Exchange, antisymmetry and Pauli repulsionCan we ‘understand’ or provide a physical basis for the Pauli exclusion principle?ESDG, 13th January 2010Mike TowlerTCM Group, Cavendish Laboratory, University of k

Pair correlation functions in siliconVMCVMC (r; r0; [n]) (r; r0; [n])g""g"#r at bond centerr at bond center0r0 in (110) planer in (110) 0*1100*11*001100*1100*Why is the parallel spin hole wider and deeper than the antiparallel one? Nobodyknows, other than to say ‘because of the Pauli exclusion principle’, or ‘due to statisticalrepulsion’, or ‘because fermions cannot be in the same state’, or whatever.

The Exclusion PrincipleLong standing, unsolved theoretical problem of atomic physics: why is that electrons within an atom donot all collect in the lowest energy orbital? In 1925 Pauli published a limited version of the ExclusionPrinciple from studies of fine structure of atomic energy levels and earlier suggestions of E.C. Stoner:Pauli’s Principle: In an atom there cannot be two or more electrons with the same quantum numbers.Then realised that the Principle applies not just to electrons but to all fermions of same type. Ifwe say quantum particles are identical when they have same mass, charge, spin, etc., then fermionsare sometimes defined to be those identical quantum particles that, when part of a quantum systemconsisting of two or more of the same particles, the system has a wavefunction that is antisymmetricalin its form. Consequent generalization of Pauli’s Principle:Exclusion Principle: In a quantum system, two or more fermions of the same kind cannot be in thesame (pure) state.The antisymmetrical form of the wavefunction is generally taken as a ‘brute fact’, i.e. as a definingcharacteristic of fermions or as a feature of nature that cannot be otherwise explained. The exclusionprinciple acts primarily as a selection rule for non-allowed quantum states and cannot be deduced as atheorem from the axioms of Orthodox Quantum Theory.References:Quantum Causality by P. Riggs (2009)Pauli’s Exclusion Principle - the origin and validation of a scientific principle, by M. Massimi (2005)“The reason why the Pauli Exclusion Principle is true and the physical limits of the principle are stillunknown.” (NASA website)

IndistinguishabilityStandard approach: justify Exclusion Principle by appealing to assumed ‘indistinguishability’ of identicalparticles. Consider two spinless non-interacting identical particles at x1 and x2 with wave functionsψA(x1) and ψB (x2). Assume composite system wave function ψ(x1, x2) ψA(x1)ψB (x2).Claim since particles indistinguishable, coords are just labels whose exchange is not meaningful. Thusrequire the 2-particle wave function to give same probability density after such exchange, i.e.2222 ψ(x1, x2) ψA(x1)ψB (x2) ψA(x2)ψB (x1) ψ(x2, x1) Not true in general! So use technique of linearly combining wave functions. Since ψA(x1)ψB (x2)and ψA(x2)ψB (x1) are both solutions of Schrödinger equation, so is any linear combination. Twopossibilities for composite system wave function:1Ψ(x1, x2) [ψA(x1)ψB (x2) ψA(x2)ψB (x1)]2If is positive, Ψ said to be symmetric with respect to coord exchange since Ψ(x1, x2) Ψ(x2, x1).If is negative, Ψ said to be antisymmetric since Ψ(x1, x2) Ψ(x2, x1).Observed fact: only symmetrical and antisymmetrical wave functions are ‘found’ in nature. Both typessatisfy required probability density equality, but only antisymmetrical ones entail the Exclusion principle(if x1 x2 then Ψ 0, i.e. there is no corresponding quantum state.)Conclusion: Exclusion Principle arises from the wave function of system of fermions beingantisymmetric (Dirac 1926, Heisenberg 1926). However, note the Exclusion Principle is not equivalentto the condition that fermionic systems have antisymmetrical wave functions (as commonly asserted)but follows from this condition. Thus, indistinguishability is not enough.

