Casimir formula

The magnetic dipole interaction and the electric quadrupole interaction, in an atom with nuclear spin I, will cause a level with angular momentum J to split into different energy levels denoted by the total angular momentum F (F=I+J), according to the Casimir formula [11],... 

Keywords Casimir effect, Fermi sea, scattering problem, Krein formula... 

For more complicated geometries, the computations become more and more involved as it is the case for the ordinary electromagnetic Casimir effect. However, Casimir calculations of a finite number of immersed nonoverlapping spherical voids or rods, i.e. spheres and cylinders in 3 dimensions or disks in 2 dimensions, are still doable. In fact, these calculations simplify because of Krein s trace formula (Krein, 2004 Beth and Uhlenbeck, 1937)... 

Formula (5.29) is the special case (odd k values) of the Casimir operator of the more general so-called unitary unimodular group SU21+1, ie. 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

The method of CFP is an elegant tool for the construction of wave functions of many-electron systems and the establishment of expressions for matrix elements of operators corresponding to physical quantities. Its major drawback is the need for numerical tables of CFP, normally computed by the recurrence method, and the presence in the matrix elements of multiple sums with respect to quantum numbers of states that are not involved directly in the physical problem under consideration. An essential breakthrough in this respect may be finding algebraic expressions for the CFP and for the matrix elements of the operators of physical quantities. For the latter, in a number of special cases, this can be done using the eigenvalues of the Casimir operators [90], however, it would be better to have sufficiently simple but universal formulas for the CFP themselves. 

Combining the above formulas, we can work out analytical expressions for the sums of the scalar products (Tk-Tk) which are generally obtained using the Casimir operators of the unitary group Ify+i and the symplectic group Sp2j+i... 

All the expressions derived so far are suitable for subshells with any value of the quantum number j. For j to be concretized, we can obtain additional relations from (23.11), which makes it possible to find, in certain cases, similar formulas for individual terms of the sums. With the scalar products (rk Tk j such expressions can be found [18] using the Casimir operators of appropriate groups (see Chapters 5, 15 and 18). Since these scalar products at odd k differ from the irreducible tensorial products j(Kk) only by a factor, we shall list the results that have been obtained for the first time (the number over the equality sign indicates for which 7-shell a given equality occurs) ... 

This paper by Ya.B. is the first published work to study the interaction of an isolated atom with the surface of a metal. The author arrived at the conclusion that the interaction energy decreases with distance as 1 Jr2. Later, H. Casimir and D. Polder1 considered this problem again using a different method, and arrived at a different result—the interaction energy is proportional to 1/r3 at distances r c/cj0 (where w0 is some characteristic frequency of the absorption spectrum of the atom and metal), and U 1/r4 at distances much greater than c/u>0. (E. M. Lifshitz2 arrived at the same result.) The difference between these results and those in Ya.B. s paper is related, however, to differences in the approximations made and in the areas of applicability of the formulas obtained. 

The dispersion nonadditivity Eib arises from the coupling of intermonomer pah-correlations in subsystems XY and YZ via the intermolecular interaction operator Vzx. This contribution can be expressed as a generalized Casimir-Polder formula,... 

The attractive Casimir force between two plates of area A can be calculated approximately using the formula F = (phcA)/(480r4), where h is Planck constant, c is the speed of light, and r is the distance between the plates. 

An alternative, yet equivalent, expression for the dipole dispersion constant is the Casimir-Polder formula (Casimir and Polder, 1948) ... 

Note that the coefficients P 2+ /2 dePend on the parameter p only through the dependence of the variable K on the product px [see Eq. (79) or (81)]. Thus, to study the squeezing properties of the field created as a result of the NSCE (non-stationary Casimir effect) it is sufficient to consider the most important special case of the parametric resonance at the double fundamental frequency 2g>i (i.e., p = 2), since the formulas for p > 2 can be obtained by a simple rescaling of the slow time (for the principal modes). In this case, only the odd modes can be excited from the vacuum, and they do exhibit some squeezing. 

The total energy (161) of the field inside the cavity (above the initial Casimir level) can be obtained by integrating the density W(x) (166) over x. The contribution of the vacuum [function Fq in (167)] and diagonal terms (given by the partial sum in (167) over n = k) can be calculated with the aid of the formula... 

Casimir s formula (17) applies equally well if is replaced by qj, the specific rotor-dependent EFG. A thorough derivation of all rotor-dependent expressions for the EFG can be found in [64,65] and need... 

We shall henceforth refer to (58) and (59) as of generalizations of the London formula [55] and the Casimir-Polder formula [56], respectively. The latter, in fact, refer to Cg dispersion coefficients for atoms expressed in terms of static (London) or dynamic (Casimir-Polder) polarizabilities, whereas (58) and (59) describe, in a completely general way, non-expanded dispersion between atoms or molecules. (59) expresses the coupling of two electrostatic interactions (l/ri2 and l/r ) involving four space points in the two molecules, with a strength factor which depends on how readily density fluctuations propagate between r and on A, r 2 and F2 on B (Fig. 4). 

It is easy to verify that with the help of the Casimir-Pol-der formula... 

At larger distances, the leading term obviously dominates and one recovers the formula derived originally by Casimir and Polder [55]. The force of interactions between atoms ean be obtained directly from the above formulas by a simple differentiation in respect to r. 

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