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RELATIVISTIC THEORY of SPONTANEOUS EMISSION Dipole Calculations

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RELATIVISTIC THEORY of SPONTANEOUS EMISSION Dipole Calculations INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS RELATIVISTTC THEORY OF SPONTANEOUS EMISSION A.O. Barut and INTERNATIONAL ATOMIC ENERGY Y.I. Salamin AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1987 MIRAMARE-TRIESTE IC/87/S ABSTRACT We derive a formula for the relativistic decay rates in atoms in a formula- International Atomic Energy Agency tion of Quantum Electrodynamics based upon the electron's self energy. Relativis- and tic Coulomb wavefunctions are used, the full spin calculation is carried out and the United Nations Educational Scientific and Cultural Organization dipole approximation is not employed. The formula has the correct nonreiativistic limit and is used here for calculating the decay rates in Hydrogen and Muonium for the transitions IP —^ lSi and 2Si —* lSi. The results for Hydrogen are: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS T(2P -> 15A) = 6.2649 x 108a~l and T{2Si. -• lSi) = 2.4946 x lO"6*"1. Our result for the 2P —* lSi transition rate is in perfect agreement with the best non- reiativistic calculations as well as with the results obtained from the best known radiative decay lifetime measurements. As for the Hydrogen 2S^ —> IS^ decay rate, the result obtained here is also in good agreement with the best known magnetic RELATIVISTIC THEORY OF SPONTANEOUS EMISSION dipole calculations. For Muonium we get: T{2P -> IS^) = 6.2382 x lO8^"1 and 15^) = 2.3997 x lO"6*"1. A.O. Barut ** and Y.I. Salamin ** International Centre for Theoretical Physics, Trieste, Italy. MIRAMARE - TRIESTE June 1987 * To be submitted for publication. ** On leave of absence from: Physics Department, University of Colorado, Boulder, CO BO3O9, USA. -I- I. INTRODUCTION of QED (ft. = c = 1, and dx ~ At present, the rate of spontaneous emission (or partial decay life-times) in atoms is not among the list of precision tests of Quantum Electrodynamics. The U) l 2~i~ and 37- decay rates of the S0 — and ^S^— states of positronium, respectively where: J* = -£¥7*** is the electron current and A^ is the total electromagnetic are part of that list. In positronium one tests the annihilation rates of the e+e~ pair, field: An — A'^ + A'^, with the superscripts e and s standing for external and self, albeit in a bound state. Whereas in Hydrogen or Muonium there is no annihilation respectively. Here A£ is treated as a given nondynamical function. On the other and we are talking about the rates of atomic transitions in, say, H* —> H + -7. hand, F,ui = A'v|(l - A'^ satisfies the Maxwell equations F^v = J" which can be The reason for excluding the rates of spontaneous emission from the list used to put equation (l), after a single integration by parts has been performed on of precision tests of QED is partly due to the absence of very accurate theoretical the last term , into the following form: calculations , because the decay rates are usually calculated in the dipole approx- imation and using nonrelativistic wavefunctions. Also, the accurate experiments W = (2) may not be easy to perform. But with the new techniques of trapped and cooled Next, we complete the elimination of A'^ from the action by inserting into (2) the atoms it may now be possible to make accurate life-time observations in Hydrogen solution of the wave equation'1-3I: and Muonium if correspondingly accurate theoretical numbers would exist. With this goal in mind , we have calculated all spontaneous decay rates in the relativistic Coulomb problem using full Dirac-Coulomb wavefunctions and without making the dipole approximation. The results are thus to all orders in Za. namely: The full spin calculation is rather cumbersome and to our knowledge has not been carried out before. In section II we give a new derivation of a general spontaneous emis- Here •D(J,,(x — y) is the causal Green's function in the covariant gauge J4"I(J = 0 , sion formula in which the decay rate, [Ynj2)l appears as the imaginary part of which we take as : a complex energy shift AEn, the real part being the Lamb-shift and the vacuum 1 3 polarization' " !. Section III contains the full spin and angular integrations as well d*k (-<*•<*-» as the radial integrations with some of the details collected in the Appendices. Fi- (3) nally , in section IV we present a number of numerical results and compare them Thus equation (2) now becomes : with the available nonrelativistic data. II. RELATIVISTIC THEORY OF SPONTANEOUS EMISSION W idp - eAl) - m]*(x) -/ A general formula for spontaneous emission from an electron in an arbi- ,2 t C ^Jfc e-«M*-») *(»)-»**(») trary external field .A™* can be derived in a very simple way directly from the action 2 y = Wo + When the Fourier expansion of the matter field * in the time variable, namely: '/ *(*) = EV>n(x)«-<a—, (5) where the Fourier coefficients are yet to be determined, is substituted into (3) and after the time integrations over /fcn>!