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Multipole Expansions in Radiation Theory of Quantum Systems Multipole expansions in quantum radiation theory M. Ya. Agre National University of “Kyiv-Mohyla Academy”, 04655 Kyiv, Ukraine E-mail: [email protected] With the help of mathematical technique of irreducible tensors the multipole expansion for the probability amplitude of spontaneous radiation of a quantum system is derived. It is shown that the found series represents the total radiation amplitude in the form of the sum of radiation amplitudes of electric and magnetic 2l-pole (l = 1, 2, 3,…) photons. All information about the radiating system is contained in the coefficients of the series which are the irreducible tensors being determined by the current density of transition. The expansion can be used for solving different problems that arise in studying electromagnetic field-quantum system interaction both in long-wave approximation and outside its framework. PACS number(s): 31.15.ag, 32.70.-n, 32.90.+a I. INTRODUCTION aligned (i.e., spin-polarized) atoms in long-wave 1 approximation . The known multipole expansion for Multipole expansions in classical electrodynamics the probability amplitude of a quantum transition are the seriess for potentials or field strengths, where accompanied by radiation of photon with definite the coefficients are the irreducible tensors dependent polarization and direction of motion is based on on the charge or current distributions of the system that representation of function e expik r, where e is is the source of the field. In every term of the multipole the vector of right(left)-hand circular polarization and k series the irreducible tensors have been convolved in is the wave vector (i.e., the momentum of the photon in scalars (for the scalar potential) or vectors (for the the units of ), in the form of the series in the vector potential or field strength) with the irreducible spherical vectors or the fields of electric and magnetic tensors specifying the corresponding fields (called the multipoles (see, e.g., [1]). The coefficients of the series multipole fields [1]). L The expression for multipole expansion in are proportional to Wigner’s DM ,1,,0 -function, electrostatics can be found in any textbook on classical where and are the azimuthal and polar angles electrodynamics (see, e.g., [2]). In that case the specifying the direction of the vector k. In the irreducible tensors specifying the source are the well interesting for us case of multipole radiation by spin- known electric multipole moments of the charge polarized atoms this expansion led to cumbersome system. The structure of multipole expansion in formulas that essentially hampered the study. That is magnetostatics is more complicated owing to the vector why we were forced to seek for other form of multipole nature of the magnetic field potential. And no wonder expansion. As a result, the multipole series had been that one can only find the first term of the expansion in derived which is highly efficient, in our opinion, in the textbook which is determined by the field of solution of different problems of quantum radiation magnetic dipole and contains the magnetic (magnetic theory both in long-wave approximation and outside its dipole) moment of the the current system as the framework. coefficient. The total multipole expansion for magnetic Note that analyzing the long-wave approximation in field of a current system containing magnetic multipole quantum radiation theory the authors of known moments has been recently derived by us [3]. textbooks on quantum electrodynamics [6,7] do at all The different forms of multipole expansions in without the multipole expansion for the amplitude of classical radiation theory are known [1,4]. Note that the photon radiation with definite momentum and expansion for the potential of radiation field given by polarization. On the first stage they limit themselves to Dubovik and Cheshkov [4] is considerably more derive the total radiation probability of electric 2l-pole convenient for applications. It has allowed to (El) or magnetic 2l-pole (Ml) photons. Considering accomplish the correct analysis of long-wave further the multipole radiation of given direction and approximation in the radiation theory and to take into polarization the authors [6] actually construct in account the contribution of toroid multipole moments artificial way the corresponding term of the multipole arising in higher orders of the approximation [4]. expansion, where the Wigner’s D-function arises again. Multipole expansion has also to work effectively in In such way one cannot furthermore take into account the quantum radiation theory, especially under the possible interference of Ml and El', l' l 1 , consideration of transitions between the states with definite values of angular momentum of the radiating system when the angular momentum selection rules take place. We needed such expansion in theoretical 1 The results of the study have been presented at ECAMP11 study of multipole radiation of light by oriented and [5]. 