Deriving the time-dependent Schrodinger m- and p-equations from the Klein-Gordon equation
aa r X i v : . [ qu a n t - ph ] S e p Deriving the time-dependent Schrödinger m - and p -equations from the Klein-Gordon equation. Paul Kinsler ∗ Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom. (Dated: Wednesday 21 st February, 2018)I present an alternative and rather direct way to derive the well known Schrödinger equation for a quantumwavefunction, by starting with the Klein Gordon equation and applying a directional factorization scheme. Andsince if you have a directionally factorizing hammer, everything looks like a factorizable nail, I also derive analternative wavefunction propagation equation in the momentum-dominated limit. This new Schrödinger p -equation therefore provides a potentially useful complement to the traditional Schrödinger m -equation’s mass-dominated limit. I. INTRODUCTION
There have been many and varied (re)derivations of theSchrödinger equation [1], based on a variety of principles –e.g. Feynman path integrals [2], stochastics (e.g. [3, 4]),utilizing axioms [5], or by applying various ad hoc approxi-mations to variants of the Klein-Gordon equation (e.g. [6]).Here I present another method, inspired by the success ofdirectionally-based factorizations of optical wave equations[7], which allow us to proceed whilst making only the bareminimum of approximations. Of course, one might say thatthe approximation used here to (re)derive the Schrödingerequation is the usual one, and so little has been achieved.However, as in its applications in optics [7, 8] and acoustics[9], the gains are threefold:1. The Klein-Gordon equation is recast without approxi-mation into a new form designed to isolate the part in-tended to be approximated away, making the nature ofthe approximation much clearer.2. That new form enables us to compare in all detailsthe exact and approximate versions side by side, eithermathematically or computationally.3. The method encourages us to explore alternate approx-imations – here, a momentum-dominated limit comple-mentary to the traditional mass-dominated one used toobtain the ordinary Schrödinger equation.Klein & Gordon started with the relativistic equation for theenergy of a massive particle, E = m c + p c , (1)and, by replacing E and p with operators using the correspon-dence principle [10]. From a mathematical perspective, thecorrespondence principle is just the process of switching be-tween one domain and its Fourier transformed counterpart.Here, the correspondence is E ↔ ı ¯ h ¶ t , (2) ppp ↔ − ı ¯ h (cid:209) , (3) ∗ Electronic address: [email protected] which allows us to directly convert eqn. (1) into the Klein-Gordon (KG) equation for the wavefunction of a single mas-sive particle, i.e. h ¯ h ¶ t + m c − ¯ h c (cid:209) i F ( rrr , t ) = , (4)or (cid:20) c ¶ t + m c ¯ h − (cid:209) (cid:21) F ( rrr , t ) = . (5)This Klein-Gordon second order wave equation can, if de-sired, be factorized using spinors to give the first order Diracequation. However, this does not allow for anything that mightalter the wavefunction behaviour away from that in a simplevacuum, so to address this lack I consider modifications in-spired by both the both the Salpeter Hamiltonian and a gravi-tiational potential. The Salpeter Hamiltonian:
It is useful – especially when de-riving the Schrödinger equations – to be able to include theeffect of a static potential within which the particle is moving.We might therefore start with the Salpeter Hamiltonian [11] H F = hp m c + p c + V ( r ) i F , (6)where the Hamiltonian can be applied twice to the wavefunc-tion F . Then, by identifying H with the energy E , we get thesquared form ( E − V ) F ( rrr , t ) = (cid:2) m c + p c (cid:3) F ( rrr , t ) , (7)which matches up to the Klein-Gordon starting point under thecondition that V =
