Position operators

Here is the position operator of atom j, or, if the correlation function is calculated classically as in an MD simulation, is a position vector N is the number of scatterers (i.e., H atoms) and the angular brackets denote an ensemble average. Note that in Eq. (3) we left out a factor equal to the square of the scattering length. This is convenient in the case of a single dominant scatterer because it gives 7(Q, 0) = 1 and 6 u,c(Q, CO) normalized to unity. 

If the P/Q operators are number conserving operators, the propagator is called a Polarization Propagator (PP). It may be viewed as the response of property P to perturbation Q. For the case where P = Q = r (the position operator), the propagator describes the response of the dipole moment (4fo r 4fo) to a linear field F = Ft. 

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 ... 

We next turn our attention to the problem of finding the position operator q, of which the localized state (t) is an eigenfunction with eigenvalue y. That this operator is not given by q = Vk in the momentum representation becomes dear upon noting that the operator... 

The above properties justify calling q the position operator for a relativistic spin 0, mass m particle. 

The correct position operator and localized amplitudes were worked out by various investigators,13 in particular Newton and Wigner14 and Foldy and Wouthuysen.15 The latter authors indicated that the amplitude... 

The components of the position operator, therefore, commute with one another furthermore they are canonically conjugate to the momentum operators... 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

In the Dirac representation, the position operator has the following representation... 

The formalism can be carried farther to discuss the particle observables and also the transformation properties of the s and of the scalar product under Lorentz transformations. Since in our subsequent discussion we shall be primarily interested in the covariant amplitudes describing the photon, we shall not here carry out these considerations. We only mention that a position operator q having the properties that ... 

Foldy, L, L., 497,498,536,539 Foldy-Wouthuysen representation, 537 "polarization operator in, 538 position operator in, 537 Ford, L. R., 259 Four-color problem, 256 Fourier transforms of Schrodinger operators, 564... 

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

We note in passing that Lemma 1 and Theorem 1 guarantee the existence of an inverse operator defined only on TZ A), the range of A, which is not obliged to coincide with H. If the range of an operator A happens to be the entire space H, TZ(A) = H, then the conditions of Lemma 1 or Theorem 1 ensure the existence of an operator A with T>[A ) = H. In particular, a positive operator A with the range TZ A) = H possesses an inverse with V[A ) = H, since the condition Ax, x) > 0 for all x Q implies that Ax yf 0 for x yf 0 and Lemma 1 applies equally well to such a setting. 

Indeed, the norm of a self-adjoint positive operator in a finite-dimensional space 17, is equal to its greatest eigenvalue j A = A,v-i- case, in... 

Suppose that the inverse operators and A exist. Moreover, we assume that A and A are self-adjoint positive operators. Substitutions of u = A f and u = A f into (15) yield... 

Theorem Let u be a solution to equation (11) and u be a solution to equation (14), where A, A and Aq are self-adjoint positive operators for which the inverse operators exist. If condition (18) and the inequality A > CjTo, Cj > 0 hold, then the estimates are valid ... 

When, in addition, A is a positive operator, these are necessary and sufficient for the p-stability in the space Ha-... 

The first analysis is connected with the case when A is a constant self-adjoint positive operator A = A > 0. As we have shown in Section 2, a necessary and sufficient condition for the stability of the weighted scheme (47) with respect to the initial data is... 

Theorem 8 Let A be a self-adjoint positive operator independent oft = nr A = A > 0. Then for the weighted scheme (47) estimate (37) is valid for... 

Summing the preceding over A = 0,1,2,.., leads to estimate (50). Lemma 5 Let A be a positive operator for which the inequality... 

Theorem 11 Let A = A t) be a positive operator and condition (55) hold. Then for scheme (46) with a > the a priori estimate... 

In our basic account A and R are taken to be constant self-adjoint positive operators and B refers to a non-self-adjoint nonnegative operator ... 

Applying Theorem 3 to scheme (65) with a constant positive operator A, it is plain to derive under conditions (68) the estimate... 

The effect of time reversal operator T is to reverse the linear momentum (L) and the angular momentum (J), leaving the position operator unchanged. Thus, by definition,... 

If we consider only the x direction, we see that a position operator equal to x multiplied by the charge e is required to move the electron. The operator then has the form... 

Before any slit operation check, write down, or save the old motor positions Operation of slits can be useful to change the beam intensity (instead of operating absorbers). Imperfect thermal stabilization of mirrors and monochromators can be compensated by proper slit operation. Before such operation is undertaken, it should be made sure that the instrument is close to thermal equilibrium. In particular after opening the main beam shutter for the first time, it may be indicated to wait for several hours. Otherwise the operator will have to follow the thermal expansion continuously. This bears the risk to destroy the adjustment or even the detector. 

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