A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum, tat is, λmw = /m v. But, te failure to obtain a reasonable teory for te matter wave frequency resulted somewat in loss of furter interest. It was expected tat te frequency of te matter wave sould correspond to te particle kinetic energy, tat is, fmw = ½ m v2/ but te resulting velocity of te matter of te particle, v = fmw λmw, is tat te matter wave moves at one alf te speed of te particle, obviously absurd as te particle and its wave must move togeter. If relativistic mass is used (as it sould in any case) te problem remains, te same mass appearing in numerator and denominator and canceling. It is no elp to ypotesize tat te total energy, not just te kinetic energy, yields te matter wave. Tat attributes a matter wave to a particle at rest. It also gives te resulting velocity as c2/v, te wave racing aead of its particle. A reinterpretation of Einstein's derivation of relativistic kinetic energy (wic produced is famous E = m c2) leads to a valid matter wave frequency and a new understanding of particle kinetics and of te atom's stable orbits. Roger Ellman, Te-Origin Foundation, Inc. 320 Gemma Circle, Santa Rosa, CA 95404, USA RogerEllman@Te-Origin.org ttp://www.te-origin.org 1
Te Matter Wave Problem A Reconsideration of Matter Waves by Roger Ellman In te early 20t Century (1924) DeBroglie proposed tat, since ligt, wic was ten considered to be a purely wave penomenon, ad been found to appear sometimes to exibit particle beavior; peraps matter, wic was accepted as being particle in nature migt sometimes exibit wave beavior. DeBroglie reasoned tat, te wavelengt of a poton being equal to Planck's constant divided by te poton's momentum, te same relationsip sould apply to a particle of matter -- it aving a wavelengt of divided by te particle momentum. Te reasoning was as follows. First considering a poton, its energy is (1) W wave = f and te energy equivalent of a mass, m, is (2) W mass = m c 2 Wile te poton's rest mass is zero it as kinetic mass corresponding to its energy. If te poton equivalent mass, m, actually appears as a wave its energy as a wave must be te same as its energy as a mass. Terefore (3) W mass = W wave m c 2 = f f m = [solving te above for m] c 2 [substituting c = λ f = for one of te c's λ c in te denominator] and, finally, (4) λ = [solving (3) for λ] m c = poton momentum recognizing tat momentum is defined as te product of mass and its velocity and te velocity of te poton is c. 2
DeBroglie ypotesized tat te wave aspect of a particle of matter sould ave an analogous wavelengt, λmw, tat sould be (5) λ mw = = particle momentum m v Tis suggestion of DeBroglie was soon verified by Davison and Germer wo obtained electron diffraction patterns and found tat te observed wavelengts of te electron matter waves corresponded well wit DeBroglie's formulation. At tat point one would tink tat te duality of matter, as of ligt, was establised and tat extensive furter investigation of matter waves would ave resulted. But tat was not te case and te reason was a fundamental problem tat could not be overcome -- te matter wave frequency. If one reasons, analogously to te derivation of λmw, tat te kinetic energy of te particle of matter sould correspond to its matter wave frequency, fmw, as (6) W k f mw = ½ m v = 2 ten te velocity of te matter wave is (7) v mw = λ mw f mw ½ m v2 1 = = v m v 2 a result tat states tat te matter wave moves at one alf te speed of te particle. Tat is obviously absurd as tey must move togeter eac being merely an alternative aspect of te same real entity. For tem not to move togeter would be as absurd as for te particle aspect of ligt to move at a different speed tan its wave aspect, te poton not arriving coincident wit te E-M wave. It is no elp in resolving tis difficulty if relativistic mass is used (as it sould be in any case) since te same mass appears in bot numerator and denominator of equation (7) were tey simply cancel out. It is also no elp to ypotesize tat it is te total energy, not just te kinetic energy, tat yields te matter wave. Suc an attempt attributes a matter wave to a particle at rest. It also gives te resulting matter wave velocity as c2/v wic as te matter wave racing aead of its particle. No, te two must keep pace wit eac oter since tey are te same ting merely looked at in one or te oter of two alternative ways. It was te inability to resolve tis problem tat led to te loss of interest in matter waves and essentially te end of furter inquiry wit regard to te wave aspect of matter. Einstein's Derivation of Relativistic Kinetic Energy Kinetic energy, KE, is defined as te work done by te force, f, acting on te particle or object of mass, m, over te distance tat te force acts, s. Tis quantity is calculated by integrating te action over differential distances. 3
(8) s KE = f ds 0 [Per above definition] s d(m v) = ds dt [Newton's 2 nd law] 0 (m v) ds = d(m v) [Rearrangement of form] dt 0 (m v) = v d(m v) [v = ds / dt ] 0 v m r v = v d [m is m r Lorentz v2 ½ contracted by v. m r is rest mass] c2 0 m r v 2 v v dv = - m r [Integration v2 ½ v2 ½ by parts] 0 (9) m r v 2 v2 ½ KE = - m r c 2 - m r c 2 [Integration v2 ½ of 2nd term] (10) v2 m r v 2 + m r c 2 [Place 2 nd term = m r c 2 over 1 st term v2 ½ denominator] m r v 2 + m r c 2 - m r v 2 = m r c 2 [Expand term v2 ½ wit brackets] m r c 2 = m r c 2 v2 ½ [Simplify] 4
