Dynamical relaxation to quantum equilibrium Or, an account of Mike's attempt to write an entirely new computer code that doesn't do quantum Monte Carlo for the first time in years. ESDG, 10th February 2010 Mike Towler TCM Group, Cavendish Laboratory, University of Cambridge www.tcm.phy.cam.ac.uk/∼mdt26 and www.vallico.net/tti/tti.html [email protected] { Typeset by FoilTEX { 1 What I talked about a month ago (`Exchange, antisymmetry and Pauli repulsion', ESDG Jan 13th 2010) I showed that (1) the assumption that fermions are point particles with a continuous objective existence, and (2) the equations of non-relativistic QM, allow us to deduce: • ..that a mathematically well-defined ‘fifth force', non-local in character, appears to act on the particles and causes their trajectories to differ from the classical ones. • ..that this force appears to have its origin in an objectively-existing `wave field’ mathematically represented by the usual QM wave function. • ..that indistinguishability arguments are invalid under these assumptions; rather antisymmetrization implies the introduction of forces between particles. • ..the nature of spin. • ..that the action of the force prevents two fermions from coming into close proximity when `their spins are the same', and that in general, this mechanism prevents fermions from occupying the same quantum state. This is a readily understandable causal explanation for the Exclusion principle and for its otherwise inexplicable consequences such as `degeneracy pressure' in a white dwarf star. Furthermore, if assume antisymmetry of wave field not fundamental but develops naturally over the course of time, then can see character of reason for fermionic wave functions having symmetry behaviour they do. { Typeset by FoilTEX { 2 Pair correlation functions in silicon VMC VMC 0 0 g r r n g r r n "" "# r at b ond center r at b ond center 0 0 r in plane r in plane 1 0.9 1 1.02 0.7 0.97 0.5 0.92 0.5 0.3 0.9 0.87 0.1 0.82 * * * * * 0.8 * * * 0 * * * * * * { Typeset by FoilTEX { 3 Required characteristics of force giving rise to `Pauli repulsion'? When we say `this is not explicable classically; it is a quantum effect' (degeneracy pressure, covalent bonding, Aharanov-Bohm effect, etc..) can we be more specific about what we mean? Strategy: Work with statistical distribution ρ since particle positions unknown. Assuming classical Newtonian trajectories, derive differential equation giving time evolution of ρ. Can we deduce from this anything about form of force in quantum case? • Probability distribution ρ must obey usual continuity equation @ρ/@t = −∇ · (ρv) so that it remains normalized as it changes shape over time (here v is velocity vector). • Assume particles obey classical dynamics. To calculate trajectories, don't use Newtonian F = ma 2 @S (rS) formulation; instead use the entirely equivalent Hamilton-Jacobi equation − @t = 2m + V - where S is related to the `action'. For convenience, combine continuity and classical Hamilton-Jacobi equations (two real equations, note) p iS into a single complex equation. To do this, introduce general complex function Ψ = reiθ = ρe h¯ with h¯ an arbitrary constant giving a dimensionless exponent. Complex equation that results is: 2 ! 2 2p @Ψ h¯ 2 h¯ r ρ ih¯ = − r + V − Q Ψ with Q = − p : @t 2m 2m ρ This is the time-dependent Schr¨odingerequation - straight out of QM - with one difference: something like a potential (`Q') is subtracted off the Hamiltonian. Note Ψ has same interpretation as in QM: a particle probability density. Tells us that if particles are to follow Newtonian trajectories, must subtract off an extra `quantum force' −∇Q (apparently due to a `wave field’ pushing the particles) from the usual classical force. { Typeset by FoilTEX { 4 Derive the same thing directly in QM 2 • Wave field evolution from Schr¨odingerequation i¯h@Ψ = PN − h¯ r2Ψ + V Ψ. @t i=1 2mi i Evolving quantum system behaves like `probability fluid’ of density jΨj2 = ΨΨ∗ h¯ ∗ with an associated time-dependent quantum probability current j = mIm(Ψ rΨ). h¯ • Suspect particle trajectories follow streamlines of current: velocity v = mImr ln Ψ (current/density). Using complex polar form Ψ = jΨj exp[iS=¯h], the wave function phase S(x1;:::; xN ; t) is given by S = ¯hIm ln Ψ (similar to velocity expression). Thus deduce trajectories xi(t) given by de Broglie guidance equation for velocity: dxi riS vi = = dt mi • Can write in 2nd-order `F = ma' form by taking t derivative: mi¨xi = −∇i(V + Q), where 2 2 P h¯ ri jΨj Q = − (quantum potential). Extra `quantum' force −∇iQ (big if large curvature i 2mi jΨj in wave field). Non-classical dynamics since particles `pushed along' by wave along trajectories perpendicular to surfaces of constant phase, as well as by classical force from other particles. • Guidance equation identical to trajectory equation in Hamilton-Jacobi theory - a standard form of classical mechanics like Hamiltonian or Lagrangian