The spin-statistics theorem and relativistic invarianceOften claimed antisymmetric form of fermionic Ψ arises from relativistic invariance requirement, is conclusively established by the spin-statistics theorem of quantum field theory (Fierz 1939, Pauli1940). Not so - relativistic invariance merely consistent with antisymmetric wave functions. Consider:Postulate 1: Every type of particle is such that its aggregates can take only symmetric states (boson)or antisymmetric states (fermion).All known particles are bosons or fermions. All known bosons have integer spin and all known fermionshave half-integer spin. So there must be - and there is - a connection between statistics (i.e. symmetryof states) and spin. But what does Pauli’s proof actually establish? Non-integer-spin particles (fermions) cannot consistently be quantized with symmetrical states (i.e.field operators cannot obey boson commutation relationship) Integer-spin particles (bosons) cannot be quantized with antisymmetrical states (i.e. field operatorscannot obey fermion commutation relationship).Logically, this does not lead to Postulate 1 (even in relativistic QM). If particles with integerspin cannot be fermions, it does not follow that they are bosons, i.e. it does not follow thatsymmetrical/antisymmetrical states are the only possible ones (see e.g. ‘parastatistics’). Pauli’s resultshows that if only symmetrical and antisymmetrical states possible, then non-integer-spin particlesshould be fermions and integer-spin particles bosons. But point at issue is whether the existence ofonly symmetrical and antisymmetrical states can be derived from some deeper principle.Actually, fact that fermionic wave function is antisymmetric - rather than symmetric or some othersymmetry or no symmetry at all - has not been satisfactorily explained. Additional postulate oforthodox QM. Furthermore, antisymmetry cannot be given physical explanation as wave function onlyconsidered to be an abstract entity that does not represent anything physically real.

Does Pauli exclusion principle need a physical explanation?“.[the Exclusion Principle] remains an independent principle which excludes a classof mathematically possible solutions of the wave equation. . the history of theExclusion Principle is thus already an old one, but its conclusion has not yet beenwritten. . it is not possible to say beforehand where and when one can expect thefurther development.” [Pauli, 1946]“ I was unable to give a logical reason for the Exclusion Principle or to deduce itfrom more general assumptions. . in the beginning I hoped that the new quantummechanics [would] also rigorously deduce the Exclusion Principle.” [Pauli, 1947]“It is still quite mysterious why or how fermions with common values in their internaldegrees of freedom [i.e spin] will resist being brought close together, as in the dramaticexample of the formation of neutron stars, this resistance resulting in an effectiveforce, completely different from the other interactions we know.” [Omar, 2005]“.The Pauli Exclusion Principle is one of the basic principles of modern physics and,even if there are no compelling reasons to doubt its validity, it is still debated todaybecause an intuitive, elementary explanation is still missing.” [Bartalucci et al., 2006]“The Exclusion Principle plays an important role in quantum physics and has effectsthat are almost as profound and as far-reaching as those of the principle of relativity.[the Exclusion Principle] enacts vetoes on a very basic level of physical description.”[Henry Margenau]

An example: electron degeneracy pressureWhen a typical star runs out of fuel it collapses in on itself and eventually becomesa white dwarf. The material no longer undergoes fusion reactions, so the star hasno source of energy, nor is it supported against gravitational collapse by the heatgenerated by fusion. It is supported only by electron degeneracy pressure. This isa force so large that it can stop a star from collapsing into a black hole, yet no-oneseems to know what it is. Which of the four fundamental forces is responsible forit? None of them seems to have the right characteristics.Degenerate matter: At very high densities all electrons become free as opposed to just conductionelectrons like in a metal. When this happens, degeneracy pressure (which is essentially independent oftemperature) becomes bigger than the usual thermal pressure.Usual explanation: Electron degeneracy pressure is a quantum-mechanical effect arising from thePauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, sonot all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energylevels. Compression of the electron gas increases the number of electrons in a given volume and raisesthe maximum energy level in the occupied band. Therefore, the energy of the electrons will increaseupon compression, so pressure must be exerted on the electron gas to compress it. This is the originof electron degeneracy pressure. [Wikipedia]All explanations apparently boil down to “because of the Pauli Exclusion Principle”, or “becausefermions can’t be in the same state”. The origin of the Pauli repulsion which prevents particles beingin the same state (that is, having identical probability distributions) is thus not understood.

