Quantum Ground States as Equilibrium Particle Vacuum Interaction States

Transcription

1 Quantu Ground States as Euilibriu article Vacuu Interaction States Harold E uthoff Abstract A rearkable feature of atoic ground states is that they are observed to be radiationless in nature despite (fro a classical viewpoint) typically involving charged particles in accelerated otions The siple hydrogen ato is a case in point This universal groundstate characteristic is shown to derive fro particle vacuu interactions in which a dynaic euilibriu is established between radiation eission due to particle acceleration and copensatory absorption fro the zero point fluctuations of the vacuu electroagnetic field [1] The result is a net radiationless ground state This principle constitutes an overarching constraint that delineates an iportant feature of uantu ground states Keywords Quantu ground states Vacuu fluctuations article vacuu interaction states Zero point fluctuations Haronic oscillator uantu ground state 1 Introduction One of the apparent paradoes of uantu theory that students often uery is the radiationless nature of atoic ground states The parado lies in the fact that radiation that ight be anticipated fro accelerated charged particle otions in atoic ground states is not observed to occur In the hydrogen ato for eaple the orbiting electron does not radiate its energy away and spiral into the nucleus The fact that during decades of successful application of uantu theory we have coe to take for granted the radiationless feature of these special botto rung stationary states does not in any way detract fro this rearkable property Fortunately a rapprocheent between classical and uantu viewpoints is possible H uthoff Institute for Advanced Studies at Austin Research Blvd Austin TX USA e ail: puthoff@earthtechorg

2 When addressed in analytical detail it becoes clear wherein the resolution to this apparent parado lies It is that one ust properly take into account how charged particle ground state otions interact with the vacuu specifically the zero point fluctuations of the vacuu electroagnetic field Although such considerations are not usually invoked in the day to day application of uantu theory to ground state specification the arguent that follows clarifies that the standard foralis leading to radiationless ground states has its genesis in the dynaics of underlying particle vacuu interactions and that the vacuu field is in fact forally necessary for the stability of atos in uantu theory As suarized in an earlier paper addressing spontaneous eission processes: The crucial role of the vacuu fluctuations eerges in the ground state of atter The stability of the ground state (ie the fact that it does not radiate) is purely a uantu effect which is due to the vacuu fluctuations [] It is sufficient for our purposes to treat such probles seiclassically on the basis of point particles interacting with a rando classical radiation field whose spectral characteristics are those of the known uantu vacuu zero point fluctuation (ZF) distribution This approach known as Stochastic Electrodynaics (SED) takes advantage of the fact that SED derivations of vacuu fluctuation driven phenoena based on ultipole/radiation field interactions parallel closely Heisenberg picture derivations in standard QED [4] Specifically in such cases calculations in SED are analogous to QED calculations with a syetric ordering of photon creation and annihilation operators [5] Before considering application on the basis of a general foralis let us apply the central arguent in detail to the siple one diensional haronic oscillator Nonrelativistic Haronic Oscillator For a one diensional haronic oscillator of natural freuency located at the origin r and iersed in the vacuu ZF field the (nonrelativistic) Abraha Lorentz euation of otion for a particle of ass and charge including radiation daping is given by [6] E (1) 6c where E is the coponent of the vacuu ZF electric field and here we neglect the force contribution fro the agnetic field (see Section however) The reuired epression for the electric field is obtained fro the electroagnetic vacuu ZF distribution whose spectral energy density is given by the Lorentz invariant epression [78] d d () c

3 which corresponds to an energy per noral ode In the SED ansatz the Fourier coposition underlying this spectru can be written as a su of plane waves E ikr itik Re dkˆ e 1 8 () A siilar epression for the agnetic field is obtained by replacing E by H ˆ by k ˆ ˆ and by In these epressions Re denotes Real part of 1 denote orthogonal polarizations ˆ and ˆk are orthogonal unit vectors in the direction of the electric field polarization and wave propagation vectors respectively k are rando phases distributed uniforly in the interval to (independently distributed for each k ) and kc Substitution of E () into E (1) leads to the following epressions for position velocity and acceleration: 1 d k e ikr itik Re ˆ ˆ 1 8 D (4) i v d k e (5) ikr itik Re ˆ ˆ 1 8 D ikr itik a Re d kˆ ˆ e 1 8 D (6) where D (7) i (8) 6 c Now for bounded steady state otion we assue stationary epectation values for the ean suare position variable and its tie derivatives Thus d k d k ' 1 ' ˆ ˆ ' * 1 ' DD'