1 amplitudes which have, generally speaking, the same order in the long-wave approximation. gl (x) J l1 2 (x) , (4) The paper is organized as follows. In Section II the 2x mathematical technique for deriving multipole expansion in quantum radiation theory is stated and the J (x) is Bessel function and Ylm is spherical multipole series for the radiation amplitude is found. function. Substituting this expansion in equation (2) Several equivalent forms of the multipole series are gives the following series for I : given in Sec. III. The possible applications of the fi derived expansion are discussed in the final section of the paper. l ˆ I fi 4 i gl (kr)j fi (r)Ylm (rˆ)drYlm (k) . (5) l, m II. MULTIPOLE EXPANSION FOR THE To derive the multipole expansion we take AMPLITUDE OF SPONTANEOUS advantage of mathematical technique of angular RADIATION momentum algebra (i.e., the technique of irreducible tensors which is based on the theory of irreducible The probability of spontaneous radiation per unit representation of the rotation group) and separate out in time of a photon with frequency , unit polarization the series (5) and respectively in the expansion of the vector e and wave vector k in spatial angle element d radiation amplitude (1), (3) the irreducible tensors is determined by the well known expression (see, e.g., being determined by the current density of transition. [6]) We use further a number of standard definitions and designations of the technique [9] which we give here 2 for the convenience. The spherical components dWfi V fi d , a , m 0, 1 , of arbitrary vector a forming the first 2c3 m rank irreducible tensor are expressed through its Cartesian components as follows: where 1 a0 az , a1 ax iay . V fi e j fi (r)exp ik rdr . (1) 2 The radiating system passes with that from the initial One can compose of two irreducible tensors Alm and state i into the final state f . The value V fi (1), Bl'm' having respectively the ranks l and l' the where j fi is the current density vector of the transition, irreducible tensors (the tensor products of the we shall call the probability amplitude of spontaneous irreducible tensors) of the ranks radiation and shall seek the multipole expansion just for it. Although the explicit structure of the current L l l', l l' 1,..., l l' density vector is not used in the derivation of multipole expansion, the corresponding formulas are given in after the rule Appendix A to complete the statement. Let us introduce the vector LM Al Bl'LM Clml'm' Alm Bl'm' , (6) m, m' I fi j fi (r)expik rdr (2) LM and write the radiation amplitude (1) in the form of where Clml'm' is the Clebsh-Gordan coefficient. If scalar product: l l' , the scalar (the zero rank irreducible tensor) is also composed. The scalar is proportional to the scalar product of two tensors that will be designated by V e I . (3) fi fi parentheses: Let us also use the known expansion [8] for the m exponential function Al Bl 1 Al,mBlm m l l 1 2l 1Al Bl 00 . (7) exp ik r 4 il g (kr)Y (rˆ)Y (kˆ) , l lm lm l0ml In particular, the zero rank tensor composed of two vectors is proportional to the ordinary scalar product of where rˆ r r and kˆ k k are the unit vectors, the vectors: 2 1 we find as a result the following series for the a b00 a b . (8) 3 probability amplitude of spontaneous radiation V fi (3): With the help of the technique analogous of that, l 2L 1 (l) ˆ used by us in derivation of multipole expansion in V fi 1 AL Yl (k) e . (11) 3 1 magnetostatics [3], one can separate out the irreducible l0 L tensors dependent on the current density of transition in the expression (5). To do it we note that the spherical Here the irreducible tensors components of the vector I fi (5) contain the A(l) 4 il g (kr)Y (rˆ) j (r) dr (12) construction of the following form: LM l l fi LM j 1m'Y (rˆ)Y (kˆ) . (9) are introduced which are determined by the current fi m l,m' lm' m' density of transition and thus specify the radiating quantum system. Note also that the index L in the internal sum (11) satisfies the triangle condition Here j is the spherical component of the current fi m l 1 L l 1 (i.e., L l, l 1 if l 0 and L 1 density vector of transition and the identity l 0 m if ) in accordance with angular momenta addition Ylm 1 Yl,m for the spherical function has been rule.
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