Required characteristics of ‘Pauli repulsion’ force supporting a white dwarf?All discussions of degeneracy pressure talk about electrons as objectively-existing point particles, so weshall also make this assumption (it then follows that the electrons must have trajectories).Strategy: Work with statistical distribution ρ since particle positions unknown. Assuming classicalNewtonian trajectories, derive differential equation giving time evolution of ρ. Can we deduce fromthis anything about form of force in quantum case? Probability distribution ρ must obey usual continuity equation ρ/ t · (ρv) so that itremains 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( S)2 formulation; instead use the entirely equivalent Hamilton-Jacobi equation S t2m V where S is related to the ‘action’.For convenience, combine continuity and classical Hamilton-Jacobi equations (two real equations, note) iSinto a single complex equation. To do this, introduce general complex function Ψ reiθ ρe h̄with h̄ an arbitrary constant giving a dimensionless exponent. Complex equation that results is: Ψih̄ t2!h̄2 V Q2mΨwith h̄2 2 ρQ .2mρThis is the time-dependent Schrödinger equation - straight out of QM - with one difference: somethinglike a potential (‘Q’) is subtracted off the Hamiltonian. Note Ψ has same interpretation as in QM: aparticle probability density. Tells us that if particles are to follow Newtonian trajectories, must subtractoff an extra ‘quantum force’ Q (apparently due to a ‘wave field’ pushing the particles) from theusual classical force. Could this ‘fifth force’ be responsible for Pauli repulsion?

Electron trajectories? What do we think about this in TCM?But do electrons really have trajectories? This is really a question of the interpretation of QM, butsince essentially no-one thinks about that here, let’s see what we actually do in practice in TCM.Density functional theory people: Create movies using ab initio molecular dynamics, where the nuclear positions are evolved usingNewton’s equations. We therefore believe that nuclei are point particles with classical trajectories. The electrons have a sort of fuzzy charge density which is a ‘solution to the Schrödinger equation’(in the Kohn-Sham sense) for a sequence of nuclear positions. So - because electrons are very light- either we don’t believe they have trajectories at all, or we believe they move much faster than thenuclei and their (presumably non-classical) trajectories are ‘smeared out’. Sometimes, for very light atoms such as H, we think that quantum effects such as zero-pointmotion or tunnelling are important. We then might do e.g. ab initio path integral moleculardynamics (e.g. Matt Probert’s implementation in CASTEP).1 The ‘exchange potential’ - which gives rise to so-called ‘quantum effects that cannot be describedclassically’ - is presumably some kind of approximation to Q?“If we were to name the most powerful assumption of all, which leads one on and on in an attempt tounderstand life, it is that all things are made of atoms, and that everything that living things do canbe understood in terms of the jigglings and wigglings of atoms.” [Feynman]1It can be shown that Feynman path-integral QM - where you sum over the infinite number of possible trajectories eachweighted by an expression involving the classical Lagrangian T V - is equivalent to just using a single term involvingthe trajectory that the electron actually follows along with the new ‘quantum Lagrangian’ T V Q. In principle onecould just calculate Q to correct the quantum H atom trajectories.

Electron trajectories? What do we think about this in TCM?Quantum Monte Carlo people: Both nuclei and electrons are treated as point particles (though the nuclei are usually clamped). We can compute forces (albeit with some difficulty) and if we bothered to implement coupledDMC-MD in CASINO [as Wagner and Mitas did with their code] then we would move the nucleialong classical trajectories just as with DFT. Widely differing timescales make it difficult to treatnuclei and electrons on the same quantum footing. In QMC the point electrons do not move along trajectories (we instead move them along astochastic random walk to sample the distribution). However it can be shown that diffusion MonteCarlo is in principle a stochastic quantum trajectory method in imaginary time (this would be moreapparent if we ever used time-dependent probability distributions).Conclusions: Neither DFT people or QMC people are Copenhagenists, since such people explicitly state - aspart of the ontology - that quantum particles do not have positions unless they are measured. It isexplicitly understood that hidden variables descriptions (and particle positions and their consequenttrajectories are hidden variables) are impossible. The ‘quantum force’ depends inversely on the mass. So when we decide whether particles followclassical trajectories, we appear to have developed a mental facility for estimating the size of Q.Clearly for heavier particles, Q will be small and the trajectories approximately classical. For lighterparticles, this is not so and we must use quantum methods to calculate their dynamics. There is no justification for saying that in TCM we do not believe in the reality of particletrajectories (either for electrons or nuclei). We may therefore proceed with a clear conscience tounderstand the Exclusion Principle using an argument based on particle trajectories.

