Equivalence of the velocity and length gauge perturbation series

arXiv:0810.0788v1 [physics.atom-ph] 4 Oct 2008 Equivalence of the velocity and length gauge perturbation series F.H.M. Faisal Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany We derive a “master” perturbation expansion for the quantum transition amplitude in a light field between the field-free initial and final atomic states in the minimal-coupling (MC) “velocity” gauge. The result is used to prove that the traditional “velocity” and “length” gauge perturbation series are equivalent infinite series representations or branches of the same amplitude function, that are equal but in a common domain of convergence (if it exists). More generally, we show that they constitute only two members of a one-parameter family of infinitely many branches of the given transition amplitude. PACS numbers: 32.80.Rm,32.80.Fb,34.50.Fk,42.50.Hz A long-standing unresolved problem of the quantum mechanical perturbation theory of light-matter interac- tion is the existence of two distinct infinite perturba- tion series for a given transition amplitude in the so- called “velocity” and “length” gauges, that have hitherto resisted a demonstration of their mathematical equiva- lence. In addition, numerical calculations of the transi- tion probability based on the two series (albeit, for prac- tical reasons, only in their truncated forms) have fre- quently shown a significant discrepancy between them and/or with various experimental data. These and re- lated difficulties have led some authors to argue in favor of the length gauge (e.g. [1, 2]) as opposed to the velocity gauge, since the former is based manifestly on a physi- cally “true” energy operator [3]. Nevertheless, the princi- ple of gauge invariance in quantum theory (e.g. [3, 4, 5]) requires that they aught to be equivalent. The purpose of this Letter is to derive a “master” per- turbation expansion of the quantum mechanical transi- tion amplitude in the minimal-coupling (MC) “velocity” gauge, and to use the result to demonstrate that the two perturbation series, that are traditionally obtained in the “velocity” and “length” gauges, are two branches of the same amplitude function and, hence, they are equal but in a common domain of convergence (if there is any). More generally, our result shows that they belong to a one-parameter family of infinitely many equivalent series representations (or branches) of the transition amplitude for a given transition process. The Schr¨odinger equation of an atomic system inter- acting with an electromagnetic field, in the minimal- coupling (MC) transverse gauge is given by: (i¯h ∂ ∂t −HMC(t))ΨMC(t) = 0 (1) where, the total Hamiltonian of the interacting system is HMC(t) = (pop −e cA(t))2 2m + eA0(r) (2) with the four-potential Aµ ≡(A0(r), A(t)) where, the scalar potential can be used to define the “atomic” po- tential, eA0(r) = Va(r), and A(t) is the transverse vec- tor potential of the light field. In the usual electric dipole approximation (e.g. Bohr-radius/ wavelength << 1), the vector potential A(t) depends only on t. As usual, the “atomic” Hamiltonian Ha ≡p2 op 2m + Va(r) (3) provides a complete set of eigen-states X allj |φa j >< φa j | = 1 (4) and eigen-energies, Ea j , which satisfy the eigenvalue equa- tion: Ha|φa j >= Ej|φa j >, all j. (5) We define the “atomic” Green’s function Ga(t, t′) by the equation (i¯h ∂ ∂t −Ha(t))Ga(t, t′) = δ(t −t′). (6) Its solution is given by, Ga(t, t′) = −i ¯hθ(t −t′)e−i ¯h Ha(t−t′) = −i ¯hθ(t −t′) X allj |φa j > e−i ¯h Ej(t−t′) < φa j | (7) This can be easily verified by its substitution in Eq. (6) and noting that the derivative of the theta-function is the delta function. Finally, we define, for later use, the total Green’s function (or propagator) GMC(t, t′) associated with the minimal-coupling Hamiltonian HMC(t), by the inhomogeneous differential equation: (i¯h ∂ ∂t −HMC(t))GMC(t, t′) = δ(t −t′). (8) The total Hamiltonian HMC(t) can always be written as a sum of two terms, in infinitely many ways: HMC(t) = Hs(t) + Vs(t); s = 1, 2, 3, · · ·∞ (9) 2 where, Vs(t) ≡HMC(t) −Hs(t), and the basis Hamil- tonians Hs(t) can be used to define the associated basis Green’s functions, Gs(t, t′), by the equations: (i¯h ∂ ∂t −Hs(t))Gs(t, t′) = δ(t−t′); s = 1, 2, 3 · · ·∞. (10) Eq. (10) can be solved in terms of the linearly indepen- dent complete set of fundamental solutions, |φ(s) j (t) >, for each s and all j, of the homogeneous Schr¨odinger equation, (i¯h ∂ ∂t −Hs(t))|φ(s) j (t) >= 0; s = 1, 2, 3, · · ·, ∞; allj. (11) Explicitly, we have: Gs(t, t′) = −i ¯hθ(t −t′) X allj |φ(s) j (t) >< φ(s) j (t′)|. (12) We may now expand the total Green’s function, GMC(t, t′), using any basis Green’ function, Gs(t, t′), in an infinite series, GMC(t, t′) = Gs(t, t′) + Z dt1Gs(t, t1)DMC(t1, t′) × Gs(t1, t′) + Z Z dt1dt2Gs(t, t1) × DMC(t1, t2)Gs(t1, t2)DMC(t2, t′)Gs(t2, t′) + · · · ; s = 1, 2, 3, · · ·, ∞. (13) where we have introduced the (inhomogeneous Schr¨odinger) operator: DMC(t, t′) ≡[(HMC(t) −i¯h ∂ ∂t

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