/o> and XQ have been performed, in this order ^w-fM*) J for convenience, we get: -\S{E, - £„ f k) l 6(E. - En - 1 [ :l>- (7) - TP (6a) E,-ETi-k and: Notice that the term proportional to i(k) + 6{--k) — 26{k) docs not contribute because of the integration over k. From here, one could proceed to the derivation of the equations of motion by minimizing the total action and subsequently solving the coupled Hartree-type equations thus obtained for the energies and wavefunctions. /' Instead of following this path though, we can avoid the nonlinear equations and use 1 the following approach. If we find the equations of motion and insert them back 6(Er -E,- k)\ into the action, it will assume its minimum value, which is W -- 0. In other words, Here P stands for the principal value integral and J] implies a sum over the an exact solution to our problem would be to find that set of wavefunctions {ipn[x}} discrete part and an integration over the continuum part of the system's spectrum. which would make Wo + Wi — 0. Now, in the absence of the nonlinear self-energy 2 In carrying out the kQ -integration, the contour is closed in the upper half plane for part IV!, which is proportional to t ,Wa vanishes precisely for the solutions of the y0 > x0 where it encloses the simple pole at k0 = — k , (k = \k\] , and in the tower Dirac equation of an electron in the external nondynamical field A^. half plane for the case ya < x0 where it encloses the pole at k0 = +k. ^-functions If we, therefore, take for {^T1(x)} the complete set of solutions of Dirac's are used in order to distinguish between the two cases. The y -integrations turn out 0 equation in such a field, {^°(x)}, with their corresponding energies {/?£} and set to be simply Fourier transforms of the C-functions which give rise to the principal En = E^ + AE,,, then as a first iteration of the action, H'n will contribute a term value integrals and the 6- functions in (6b). 2jrX3tl A£r, and Wi is evaluated with the functions {V'iiM}' Thus we get from the Now , the (^-function, 6{En - Em + Er - E,), can be satisfied by the two vanishing of the action in the first iteration: choices I21; l> (1) n — m and simultaneously t = s. W\ ^ -2 (8) (2) n — a and simultaneously r = m. where the superscript on W[ is added to indicate that we are considering a first With this, Wi becomes: iteration of the action. In particular, for our problem A'^ is a Coulomb field and {V£M} and {E"J arc therefore the sets of Dirac-Coulomb wavefunctions and be elevated to a higher state s. We choose the second ^-function for treating the eigenenergies, respectively. From (7) and (8) we immediately identify the shift phenomenon of photoexcitationM. The fact that both of these terms come out in in the nth energy level as a sum of three terms having the following physical inter- a single equation is one of the advantages of using an action approach. pretations. (From here on we shall drop the superscript c on ipn). We make two remarks at this point. First , it should be emphasized that (l) Vacuum Polarization: the choice of ^-function we have just made is in no way arbitrary as it may sound at first sight. In fact, it is dictated by the remaining fc-integration over the interval (0,oo) and choosing one of the two functions automatically precludes the other. If e<M«-y) .(y) it is an emission process that we study, then En > E, and, since it is positive, only the function 6{E. - En + k) contributes and not S[E. - En - k). Conversely , in (9) the case of absorption, the other ^-function will contribute. (2) Spontaneous Emission and Absorption: B The second remark concerns the relation of AEft to the decay rate of the nth level. When the atomic state of some system of energy t decays in time, the time dependence of its wavefunction is written as'5': e"*'1'"1^)' = e~ic'te~$t, d3k r« ' "trr 6{E. -En- k)\) (10) where P is the decay rate of the state or twice its inverse mean life-time. In other words: (3) The Lamb-Shift: r = -2Im(e) (12) So, taking the right ^-function in (10) and using (12), we get the following general formula for the decay rate of the nth level: /^ l (11) 2Jt E,-En -k £.-£„• The vacuum polarization term has been treated elsewhere™ and so has the Lamb-shift term'4!.
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