Indistinguishability and wave field overlapUnfortunately, if particles have a continuous existence, then the usual way of arguing in terms ofpermutation invariance and so on becomes invalid. We can no longer assume identical particles arealways indistinguishable, since they may be distinguished by their spatial relations (trajectories).Permutation invariance postulate: If Ψ is the state of a composite system whose components areidentical particles, then expectation value of any obervable A is the same for all permutations of Ψ.This allows for quantum states that are symmetric, antisymmetric, and of higher symmetry, and so wemust supplement this with the following (experimentally-derived) postulate:Symmetrization postulate: The only possible states of a system of identical particles are described bystate vectors (wave functions) that are either completely symmetrical or completely antisymmetrical.In a realist approach the wave function antisymmetry is a conceptual problem since, if the wave fieldis a physical field that propagates through space, it should be representable by functions without anyparticular symmetry. In our arguments we instead use the criterion that particles are indistinguishableif their individual wave fields spatially overlap (either now or at some particular time in the past).Justification: work out expectation value of square of distance between two particles for product wavefunction ψA(x1)ψB (x2) and for an antisymmetrized one 12 [ψA(x1)ψB (x2) ψA(x2)ψB (x1)]:2222h(x1 x2) i hx iA hx iB 2hxiAhxiB 2 hxiAB Rwhere hxiAB xΨ A(x)ΨB (x) dx is measure of overlap between wave fields ΨA and ΨB , ande.g. hxi is the expectation value of x in the (single particle) state denoted A. If no overlap, then theantisymmetrized result (blue green) reduces to the product one (blue only). The fermions are thendistinguishable, in which case particles must be widely separated and have remained so.

SpinTo do this properly we need a realistic explanation of spin, since the Exclusion Principle prescribesthat if the fermions of a particular physical system share the same set of quantum numbers (and thisincludes the spin quantum number) then they cannot be at the same location.Initial concept of spin had its origin in the experiments of Sternand Gerlach in which a beam of silver atoms was split in two bypassage through a non-uniform magnetic field. In 1925 Uhlenbeckand Goudsmit proposed that an electron had a magnetic dipolemoment which they explained using the classical idea of anextended particle (in this case, an electron) spinning about anaxis through its centre. They used this idea to explain the resultsof the Stern-Gerlach experiments.It has become clear that what is called the ‘spin of a quantum particle’ cannot be the rotationalangular momentum of a spinning particle. In other words, spin cannot be due to an extended bodyrotating about an axis through its centre of mass. The reasons against the axial rotation explanationare readily provided: the rotation of an extended particle would not require an additional variable for its specification; the spin’s vector does not depend on the particle’s position and momentum; angular momentum due to rotation about the centre of mass cannot take half-odd-integer values; the rate of rotation required to give results in agreement with experiment would need tangentialvelocities exceeding the speed of light in vacuum.

Pauli theory of spinIn order to meet the need for incorporating spin into Orthodox QM, much attention was given todeveloping spinor representations and spin algebra as a way of dealing with an aspect of quantumsystems (i.e. spin) that was not properly understood. E.g., the Pauli equation ih̄( Ψ/ t) HΨ)for a single spin- 21 particle has a two-component spinor wavefunction Ψ and the following Hamiltonian:»–2 h̄2ieH A µB · σ eA0 V2mh̄cwith A and A0 being the electromagnetic potentials, B A an external magnetic field and Va (classical) scalar potential. The vector quantity σ has Pauli’s ‘spin matrices’ as its components:„«„«„«0 10 i10σx , σy , σz 1 0i00 1where σ 2 σx2 σy2 σz2. These matrices are operators that represent the spin observables, e.g.‘down’ spinz -component of spin given by sz 12 h̄σz . Eigenfunctions of spin representing ‘up’ and given by following two-component spinor wave functions: χ1 and χ2 1 00 1 .General expression for system not in eigenstate is the superposition χ aχ1 bχ2 where a, withcomplex b. These functions give required measured values of spin, i.e. (h̄/2) with certainty whensystem is in an eigenstate, or with probability a 2 for up and b 2 for down when in a superposition.Although it is the case that spinor methods have been formally successful, they are really a technicalmeans of not addressing the underlying nature of the spin phenomenon. Indeed, the Pauli equationdoes not provide any insight into the origin or characteristics of spin:Pauli’s theory does not explain the origin of the spin, nor does it give any reason for its magnitude. Itmerely provides a method for incorporating it into quantum mechanics. [Lindsay and Margenau, 1957]