dynamics. There S is indefinite integral of classical Lagrangian with respect to t (note the `action' is the definite integral with fixed endpoints). Suggests immediately how to obtain the classical limit, i.e. when Q = rQ = 0 the wave component of matter is passive and the particles follow classical trajectories (impossible in orthodox QM!). { Typeset by FoilTEX { 5 It's a dead theory! Stationary electrons \The biggest problem is the electron in hydrogen not moving - Copenhagen does not need to invoke QED to explain lack of radiation from atoms and the spherical symmetry of the charge. Up to that point then I did not think that [the theory] was dead just that it was just as flawed as everything else - but I do find it difficult to see how a single electron can be `aware' of the wave and yet have nothing really to do with. [sic]" [MCP email to MDT] Strictly speaking, Copenhagen states Nature is inherently probabilistic; cannot pick and choose which particles it applies to as we do in TCM where most nuclei not only objectively exist but follow Newtonian trajectories. Despite Bohr, it is perfectly consistent to say that electrons/light nuclei exhibit `quantum effects’ because they follow non-classical trajectories due to the influence of the wave field. In the extremely unlikely event of preparing a proton-electron stationary state (H!) in completely field- free space then indeed the particles are motionless, held in equilibrium by opposing forces (classical and quantum). In classical electrostatics, this is a dipole, and a nearby `test charge' will be repelled. Two possible explanations for QM behaviour: (1) say the electron simultaneously occupies every position it could possibly be measured in; the resulting spherical symmetry implies no dipole force, or (2) there is another equal and opposite force due to the wave field which holds the test charge in equilibrium. At this level (apart from the first option being a bit weird) there is no reason to prefer either option. Recall though, that Schr¨odingertheory is for spin-zero particles (we can only do spin-polarized calculations through a kind of cheating). If you take the non-relativistic limit of a relativistic wave equation such as the Dirac equation which correctly incorporates spin one finds an extra spin-dependent term in the guidance equation, and the classical and quantum forces never balance.. { Typeset by FoilTEX { 6 It's a dead theory! Radiation \The biggest problem is the electron in hydrogen not moving - Copenhagen does not need to invoke QED to explain lack of radiation from atoms and the spherical symmetry of the charge. Up to that point then I did not think that [the theory] was dead just that it was just as flawed as everything else - but I do find it difficult to see how a single electron can be `aware' of the wave and yet have nothing really to do with. [sic]" [MCP email to MDT] • The wave field can be thought of as a time-dependent distribution of energy-momentum in space. Because the wave field is a repository of energy it can exert a force on the particles. Depending on the circumstances, some (or all) of a particle's energy-momentum can be transferred and temporarily stored in its wave field. Once stored in the field, energy-momentum can be returned to the particle if circumstances change, and its KE will increase (it will accelerate). The motion of a quantum particle thus need not be in a straight line even if there is no external field present. • If the pilot-wave passes an obstacle (such as a couple of slits) then its form will change (it will develop an interference pattern in this case) and energy will be transferred to and from the particle travelling through it; the electron trajectory will then deviate from its classical (Newtonian) path. It gets guided into places where there is constructive interference in the wave.. • When classical equations of motion (for charges, electromagnetic fields) are satisfied then accelerated charges necessarily radiate. However, the laws of motion discussed here are not the classical ones. The wave field affects not only the electrons, but photons as well. It turns out this influence is such that it completely prevents radiation when the wave field is an energy eigenstate. • As usual, Copenhagen pretends to solve a problem by refusing to address the issue of what actually exists (after all, the `acceleration operator' usually has a non-zero mean, so..) { Typeset by FoilTEX { 7 Usual objections to particle trajectories Because von Neumann proved `hidden variables' were impossible. However, `such was the momentum of the Copenhagen interpretation and von Neumann's reputation that when Grete Hermann pointed out in 1935 that the supposed proof contained a blatant and devastating fallacy, she was simply ignored, and the Copenhagen interpretation remained the almost unquestioned accepted interpretation for decades.' It was left to Bell to rediscover the flaw in 1966.