0. As would be expected, the same as theKlein-Gordon equation in a Coulomb potential if V = − e / r .In the following, I will call the potential V the “Salpeter po-tential” to specify its conceptual origin. Gravitational potential:
Although it might seem unlikelythat gravitational potentials have sufficient variation in eitherspace or time to produce effects that apply to quantum phe-nomena, it is nevertheless interesting to see how gravity mightappear in the Schrödinger equation. In general relativity, theNewtonian limit for a gravitational potential X ( rrr , t ) gives anexpression for E which is [12, 13] E = m c [ + X ( rrr , t )] + p c . (8)In an operator form, applied to some wavefunction F , thiswould then be E F ( rrr , t ) = (cid:8) m c [ + X ( rrr , t )] + p c (cid:9) F ( rrr , t ) . time dependent X term elegantly as part of the energy (i.e. on the LHS), itis best left as a perturbation and treated in the same way asthe momentum – both are small in the non-relativistic limits.Note that when properly scaled, we can also use X ( rrr , t ) as aproxy for any other space and time dependent potential thatmight affect our system. Combined potentials:
So we only have to perform the fol-lowing calculation once , I will combine the Salpeter energyexpression eqn. (7) with that allowing for a gravitational po-tential eqn. (8). For an operator-like form, applied to a wave-function F , we have [ E − V ( rrr )] F ( rrr , t ) = (cid:8) m c [ + X ( rrr , t )] + p c (cid:9) F ( rrr , t ) (10) = (cid:2) m c + m c X ( rrr , t ) + p c (cid:3) F ( rrr , t ) . (11)Here the potential V ( rrr ) has no t dependence, because allowingthat would complicate the transformation of the LHS into atime derivative. However, any time dependent part of a moregeneral V could easily be merged into X ( rrr , t ) . In a Klein-Gordon wave equation form, this is (cid:26) c ¶ t + m c ¯ h [ + X ( rrr , t )] − (cid:209) (cid:27) F ( rrr , t ) = . (12)In frequency space ( w , rrr -space), this becomes (cid:26) − w c + m c ¯ h (cid:2) + X ( rrr , w ) (cid:3) − (cid:209) (cid:27) F ( rrr , w ) = , (13)where the breve (here ˘ X ) tells us to convolve X with F over w .Alternatively, in wavevector space ( t , kkk -space), this becomes (cid:26) c ¶ t + m c ¯ h (cid:2) + X ( kkk , w ) (cid:3) + k (cid:27) F ( kkk , w ) = , (14)where the hat (here ˆ X ) tells us to convolve X with F over kkk . Both types of convolution play no interesting role in thefollowing calculations, and are merely an intermediate stagewhich disppears when the equations being used are convertedback into their primary t , rrr domain. Method:
In what follows, I use eqn. (12) which contains twodifferent types of potential, to derive approximate equationswhich have only first order derivatives in the propagation vari-able; i.e. t for the usual temporally propagated Schrödingerequation. To complement the Schrödinger equation deriva-tion, I also derive a spatially-propagated version, which is ap-plicable in a different limit. For a more systematic look at thedifferences between temporal propagation and spatial propa-gation, the reader is referred to Ref. [9]. Further, althoughhere we factorize in Cartesian coordinates, this is not the onlypossible choice [8]. Finally, note that my original source forthe factorization method used was by Ferrando et al. [14]. II. MASS DOMINANT: THE SCHRÖDINGER EQUATION