Spin with trajectories We have seen that if we accept particles have a continuous objective existence, then in QM itappears as if they are acted on by a force V Q, with the Q bit having its origin in anaccompanying wave field mathematically represented by Ψ. The wave field ought therefore to be areservoir of potential energy, and can receive or impart energy and momentum to the particles. Q is the potential energy of the wave field and represents the amount of energy available to theparticle/configuration at its specific position in the field. There is no spin in the non-relativistic Schrödinger theory. However, if we take the non-relativisticlimit of the relativistic trajectory equations, we find that Q develops a spin dependence.“In classical physics the aim of research was to investigate objective processes occurring in spaceand time. In the quantum theory, however, the situation is completely different. The very fact thatthe formalism of quantum mechanics cannot be interpreted as visual description of a phenomenonoccurring in space and time shows that quantum mechanics is in no way concerned with the objectivedetermination of space-time phenomena” [Heisenberg, 1965]. Hmmm.

An important inference about the nature of spin We have seen that spin cannot arise from electron rotation, nor do electrons appearto have internal structure. Moreover, the fact that the quantum potential Q (whichrepresents a portion of the wave field’s energy) has a spin dependence implies thatspin must be a property of the wave field. Is there a precendent for this? Yes! In electromagnetic theory - spin is part of anelectromagnetic wave’s angular momentum, the part which is dependent on thewave’s polarization (see e.g. Jackson electromagnetism textbook 1975, p. 333,Ohanian 1986 - see next page).Consider, for example, a circularly-polarized plane electromagnetic wave with a vectorpotential A given by: h E0x iA (x̂ iŷ) iexp iω t ωcwhere E0 is the electric field strength, ω is the angular frequency, and x̂ and ŷare Cartesian unit vectors. The polarization-dependent part of the wave’s angularmomentum (i.e. its spin s) is:Z 21E0 3s ẑ d x2µ0 cω

Nice paperSee also “What is spin?”, A. Gsponer, arXiv:physics/0308027v3 (2003).

Spin and polarization“The lack of a concrete picture of the spin leaves a grievous gap in our understanding of quantummechanics . spin could be regarded as due to a circulating flow of energy, or a momentum density inthe electron wave field . this picture of the spin is valid not only for electrons .” [Ohanian, 1986] In this picture wave fields must have states of polarization, similarly to the caseof an electromagnetic wave. However, it is obvious that in non-relativistic QMwavefunctions are scalar waves describing spinless quantum states. It mightthus be objected that if wave fields have states of polarization, then QM wavefunctions would have to represent vector waves, and this might conflict with therepresentation of quantum systems with spin by spinors. However, there is more than one formal way to achieve this representation. Inparticular, either vector waves or scalar waves plus spinors can be used. Indeed,spinors are used this way in classical wave theory [see e.g. Rogalski/PalmerAdvanced University Physics p.401-403 (2006)].As previously noted, therepresentation of spin by spinors is only a method of dealing with the spinphenomenon without needing an understanding of its fundamental nature. The explanation of spin as the polarization-dependent part of the wave field’sangular momentum has not only not been accepted by most physicists who areaware of this explanation, it is almost universally ignored. The principal reason forthis is probably that, in orthodox QM, the wave field is generally not considered tobe a real field, but to be a representation of our knowledge (or something similar).

The exclusion principle in a trajectory theory Let’s delay discussion of why fermionic wave functions are antisymmetric, and forthe moment just accept that they are. What then is the causal mechanism whichexplains the Exclusion Principle? Normally ones says things like “A system in an antisymmetric state exhibits whatis called statistical repulsion” [Park, 1974]. Strange notions such as ‘statisticalrepulsion’ come from dismissing any possibility of a realistic, causal description ofquantum phenomena and leaves this kind of correlated particle motion completelyunexplained. However, if we accept that electrons have trajectories.“The symmetrization or antisymmetrization of the wavefunction has nothing todo with the ‘indistinguishability’, but in fact, implies the introduction of forcesbetween the particles making up the system, which bring about correlations in theirmotion.” [Holland, 1993] When we analyze possible trajectories, we find they cannot pass through the nodalsurface of the wave field (where the wave amplitude is zero) because the quantumforce is always directed away from these nodes. Antisymmetrical wave fields havenodal surfaces, symmetrical ones do not. This is the basis of the exclusion principlefor fermions.Let’s try to show this mathematically!