(11) KE = m v c 2 m r c 2 [m v is total mass at v 0 Tis result states tat: m r is total mass at v = 0 m v = m r Lorentz transformed] {Kinetic Energy} = {Total Energy} - {Rest Energy} or {Total Energy} = {Kinetic Energy} + {Rest Energy} Te appearance in tis result tat te energies are te product of te masses times c 2, te speed of ligt squared, was te origination of tat concept, te famous Einstein's E = m c 2. Te concept falls out naturally from applying te Lorentz transforms to te classical definition of kinetic energy. It is somewat surprising tat Einstein was te first to do tat inasmuc as it was Lorentz wo developed te Lorentz transforms and te Lorentz contractions. Alternative Treatment of te Same Derivation If in te above original derivation one proceeds differently from equation (9)on, as below, a sligtly different result is obtained. (9) m r v 2 v2 ½ KE = - m r c 2 - m r c 2 [Repeat (9) v2 ½ to start] (12) m r v 2 v2 ½ KE + m r c 2 = - m r c 2 [Move te v2 ½ "- m r c 2 "] Considering and evaluating te tree terms of equation (12): (13) KE + m r c 2 = Kinetic plus rest energies = Total Energy = m v c 2 (14) m r v 2 A relativistically increased = energy of motion wic equals v2 ½ zero wen v = 0. = m v v 2 (15) v2 ½ A relativistically reduced m r c 2 = rest energy wic equals te at rest energy wen v = 0 = Equation(13) Equation (14) = m v c 2 - m v v 2 te result is tat equation (12) is equivalent to 5
(16) Total Energy in Energy in Energy = Kinetic Form + Rest Form m v c 2 = m v v 2 + m v (c 2 - v 2 ) and (dividing te above energy equation by c 2 to obtain an equation in mass) (17) Total Mass in Mass in Mass = Kinetic Form + Rest Form m v = m v v 2 / c 2 + m v (1 - v 2 / c 2) Wy is te formulation for classical Kinetic Energy KE = ½ m v2 but Energy in Kinetic Form is simply m v2 witout te ½? Wen dealing wit quite small velocities (v very small relative to c) te excursion of total energy above rest energy and te excursion of energy in rest form below rest energy are bot essentially linear. In tat case te portion above te rest case is essentially alf of te total excursion above and below te rest case. Te classical kinetic energy is ten alf, ½ m v2, te total energy in kinetic form, m v2, for [v/c] quite small. Application to te Problem of te Matter Wave Tus te traditional view of kinetic energy as te energy increase due to motion may not be valid as a description of te processes taking place. Before te encountering of te relativistic cange in mass wit velocity te traditional view did not lead to problems in spite of its apparently being an over-simplification. Using mass- and energy-in-kinetic-form to obtain te frequency of te matter wave proceeds as follows. (18) m v v 2 [equation (6), but using W v, f mw = energy-in-kinetic-form, for W k, kinetic energy] Using tis result for matter wave frequency and using te same relativistic mass, m v, in equation (5) for te matter wavelengt te velocity of te matter wave ten is (19) v mw = f mw λ mw m v v 2 = m v v = v and te wave is traveling wit and as te particle. On tat basis te wave aspect of matter is ten establised bot experimentally (Davison and Germer and teir successors) and teoretically (te above development). Tat gives new significance to te fact, observed at te time of Bor's development of te relationsip between atomic line spectra and atomic orbital structure, tat te stable orbits of atomic electrons are an integer multiple of te orbital electron's matter wave lengt. Te fact of te stable orbits as long been accepted witout a specific reason, a specific operative cause, for tose orbits and only tose orbits being stable. Te matter wave of te orbiting electron now provides an operative reason, as follows. 6
For te orbit to be stable it must be te same for eac pass, pass after pass. If eac pass includes exactly an integer number of te orbital electron's matter wave lengts ten eac pass is te same in tat regard. But if, for example, te orbital pat lengt contains only 9/10 of a matter wave lengt, 9/10 of te matter wave period, ten te next pass will contain te missing 1/10 of te matter wave lengt or wave period plus 8/10 of te next, and so on. Te matter wave being sinusoidal in form, te successive orbital passes will be all different. It is tis beavior wic operatively causes te "stable orbits", and only tose orbits, to be stable. It as noting to do wit angular momentum nor quantization of angular momentum. For te angular momentum ypotesis tere is no underlying reason nor mecanism to produce stability or instability. Te quantization of angular momentum concept is merely a defined condition, witout operative cause, just as were te "stable orbits" it seeks to explain until teir ere being justified in terms of te operative matter wave beavior Te statement tat te orbital electron's angular momentum is quantized, as in te following traditional equation (20) m v R = n [n = 1, 2, ] 2π is merely a mis-arrangement of (21) 2π R = n = n λ mw [n = 1, 2, ] m v a statement tat te orbital pat lengt, 2π R, must be an integral number of matter wavelengts, n λ mw, long. Te latter statement as a clear, simple, operational reason for its necessity. Te former statement is arbitrary and is justified only because it produces te correct result, even if witout an underlying rational reason. References [1] Tis paper is based on development in R. Ellman, Te Origin and Its Meaning, Te-Origin Foundation, Inc., ttp://www.te-origin.org, 1997, in wic te development is more extensive and te collateral issues are developed. 7