We can see from the correspondence principle describedabove that the energy E is related to evolution in time t , while transmittedreflectioninput x t ev o l u t i on propagation F I NA L S T A T E I N I T I A L C O ND I T I O N S FIG. 1: For temporal propagation (right), initial conditions cover allspace at an initial time t i ; the final state at t f also covers all space.The effect of a reflective interface is also indicated, since it makesthe important distinction between propagation and evolution clearer. also noting that in non-relativistic scenarios the bulk of a mas-sive particle’s energy is frozen in its rest mass. Thus to re-duce the second-order-in-time KG equations down to the first-order-in-time Schrödinger equation we need to manipulate thestarting equations while focussing on the energy E , and therest mass m .To proceed I will follow the directional factorizationmethod recently popularized in optics [7], albeit with an al-ternate physical focus on temporal propagation (see e.g. [9]).This is the most physically motivated factorization, and we de-compose the system behaviour (waves) into directional com-ponents that then evolve either forward or backward in space,as shown in Fig. 1. To analyse temporal propagation, we needa useful reference paramater to characterise it, and it shouldpreferably be one that remains constant. In this case, a fre-quency domain analysis is called for: we might therefore useeither an energy or a frequency w . This means that the partsof the physics we wish to ascribe to the role of “referencepropagation” must be time independent.Start by defining E ′ = E − V ( rrr ) = ¯ h w to work in a scaledfrequency ( w ) space, so that¯ h w F ( rrr , w ) = h m c + m c ˘ X ( rrr , w ) − ¯ h c (cid:209) i F ( rrr , w ) (15) (cid:2) ¯ h w − m c (cid:3) F = h m c ˘ X − ¯ h c (cid:209) i F (16) (cid:0) ¯ h w − mc (cid:1) (cid:0) ¯ h w + mc (cid:1) F = h m c ˘ X − ¯ h c (cid:209) i F (17) F = ˘ Q ( ¯ h w − mc ) ( ¯ h w + mc ) F (18) F = (cid:20) / mc ¯ h w − mc − / mc ¯ h w + mc (cid:21) ˆ Q F , (19)where ˘ Q ( rrr , w ) = m c ˘ X ( rrr , w ) − ¯ h c (cid:209) . F evolves according to two complementaryparts of differing sign. The term proportional to ( ¯ h w − mc ) − generates a forward-like evolution, and that proportional to ( ¯ h w + mc ) − generates a backward-like evolution [14]. As aresult we can likewise split the wavefunction into correspond-ing pieces, with F ≡ F + + F − . When we transform back intothe time domain, these will (must!) propagate forward in time t , all the while holding information about the wavefunction asa function of rrr . To avoid notational clutter, we use this fact asan excuse to omit the time argument, and only the rrr argumentof F ± will be given.Further, since the F + forward evolving component is bydefinition propagating to later times, its excitations thereforemust (also) be understood to be evolving forward in space( rrr → ¥ ). In contrast, the F − backward evolving compo-nent (also propagating to later times), therefore has excitationsevolving backward in space ( rrr → − ¥ ).Continuing the separation of F + and F − , we see that F + ( rrr ) + F − ( rrr ) = (cid:20) / mc ¯ h w − mc − / mc ¯ h w + mc (cid:21) ˘ Q F ( rrr ) (21) F ± ( rrr ) = ± (cid:16) m c ˘ X − ¯ h c (cid:209) (cid:17) ( mc ) ( ¯ h w ∓ mc ) [ F + ( rrr ) + F − ( rrr )] (22) F ± ( rrr ) = ± (cid:16) mc ˘ X − ¯ h (cid:209) m (cid:17) ¯ h w ∓ mc [ F + ( rrr ) + F − ( rrr )] , (23)which enables us to write (cid:0) ¯ h w ∓ mc (cid:1) F ± ( rrr ) = ± mc ˘ X − ¯ h (cid:209) m ! [ F + ( rrr ) + F − ( rrr )] (24) (cid:0) E − V ∓ mc (cid:1) F ± ( rrr ) = ± mc ˘ X − ¯ h (cid:209) m ! [ F + ( rrr ) + F − ( rrr )] , (25)and finally E F ± ( rrr ) = ± mc F ± ( rrr ) + V F ± ( rrr ) ± mc ˘ X − ¯ h (cid:209) m ! [ F + ( rrr ) + F − ( rrr )] . (26)If F − is set to zero, and only the F + is considered, we see thatin terms of momentum p = ¯ hk , and for a time-independent X ,this will have the dispersion relation E = E ( p ) = mc + V + mc X ( k ) + p / m .By using the correspondence principle [10] to replace