

V. Operators That Commute and the Experimental Implieations Two hermitian operators that eommute... 

These two operators are not Hermitian operators (although Jx and Jy are), but they are adjoints of one another ... 

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. 

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 ... 

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... 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

In order to construct solutions corresponding to a particle in motion, consider the hermitian operators... 

Due to the 8-function commutation rules that the cA(k) operators satisfy, Eqs. (9-636) and (9-637), we can interpret the hermitian operators... 

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... 

Heisenberg-type descriptions for two observers, 667, 668 Heitler, W., 723 Helicity operator, 529 Hermitian operator, 393 Hermitian operator Q describing electric charge properties of particles, 513... 

Hermitian operators for electric and magnetic field intensities, 561 Herzfeld, C. M., 768 Hessenberg form, 73 Hessenberg method, 75 Heteroperiodic oscillation, 372 Hilbert space abstract, 426... 

The product of two hermitian operators may or may not be hermitian. Consider the product AB where A and B are separately hermitian with respect to a set of functions xpi, so that... 

By setting B equal to A in the product AB in equation (3.11), we see that the square of a hermitian operator is hermitian. This result can be generalized to... 

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