4 1 Re ep i ' i ' ti i ' ' k k r k k (9) where use of the cople conjugate and the notation 1Re derive fro use of eponential notation With dk dk dkk and averaging over rando phases Re ep i k k' r i ' t i k i k' ' ' ' k k ' (1) Euation (9) can therefore be siplified to 1 d ˆ ˆ k dkk * 1 8 DD (11) We further note that with the su over polarizations given by ˆ k k k k (1) 1 the angular integration in k takes the for ˆ ˆ ˆ ˆ ˆ ˆ ˆ i j ij i j 8 d k 1 k (1) 1 ˆ ˆ d ˆ ˆ k Substitution of E (1) into E (11) and a change of variables to kc then leads to 6 6 d d * 6 c DD c (14) Due to the sallness of for charge to ass ratios of interest the integrand in E (14) is sharply peaked around We therefore invoke the standard resonance approiation etending the liits of integration and replacing by in all but the difference ter This yields with substitution of the definition of fro E (8) the ean suare fluctuation in position as since the (Lorentzian lineshape) integral is unity d 1 (15)

5 Calculation of the ean suare fluctuation in velocity 1 DD * i Di D * DD * yielding v follows as above ecept that v (16) A siilar calculation for the ean suare fluctuation in acceleration a yields a (17) With the above calculations in hand we are now in a position to characterize the ground state of the haronic oscillator First the ean suare fluctuation in position given by E (15) atches that obtained in the usual uantu echanical treatent Second the ean suare fluctuation in oentu given by p v (18) also atches that obtained fro the standard QM treatent The haronic oscillator s ground state energy kinetic plus potential is given by also in agreeent with the QM result 1 1 E v (19) Since the ean position and ean oentu p of the stationary state oscillator are zero we also calculate the uncertainty relationship as p p p p p () again in agreeent with the known QM result for the haronic oscillator ground state Now in accordance with the arguent being pursued here we copare the average power being absorbed fro the vacuu fluctuation distribution with that radiated due to accelerated otion to deterine their relative agnitudes The power absorbed fro the electric field due to the driving force F E is given by abs Fv E (1)

6 With derivation of E and given by Es () and (5) respectively the calculation carries through as in the above to yield abs 1 c () The power radiated due to accelerated otion is given by the standard Laror epression as [9] a rad 6 c 6 c () which with substitution fro E (17) and coparison with E () yields rad abs (4) Thus we find that the ground state paraeters of the uantu echanical haronic oscillator can be accounted for on the basis of interaction between a haronically bound point particle and the vacuu electroagnetic zero point fluctuations Specifically the stationary ground state thus established derives fro an average balance of power between that absorbed fro the vacuu fluctuations and that lost by radiation due to accelerated otion [1] It can be noted in passing that even in the liit (uncharged oscillator) this outcoe reains the sae as cancels out in the Generalized Approach rad abs relationship Having derived the above relationship between absorbed and radiated powers for the haronic oscillator s ground state we now inuire as to whether this balance is specific to the haronic oscillator by virtue of its siple linear restoring force or can be etended to ore general cases (eg nonlinear oscillator hydrogen ato particle in a bo etc) We begin with the generalization of E (1) et r r E r B F (5) et and we assue F V for a broad class of cases of interest with V a tie independent confining potential Multiplication of E (5) by r taking into account atheatical siplifications (eg r r B r 1 ddt r r dv dt V t rvr V for a tie independent potential) followed by collection of ters leads to

7 d 1 r r r r E r (6) dt V For a stationary ground state the second ter on the left vanishes and thus the average power radiated due to accelerated otion (Laror radiation) is balanced by the average power absorbed fro the vacuu fluctuations Substituting the definition of fro E (8) we obtain 6 c a E v (7) Thus the stationary ground state although involving accelerated charged particle otion and hence possessing an associated Laror radiation loss is nonetheless observed to be overall radiationless in nature due to the copensatory absorption fro the background electroagnetic vacuu zero point fluctuations The balance so obtained also accounts for the well known fact that an oscillator or ato in its ground state does not on net absorb zero point radiation and therefore reains in its ground state Finally we note that this general result is independent of the for of the (tie independent) confining potential V and is thus applicable to a wide range of probles 4 Concluding Rearks Addressed is the seeing parado that even though uantu ground states typically involve charged particles in accelerated otions such states are nonetheless observed to be radiationless in nature Though this feature is overlooked in everyday application of uantu theory to ground state description nonetheless this rearkable property is worthy of soe discussion and clarification Such is at hand when one recognizes that ground state atoic structures are not isolated entities in an epty background but are perforce iersed in a background of vacuu fluctuations that of the vacuu electroagnetic zero point fluctuations being the priary coponent of interest with regard to the behavior of charged particles Atos therefore constitute open systes engaged in dynaic interactions with the surrounding vacuu states Specifically the on net radiationless characteristic of the ground state is shown here to derive fro particle vacuu interactions in which a dynaic euilibriu is established between radiation eission due to particle acceleration and copensatory absorption fro the zero point fluctuations of the vacuu electroagnetic field Thus the vacuu field is forally necessary for the stability of atoic structures and this underlying principle therefore constitutes an iportant feature of uantu ground states

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