E ↔ ı ¯ h ¶ t , (27)we see that back in t , rrr space the convolution vanishes, and we get a pair of coupled differential equations, ı ¯ h ¶ t F ± ( rrr ) = ± mc F ± ( rrr ) + V F ± ( rrr ) ± mc X − ¯ h (cid:209) m ! [ F + ( rrr ) + F − ( rrr )] . (28)This is a pair of first order wave equations coupled only bythe gravitational potential X ( rrr , t ) and the momentum squaredterm (i.e. that (cid:181) (cid:209) ); the potential V does not couple the twobecause it was chosen to be time independent. Those cou-plings, along with the rest mass, the wavefunction(s), andtheir spatial derivatives, then tell us how F ± ( rrr ) will changeon propagating forward in time.If both X and the momentum are small compared to the(dominant) mass term, as is true in Newtonian and non-relativistic scenarios, then any finite F + will only weaklydrive F − , and any finite F − will only weakly drive F + . Fur-ther, the two components evolve very differently, one “for-wards” in space at w ∼ mc / ¯ h and the other “backwards” at w ∼ − mc / ¯ h . Thus any finite cross-coupling that does occurwill be very poorly phase matched, and will almost certainlyaverage out to zero . This smallness criteria, viz. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" X − ¯ h (cid:209) m c F ∓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) ± Vmc (cid:21) F ± (cid:12)(cid:12)(cid:12)(cid:12) , (29)is therefore the minimum criteria which must hold for theSchrödinger equation to be valid; although we should alsobe sure that any periodicities in X ( rrr , t ) or V ( rrr ) do not phasematch the cross-coupling terms and allow them to accumulateto a significant level.Assuming for now that this is true, as is indeed likely fornon-relativistic low-momentum situations, we get ı ¯ h ¶ t F ± ( rrr ) = ± mc F ± ( rrr ) + V F ± ( rrr ) ± mc XF ± ( rrr ) ∓ ¯ h (cid:209) m F ± ( rrr ) . (30)Next we can choose to – but are not compelled to – factorout the fixed rest-mass part, which gives rise to fast oscilla-tions induced by the energy of the particle’s rest mass m . Thisis done by introducing F ± ( rrr ) = y ± ( rrr ) e ± ımc t / ¯ h , (31)so that ı ¯ h ¶ t y ± ( rrr ) = + V y ± ( rrr ) ± mc X y ± ( rrr ) ∓ ¯ h (cid:209) m y ± ( rrr ) . (32)Then we can choose our preferred direction – forwards intime – as indicated by a choice of upper signs, so that ı ¯ h ¶ t y + ( rrr ) = + (cid:2) V ( rrr ) + mc X ( rrr , t ) (cid:3) y + ( rrr ) − ¯ h (cid:209) m y + ( rrr ) , (33) See appendix B of [7], and also e.g. [15]. m is dominant, we might denote it the Schrödinger“ m -equation”.If we were to consider propagating the wavefunction y + forward in time, we might divide both sides by ı ¯ h to get ¶ t y + ( rrr ) = − ı ¯ h (cid:2) V ( rrr ) + mc X ( rrr , t ) (cid:3) y + ( rrr ) + ı ¯ h (cid:209) m y + ( rrr ) . (34)It is worth noting that the last term in eqn. (33) (or in-deed eqn. (34)) is a diffusion term, and causes wavefunctionsto spread outwards. While this is the usually expected be-haviour, it is worth noting that being a diffusion does generatea causal problem – if starting from a strictly bounded wave-function, the diffusion term immediately generates some non-zero wavefunction values at arbitrarily large distances. Thusparts of the wavefunction have propagated faster than light-speed! Of course, this simply an artifact introduced by ourmass-dominated non-relativistic approximation; it is not a fea-ture of the initial Klein-Gordon wave equation, which remainsproperly causal [16]. Having made such an approximation, weshould certainly not expect it to give useful (or even sensible)results for any effects propagating at or near lightspeed. Theartifacts are outside the scope allowed by the approximationsused, and however annoying, they do not represent inherentphysical failings. If those artifacts are problematic in a partic-ular case, then the conclusion should be that the Schrödingerequation is too approximate to use.Lastly, we can extract dispersion relations rather directlyfrom eqns. (30) or (34), by returing to the wavevector kkk do-main and using ppp = ¯ hkkk . It is E − mc = V ( kkk ) + mc X ( kkk , t ) + p / m , (35)as expected in the mass-dominated limit considered. III. MOMENTUM DOMINANT: THE p -EQUATION In contrast to the intent of the Schrödinger derivation, herewe focus on momentum-dominated systems, which naturallypropagate with a strong spatial orientation. This means wemust aim to reduce the second-order-in-space KG equationsdown to the first-order-in-space “ p -equation” by manipulatingand approximating the starting equations treating momentumas the quantity of primary importance. Such a treatment typ-ically makes most sense with very light or massless particles,and indeed a spatially propagated description is very widelyused in optics (see e.g. [7] and references therein). Thisfactorization assumes a propagation forward in space, whilstdecomposing the system behaviour (waves) into componentsthat evolve either forward or backward in time, as shown infig. 2. The consideration of spatial propagation means thatthe result I present in this section is somewhat related to the tx reverse−reflectioninput transmitted p r op a g a t i on evolution FINAL STATEINITIAL CONDITIONS
FIG. 2: For spatial propagation, initial conditions cover all timesat an initial location x i ; the final state at x f again covers all times.The effect of a reflective interface is also indicated, since it makesthe important distinction between propagation and evolution clearer.Note that in this case, any “reflections” generated are not like thosewe would normally expect, although they are closely related. This re-sults from our insistence that all wave components must travel (prop-agate) forward in space. “spacelike counterpart of the Schrödinger equation” as previ-ously derived by by Holodecki [17] . However, the derivationpresented here is necessarily aimed at the limit where the ef-fect of the particle mass is a only small correction, in contrast,Holodecki’s result is limited to the non-relativistic regime.To analyse spatial propagation, we need a useful referenceparameter to characterise it, and it should preferably be onethat remains constant. In this case, a spatial frequency do-main analysis is called for: we might therefore use either lin-ear momentum ppp or a wavevector kkk . This means that the partsof the physics we wish to ascribe to the role of “referencepropagation” must be independent of the primary propagationdirection.For the procedure to work, we need to assume a directionalong which the waves will primarily propagate. Without lossof generality, we will assume this to be the z -axis, with the x and y -axes to account for any transverse properties. Thuswe will focus on the p z momentum component, and relegate p x and p y to the status of corrections. After defining ¯ E = E − m c = ¯ h w − m c , and p T = p x + p y , we work in aspatial frequency (wavevector) space kkk . Remembering that ppp = ¯ hkkk , we proceed in the following way (cid:2) E − ˆ V ( kkk ) (cid:3) F ( kkk , w )= (cid:2) m c + m c ˆ X ( kkk , w ) + ¯ h c k (cid:3) F ( kkk , w ) (36) (cid:2) ¯ h c k z − (cid:0) E − m c (cid:1)(cid:3) F = (cid:2) − ¯ h c k T + ˆ V − (cid:0) E ˆ V + ˆ V E (cid:1) − m c ˆ X (cid:3) F (37) Thanks to S.A.R. Horsley for the reference. (cid:2) ¯ h c k z − ¯ E (cid:3) F = ˆ W F (38) ( ¯ hck z − ¯ E ) ( ¯ hck z + ¯ E ) F = ˆ W F (39) F = ( ¯ hck z − ¯ E ) ( ¯ hck z + ¯ E ) ˆ W F (40) F = (cid:20) / E ¯ hck z − ¯ E − / E ¯ hck z + ¯ E (cid:21) ˆ W F , (41)where for convenience I have definedˆ W ( kkk , w ) = − c p T + ˆ V − E ˆ V − ˆ V E − m c ˆ X , (42)retaining the ordering of E and V as is needed when E is(re)turned to operator form (i.e. as a time derivative).We can see from the term in square brackets on the RHSof eqn. (41) that F evolves according to two complementaryparts of differing sign. The term proportional to ( ¯ hck z − ¯ E ) − generates a forward-like evolution, and that proportional to ( ¯ hck z + ¯ E ) − generates a backward-like evolution [14]. As aresult we can likewise split the wavefunction into matchingpieces, with F ≡ F + + F − . When we transform back into thespatial domain, these will (must!) propagate forward in space z , all the while holding information about the wavefunction asa function of x , y , t . To avoid notational clutter, we use this asan excuse to omit the spatial argument z , and only the x , y , t arguments of F ± will be given.Further, since the F + forward evolving component is bydefinition propagating to larger z , it therefore must (also) beunderstood to have excitations that evolve forward in time( t ). In contrast, the F − backward evolving component (alsopropagating to larger z ), will contain excitations that evolvebackward in time. While the notion of treating waves thatevolve backward in time would (or perhaps should) typicallybe viewed with suspicion, it can nevertheless be defended asa useful approximation in many circumstances – notably, thispicture allows a remarkably powerful way of treating disper-sion [9].Continuing the separation of F + and F − , we see that F + ( x , y , w ) + F − ( x , y , w ) = (cid:20) / E ¯ hck z − ¯ E − / E ¯ hck z + ¯ E (cid:21) ˆ W F ( x , y , w ) (43) F ± = ± / E ¯ hck z ∓ ¯ E ˆ W (cid:2) F + ( x , y , w ) + F − ( x , y , w ) (cid:3) , (44)and this enables us to write ( ¯ hck z ± ¯ E ) F ± ( x , y , w ) = ± ˆ W E (cid:2) F + ( x , y , w ) + F − ( x , y , w ) (cid:3) (45)¯ hck z F ± ( x , y , w ) = ± ¯ E F ± ( x , y , w ) ± ˆ W E (cid:2) F + ( x , y , w ) + F − ( x , y , w ) (cid:3) . (46) By again using the correspondance principle to convert backfrom an w , kkk based description, into a t , rrr form, and with (cid:209) T =( ¶ x , ¶ y , ) , we get a pair of coupled differential equations, − ı ¯ hc ¶ z F ± ( x , y , t ) = ∓ ı ¯ h ¶ t F ± ( x , y , t ) ± ¯ h c (cid:209) T E (cid:2) F + ( x , y , t ) + F − ( x , y , t ) (cid:3) ±
12 ¯ E (cid:2) V − V ı ¯ h ¶ t − ı ¯ h ¶ t V − m c X (cid:3) × (cid:2) F + ( x , y , t ) + F − ( x , y , t ) (cid:3) (47) ¶ z F ± = ± c − ¶ t F ± ± ı ¯ hc (cid:209) T E (cid:0) F + + F − (cid:1) ± ı hc ¯ E (cid:2) V − ı ¯ hV ¶ t − ı ¯ h ¶ t V − m c X (cid:3) × (cid:2) F + + F − (cid:3) . (48)On combining the time derivative terms, this becomes ¶ z F ± = ± c (cid:20) + V ¯ E (cid:21) ¶ t F ± ± Vc ¯ E ¶ t F ∓ ± ı ¯ hc (cid:209) T E (cid:0) F + + F − (cid:1) ± ı hc ¯ E (cid:2) V − ı ¯ h ( ¶ t V ) − m c X (cid:3) (cid:0) F + + F − (cid:1) . (49)Whichever of eqns. (48) or (49) you might prefer, eitherconsists of a pair of first order wave equations coupled onlyby the potentials V and X , as scaled by the mass-compensatedenergy component ( ¯ E ). Those couplings, along with the wave-function(s), then tell us how F ± will change on propagatingforward in space z . Note that unlike in the Schrödinger ( m -)equation case, there is consequently no explicit mass depen-dent oscillation; the effect of the mass appears solely as a cor-rection to the effect of the potential; although the rest-massoscillation remains a legitimate contribution to F ± .If this potential-based coupling is small, as is perhaps likelyfor light or massless particles, then any finite F + will onlyweakly drive F − , and any finite F − will only weakly drive F + . Further, the two components evolve very differently, one“forwards” in time at p ∼ E / c and the other “backwards” at p ∼ − E / c . Thus any finite cross-coupling that does occurwill be very poorly phase matched, and will almost certainlyaverage out to zero. This smallness criteria, viz. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( V ¯ E + ı ¯ hc (cid:209) T E + ı (cid:2) V − ı ¯ h ( ¶ t V ) − m c X (cid:3) h ¯ E ) F ∓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) + V ¯ E (cid:21) F ± (cid:12)(cid:12)(cid:12)(cid:12) , (50)is therefore the minimum criteria which must hold for this p -equation to be valid; although we should also be sure that anyperiodicities in X ( rrr , t ) or V ( rrr ) ¶ z F ± = ± c − ¶ t F ± ± ı ¯ hc (cid:209) T E F + ± ı hc ¯ E (cid:2) V − ı ¯ hV ¶ t − ı ¯ h ¶ t V − m c X (cid:3) F ± . (51)Or, with combined time derivatives, ¶ z F ± = ± c (cid:20) + V ¯ E (cid:21) ¶ t F ± ± ı ¯ hc (cid:209) T E F + ± ı hc ¯ E (cid:2) V − ı ¯ h ( ¶ t V ) − m c X (cid:3) F ± . (52)In this last form, we see that the typical (or “reference”)wavevector K for a wavefunction component evolving withfrequency w is K ( w ) = w c [ + V ( w / c ) / ¯ E ] . (53)Again we have a diffusion-like term in our first order waveequation (52), here dependent on (cid:209) T . Now, however, becausewe are propagating a wavefunction known as a function oftime forward in space, the extremes of the diffusion behaviour(which is in this case actually a diffraction ) correspond to veryslow processes, and therefore are not acausal artifacts . IV. SUMMARY
I have shown how to derive the Schrödinger equation for aparticle of mass m , starting from the Klein-Gordon equation, while taking into account the possible effects of both staticand/or dynamic potential landscapes influencing the evolu-tion of the wavefunction. This “ m -equation” is found usingan approximation which assumes that the object’s energy isdominated by its rest mass – i.e. that it is moving as non-relativistic speed. The method is an adaption [9] of a factor-ization scheme recently applied in optics [7, 8, 15], but notoriginating from there.Further, I also derive an alternative to the Schrödinger equa-tion in a different and complementary limit, i.e. that of largemomentum p . This equation does not propagate the waveequation forward in time, as the usual Schrödinger equationdoes, but forward in space. This alternate “ p -equation” is pre-sented here primarily as an exercise in technique, and discus-sions of its possible utility are left for later work. [1] J. S. Briggs and J. M. Rost, Found. Phys. , 881 (1996),doi:10.1119/1.18114[3] E. Nelson, Phys. Rev. , 1079 (1966),doi:10.1103/PhysRev.150.1079.[4] I. M. Davies, J. Phys. A , 3199 (1989),doi:10.1088/0305-4470/22/16/010.[5] A. Sanayei (2013), arXiv:1309.1787.[6] D. W. Ward and S. M. Volkmer (2006), arXiv:physics/0610121.[7] P. Kinsler, Phys. Rev. A , 013819 (2010),arXiv:0810.5689, doi:10.1103/PhysRevA.81.013819.[8] P. Kinsler (2012), 1210.6794, arXiv:1210.6794.[9] P. Kinsler (2012), 1202.0714, arXiv:1202.0714.[10] N. Bohr, The Correspondence Principle (1918-1923) (Elsevier, Amsterdam, 1976).[11] E. E. Salpeter and H. A. Bethe, Phys. Rev. , 1232 (1951),doi:10.1103/PhysRev.84.1232.[12] B. F. Schutz, A first course in general relativity (CambridgeUniversity Press, 1986), ISBN 0-521-27703-5.[13] S. M. Carroll (1997), arXiv:gr-qc/9712019.[14] A. Ferrando, M. Zacares, P. F. de Cordoba, D. Binosi, andA. Montero, Phys. Rev. E , 016601 (2005),doi:10.1103/PhysRevE.71.016601.[15] P. Kinsler, J. Opt. Soc. Am. B , 2363 (2007),arXiv:0707.0986, doi:10.1364/JOSAB.24.002363.[16] P. Kinsler, Eur. J. Phys. , 1687 (2011),arXiv:1106.1792, doi:10.1088/0143-0807/32/6/022.[17] R. Horodecki, Nuovo Cimento B , 27 (1988),doi:10.1007/BF02728791., 27 (1988),doi:10.1007/BF02728791.