Comment on ‘‘Quantum oscillator in a non-self-interacting radiation field: Exact calculation of the partition function’’

Abstract

CommentsComments «e sh«t papers which comment on papers of other authors preuiously published tn the phys&eai Revievr Comment should state clearly to which paper it refers and must be accompanied by a brief abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors.In their recent paper [Phys. Rev. A 35, 4122 (1987)]Castrigiano and Kokiantonis claim to present an exact calculati. on of the partition function for a quantum oscillator interacting with the blackbody radiation field. In this Comment it is shown that their result is in fact wrong, due essentially to their neglect at the outset of the A term in the Hamiltonian. It is further shown how their error can be repaired to give the correct result given by us in an earlier publication [Phys. Rev. Lett. 55, 2273(1985].In their paper' Castrigiano and Kokiantonis claim to present an exact calculation of the partition function and, hence, the free energy of a quantum harmonic oscillator interacting with the radiation field. We wish to point out here that their calculation is wrong and that when the error is repaired a calculation such as theirs gives the correct result.The point is this. They begin with the correct Hamiltonian for a three-dimensional charged oscillator of (bare) mass m and spring constant K interacting via dipole coupling with the radiation 6eld 0= ' p--'A +-, 'K:'+H, .' 2@i tonian (1), takes the form "2 2&e 2VX Finally they interchange the meaning of coordinate and momentum with the canonical transformation: r~p, p~-r. The result is the Hamiltonian H"= -Kp + r -=1 2 1 2 2 2@i ' ]/2 2~e r + gAco a& + ei, fico V where the vector potential in dipole approximation is ' 1/2 2vTRc A= g e"(a"+ai", ), co V (2) ' 1/2 2me r X &g + eg + %co V and the radiation field Hamiltonian is H"d --g fico(a& az +-, ') . k, cr But at the outset they drop the term e A /2m' Then, at a later stage, they add a term [the "counterterm" appearing in the text following Eq. (10)] that, when expressed in terms of the variables of the original HamilThis Harniltonian does not correspond to the oscillator coupled to the radiation field. In fact, it corresponds to a Drude model with a frequency-independent friction constant equal to 2Ke /3mc . This, then, is their error: they calculate from the outset with the wrong Hamiltonian. Indeed, the "main result" of Castrigiano and Kokiantonis, their Eq. (11), is equivalent with the expression for the free energy for such a Drude model given in Ref. 2, Eq. (25). Thus their path-integral calculation is correct; it is their physics which is wrong. Here we should 37 3609 O~1988 The Axnerican Physical Society

Fourth-order master equation for a charged harmonic oscillator interacting with the electromagnetic field

“…The diamagnetic interaction A 2 is generally neglected [36,37], because its inclusion makes calculation of path-integral based master equation difficult. But its inclusion is shown to be important for the correct derivation of the partition function of a charged oscillator in blackbody field [34,38] and we keep it here.…”

“…We consider a charged harmonic oscillator in a blackbody radiation field [14,33,34]. The total Hamiltonian of the closed system formed by the oscillator and field can be written as:…”