He II-He I Transition in a Heat Current: Model Calculations

Abstract

Nc,(f ) = (hE, --, ' ihl" ) c (f) =Z z (~,z-&NI' )8lcz (t) I'c (f), wherẽ . , --, ' iI-r. , (75) tr~g 1e ceo &R/c "tr (77) This gives a general formula for the dynamical level shift Z; z~q I c~I and the dynamical width gql', 6Ic&I . For two levels, the width formula is equivalent to Eq. (50) for the time rate of change tulate of phase incoherence of components of the Schrodinger wave function made in quantum statistical mechanics. Since the total radiation density obeys Planck's law, there is no conflict with experimental data on blackbody radiation in which total radiant energy is measured.As has been shown by Jaynes and co-workers, ' the semiclassical theory describes dynamical radiative level shifts as well as radiative lifetimes due to spontaneous emission. These two phenomena both follow from Eq. (48), when applied to the total effect of all levels Ez on a given level E . This equation becomes of l~1l'.Crisp and Jaynes' found that for electric dipole radiation the 1s-2P level shift obtained from Eq. (77) for nonrelativistic wave functions agrees within experimental error with the experimental value, assuming that I c"I is unity. The 2s-2p level shift (the Lamb shift) is roughly two-thirds of the experimental value when computed with nonrelativistic wave functions, assuming that I c2, I is unity under the conditions of the Lamb-shift experiments. These calculations need to be refined before they could be considered to provide a definitive test of the semiclassical theory.It is important to note that irreversibility is built into the theory of spontaneous emission given here by assuming retarded solutions of the inhomogeneous wave equation, in Eq. (15). In consequence, the rate of energy production given by Eq. (21) is positive definite, at least for electric dipole radiation. If advanced potentials had been used, the energy production rate would change sign, and it vanishes if half-advanced, half-retarded potentials are used. Thus the choice of retarded potentials establishes an "arrow of time, " in that energy is dissipated from an excited material system as time progresses, and an eventually established equilibrium would not be disrupted by reversing the time sense.'E.The Ginzburg-Pitaevskii phenomenological theory has been extended to include the effects of vortex lines. The model calculations are compared with some of our recent experimental results on the He II -He I transition in the presence of a heat current.