Fermionic repulsion in a trajectory theoryA total antisymmetrical wave function for a many-electron system can occur in a number of ways. For2 electrons there are 3 states of interest where the electrons ‘avoid each other’. Collectively called the‘triplet state’ with total z -components of spin h̄, h̄, 0. Their wave functions (which are products ofspace and spin) all have antisymmetrical spatial components so Ψ 0 if x1 x2 and are given by:Ψ {ψA(x1)ψB (x2) ψA(x2)ψB (x1)} α(1)α(2)Ψ {ψA(x1)ψB (x2) ψA(x2)ψB (x1)} β(1)β(2)Ψ {ψA(x1)ψB (x2) ψA(x2)ψB (x1)} {α(1)β(2) α(2)β(1)}SNow let the spatial part be written in complex polar form: ψA(x1)ψB (x2) ψA(x2)ψB (x1) Rei h̄ .SWhen this is zero the amplitude R must be zero (since ei h̄ cannot be zero by definition). Thus, as anodal region of the wave field is approached, the value of R will tend to zero. The (repulsive) quantumforce on each particle is Fi (dpi/dt) iQ where Q h̄2/2mR( 21R 22R) spin-dependent terms. Finding the negative gradient of Q (ignoring the spin-dependent terms sincethe spatial terms will dominate as R tends to zero) gives:2ih̄2 X h22Fi R i( j R) ( i R)( j R)2mR2 j 1It can be seen that as R 0 then Fi . The ‘Pauli repulsion’ force Fi exerted by thewave field on the two fermions prevents them coming into close proximity of each other when their‘spins are the same’ (i.e. in cases where the spatial part of Ψ is antisymmetric). More generally, thedynamics as shown by this trajectory theory prevent fermions occupying the same quantum state.

Towards a fundamental basis for the exclusion principleMust now explain why Ψ in a fermionic system takes an antisymmetric form. Someconsiderations: In a realist theory where the wave field objectively exists, the antisymmetrical formshould be explicable in terms of the well-established behaviour of physical waves. Iffermions are localized particles as we postulate, then the Exclusion Principle mustalso be a non-local effect. The Exclusion Principle is normally assumed to apply to all physical situationsinvolving fermions of the same kind, but it has been argued that its applicabilityshould be restricted; there are certain kinds of non-stationary state for which itproduces paradoxical results. Reasonable statement: Exclusion Principle appliesto a system of identical fermions that has constraints imposed upon it which arenecessary but not always sufficient for the establishment of a stationary state.Physically, a stationary state results when two travelling waves that are propagatingin opposite directions superpose (due to restriction in a finite spatial region such as abox or an atom). Clearly the Exclusion Principle does not apply to widely-separatedfermions of the same kind; need wave-field overlap.But if only one of these waves was π out of phase with the other then wouldn’t theirsum be an antisymmetric function?

Modelling of fermionic wave fieldsConsider system of two neutrons (to avoid electrical interaction) in a large box. Particles initiallymoving and well-separated with non-overlapping travelling wave fields. Initial wave function of productform. Let us make the radical assumption that the antisymmetrical form of the wave function developsover the course of time, rather than being fundamental.So what happens when the neutrons come close enough together for wave field overlap? Withoutinvoking the antisymmetry assumption, there is no obvious expression for the form of the two-neutronwave function when the individual wave fields first overlap.Speculation: After some time, the initial wave field ΨI of the two neutron system will be successivelyreflected from the ends of the box. In the case of a fermionic wave field, reflection at a rigid wallcauses a change of the wave field’s phase of π radians. This is a well-known effect when a physicalwave is reflected from a fixed boundary. However, it is the polarization of the incident wave field(and not the total spin) that determines whether there is a change of phase on reflection. (Spin partof wave field doesn’t change on reflection). Interference between incident and reflected wave fieldsproduces stationary state within the box, and the total wave function will have an antisymmetric formdue to the negative sign in the reflected wave. (A similar argument can be made for atoms).Difficult to analytically model this behaviour, since do not yet know valid mathematical description ofthe initial overlap of individual non-station

formulation; instead use the entirely equivalent Hamilton-Jacobi equation @S @t (rS)2 2m V-where Sis related to the 'action'. For convenience, combinecontinuityandclassical Hamilton-Jacobiequations (two real equations, note) into asingle complex equation. To do this, introduce general complex function rei p ˆe iS h