Why Professor Richard Feynman was upset solving the Laplace equation for spherical waves? Anzor A. Khelashvili a)
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1 Why Pofesso Richad Feynman was upset solving the Laplace equation fo spheical waves? Anzo A. Khelashvili a) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9, 009, Tbilisi, Geogia and St. Andea the Fist-called Geogian Univesity of Patiachy of Geogia, Chavchavadze Ave. 53a, 06, Tbilisi, Geogia Teimuaz P. Nadaeishvili b) Iv. Javakhishvili Tbilisi State Univesity, Faculty of Exact and Natual Sciences, Chavchavadze Ave. 3, 079, Tbilisi, Geogia and Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9, 009, Tbilisi, Geogia Abstact. We take attention to the singula behavio of the Laplace opeato in spheical coodinates, which was established in ou ealie wok. This singulaity has many nontivial consequences. In this aticle we conside only the simplest ones, which ae connected to the solution of Laplace equation in Feynman classical books and Lectues. Feynman was upset looking in his deived solutions, which have a fictitious singula behavio at the oigin. We show how these inconsistencies can be avoided. Keywods: Laplace equation, Spheical and Catesian coodinates, bounday condition. PACS numbes: w, Ca, Ta I. INTRODUCTION R.Feynman in his Lectues discussed the deivation of spheical waves on the basis of wave equation in spheical coodinates. Deived solution has a singulaity at the oigin = 0. He wote about this solution the following: Ou solution must epesent physically a situation, in which thee is some souce at the oigin. In othe wods, we
2 have inadvetently made a mistake. We have not solved the fee wave equation eveywhee; we solved it with zeo on the ight eveywhee except at the oigin. Ou mistake cept in because some of the steps in ou deivation ae not legal when = 0.. R. Feynman pobably had meant singula behavio of Laplace opeato at the oigin. Nowadays we know that Laplacian is indeed singula in spheical coodinates,3 and a moe caution teatment is necessay. Ou aim in this aticle will be caeful investigation of Feynman s poblem. This aticle is oganized as follows. In Sec. II we conside the electostatic poblem. In Sec. III we conside the Yukawa potential. Sec. IV we give concluding emaks. II. ElECTROSTATIC PROBLEM Let us begin, follow Feynman, by electostatic poblem, whee the same mistake occus. R.Feynman mentioned: Let s show that it is easy to make the same kind of mistake in an electostatic poblem. Suppose we want a solution of the equation fo an electostatic potential in fee space, ψ = 0. In an explicit fom this equation looks like ψ ( ) = d ψ dψ + = 0 d d () R.Feynman continued: It is often moe convenient to wite this equation in the following fom d ψ = ( ψ ) () d If you cay out the diffeentiation indicated in this equation, you will see that the ight hand side is the same as in pevious equation
3 We want to emphasis that exactly this statement fails at = 0 4. It was shown in 4 that the coect elation looks like d ψ = d ) (3 ( ψ ) 4πδ ( ) ( ψ ) (3) Theefoe some of elations of Feynman s book will undego elevant coections. If we intoduce the epesentation of 3-dimensional delta function in spheical coodinates, namely, and use taditional shot elation ( ) ( 3 ) δ δ () =, (4) 4π u ( ) ψ ( ) we deive the following fom of Laplace equation = (5) d u δ ()() u = 0 (6) d It seems that afte tansition to function u( ) thee appeas souce-like tem in the Laplace equation δ ()() u = δ ()() u 0. It is caused by singula chaacte of Jacobian of tansfomation fom Catesian to spheical coodinates, J = sinθ at the oigin. (usually the singulaity with espect to θ is avoided by the equiements of discontinuity and uniqueness, which esults in appeaance of spheical hamonics ( θ, ϕ) How can we eject this exta tem fom equation? It depends on value of ( 0) