aa r X i v : . [ phy s i c s . g e n - ph ] M a r A Two-Page “Derivation” of Schr¨odinger’s Equation
C. Baumgarten ∗ (Dated: March 13, 2020)We give an exceptionally short derivation of Schr¨odinger’s equation by replacing the idealizationof a point particle by a density distribution. I. INTRODUCTION: POINT PARTICLES
Some physicists may be happy just tohave a set of working rules leading to resultsin agreement with observation. They maythink that this is the goal of physics. Butit is not enough. One wants to understandhow Nature works. There is strong reason tobelieve that Nature works according to math-ematical laws. All the substantial progress ofscience supports this view.P.A.M. Dirac [1]The prevalent narrative of quantum mechanics (QM)makes intense use of the notion of the classical point par-ticle – mostly by refusing it. Often the depiction seems tosuggest that quantum particles resist to behave like theirclassical description suggests. It is, in context of QM,rarely discussed, that the “classical point particle” neverwas, within classical mechanics, regarded as an undis-puted and self-consistent idea. The literal form of the“material point” was mostly promoted by Boscovich [2],a Serbian Jesuit and scientist. “According to Boscovitchan atom is an indivisible point, having position in space,capable of motion, and possessing mass. [...] It has noparts or dimensions: it is a mere geometrical point with-out extension in space: it has not the property of impen-etrability, for two atoms can, it is supposed, exist at thesame point” [3]. But, as Glazebrook pointed out, JamesClerk Maxwell (for instance) disagreed with Boscovich:“We make no assumption with respect to the nature ofthe small parts – whether they are all of one magnitude.We do not even assume them to have extension and fig-ure. Each of them must be measured by its mass, andany two of them must, like visible bodies, have the powerof acting on one another when they come near enough todo so. The properties of the body or medium are deter-mined by the configuration of its parts.”These two quotes exemplify that there was, in classicalphysics, no agreement about the nature of “the smallestparts”. Rohrlich critically reviewed the idealization ofthe classical point particle [4]. He wrote that “in thepoint limit, classical physics cannot be expected to makesense at all”. As he explains, a point charge, regardedfrom the standpoint of classical electrodynamics, impliesinfinite self-energy. Hence, according to Rohrlich, “the ∗ [email protected] concept of a ”classical point particle” is, in view of quan-tum mechanics, an oxymoron. Quantum mechanics tellsus that below a certain magnitude of distance, usuallycharacterized by a Compton wavelength, classical physicsceases to be reliable; predictions made by classical me-chanics or classical electrodynamics must be replaced byquantum mechanical predictions.”It is the thesis of this note that Rohrlich’s argumentcan be reversed: One may argue with Rohrlich, that itis a consequence of quantum mechanics that the classicalpoint particle looses validity “below a certain distance”.But as Rohrlich argued the point particle is also question-able on the basis of classical electrodynamics. Providedone is willing to follow this argument, Schr¨odinger’s equa-tion can indeed be derived in few steps. II. EXTENDED PARTICLES ANDSCHR ¨ODINGER’S EQUATION
Consider that an extended particle is described by anormalizable spatial density distribution ρ ( t, ~x ) Z ρ ( t, ~x ) d x = 1 . (1)A density is a positive definite quantity: ρ ≥
0. In orderto fulfill this (nasty) requirement, one can express thedensity, for instance, by the square modulus (or someeven power) of (a sum of) auxiliary functions ψ i ( t, ~x )such that ρ ( t, ~x ) = X i ψ ni ( t, ~x ) . (2)For simplicity we use complex numbers and chose thefollowing positive semidefinite expression ρ ( t, ~x ) = ψ ⋆ ψ . (3)The auxiliary function ψ ( t, ~x ) is then due to Eq. 1 squareintegrable. Therefore it’s Fourier transform ψ ( t, ~k ) exists: ψ ( t, ~x ) ∝ Z ˜ ψ ( ω, ~k ) exp [ − i ( ω t − ~k · ~x )] d k dω. (4)But the Fourier transformation alone, without any fur-ther constraint, does not yield a physically model of any-thing. All known physical waves are characterized by arelation between frequency and wavelength, i.e. by a dis-persion relation. As pointed out initially by Sir Hamilton,the velocity of an ensemble of waves, a “wave packet”, isgiven by the so-called group velocity [5–7] ~v gr = ~ ∇ k ω ( ~k ) = ( ∂ω∂k x , ∂ω∂k y , ∂ω∂k z ) T , (5)where ω ( ~k ) is the mentioned dispersion relation. Eq. 5has precisely the form of the velocity equation of classicalHamiltonian mechanics, which relates the velocity to thegradient of the energy (i.e. the Hamiltonian function) inmomentum space: ~v = ~ ∇ p H ( ~p ) (6)Hence, if the wave packet is supposed to provide a de-scription of a classical particle, the (average) velocity ofthe wave packet must agree with the Hamiltonian expres-sion : ~ ∇ k ω ( ~k ) = ~ ∇ p H ( ~p ) . (7)A solution that complies with Eq. 7, where energy andmomentum have the classical units, requires the intro-duction of a proportionality constant with the unit ofaction, let’s call it ~ . Since Eq. 7 equates two deriva-tives, one has to allow for “integration constants”, i.e.functions (potentials) that do not depend on momentumor wave-number. We therefore use ~ ~k = ~p + q ~A ( ~x ) ~ ω ( ~k ) = H ( ~p ) + q φ ( ~x ) (8)with some arbitrary constant q . Note that this is nota physical hypothesis, but a formulation of the linearrestriction that the wave ensemble has to comply with, ifit is supposed to consistently represent a classical particlein some way. The “integration constants” φ and ~A arewell known in classical mechanics and can assumed to bezero for a free particle.The total normalization must of course be preservedand this requires that the wave motion is non-dispersiveor adiabatic . But this is a plausible assumption. MaxBorn referred to the classical adiabatic invariance of thephase space volume Φ = const and to the fact that energy(change) and frequency (change) are, in such processes,proportional to each other δ E = Φ δω [11]. Furthermoreit is long known that the real and imaginary compo-nents of the wave function are subject to Hamiltonianmotion [12, 13].Instead of showing that Schr¨odinger’s equation impliesHamiltonian notions, we consider the reverse argument:if one presumes the validity of energy conservation and Several authors of standard textbooks on QM, for instance Mes-siah [8], Schiff [9] as well as Weinberg [10] use this equation, notto derive Schr¨odinger’s equation, but merely to make it plausible. hence of classical Hamiltonian notions in wave dynam-ics, then Eq. 7 is automatically valid. Hence one arrivesunceremoniously at the de-Broglie relations: E = ~ ω~p = ~ ~k . (9)Inserting this into the Fourier transform yields: ψ ( t, ~r ) ∝ Z ˜ ψ ( E , ~k ) exp [ − i ( E t − ~p · ~x ) / ~ ] d p d E . (10)Once this is written, it is obvious that the energy is givenby the time derivative, and momentum by the spatialgradient. That is, the canonical “quantization” rules di-rectly follow: E ψ ( t, ~r ) = i ~ ∂∂t ψ ( t, ~r ) ~p ψ ( t, ~r ) = − i ~ ~ ∇ ψ ( t, ~r ) . (11)Using these relations to express the classical (kinetic) en-ergy of a free particle E = ~p m results in Schr¨odinger’sequation for a free particle: i ~ ∂∂t ψ ( t, ~x ) = − ~ m ~ ∇ ψ ( t, ~r ) . (12)Adding a potential energy (density) ρ ( t, ~x ) V ( ~x ) readilyyields Schr¨odingers equation for a particle in potential V ( ~x ): i ~ ∂∂t ψ ( t, ~x ) = (cid:18) − ~ m ~ ∇ + V ( ~x ) (cid:19) ψ ( t, ~r ) . (13)The “derivation” of Schr¨odinger’s equation - as presentedhere - is short and physically rigorous. It provides anew perspective on the relationship between classical me-chanics and quantum theory and shows that, contrary tousual assertions, these theories are not mathematicallydisjunct. Since the Copenhagen view suggests that quan-tum theory is merely a formalism to predict measurementoutcomes (“detector clicks”), nothing should prevent usfrom deriving Schr¨odinger’s equation in the simplest pos-sible way. III. SUMMARY AND CONCLUSIONS
As Rohrlich’s analysis reveals, the alleged intuitive-ness and logic of the notion of the point particle fails,on closer inspection, to provide a physically and logicallyconsistent classical picture. If we dispense this notion,Schr¨odinger’s equation can be easily derived and mightbe regarded as a kind of regularization that allows tocircumvent the problematic infinities of the “classical”point-particle-idealization.Our presentation demonstrates that the “Born rule”,which states that ψ ⋆ ψ is a density (also a “probabilitydensity” is positive semidefinite), can be made the initialassumption of the theory rather than it’s interpretation.However, as well-known, Schr¨odinger’s equation is notthe most fundamental equation, but is derived from theDirac equation. Only for the Lorentz covariant Diracequation we can expect full compatibility with electro- magnetic theory. We have shown elsewhere how theDirac equation can be derived from “first” (logical) prin-ciples [14–16]. The derivation automatically yields theLorentz transformations, the Lorentz force law [17–19]and even Maxwell’s equations [14] in a single coherentframework. [1] P.A.M. Dirac “Does renormalization make sense?” AIPConf. Proc. Vol. 74 (1981), pp. 129-130.[2] Roger Joseph Boscovich, S.J. “A Theory of NaturalPhilosophy” Open Court Publ. Chicago/London (1922). https://archive.org/details/atheoryofnaturalphilosophy/ See also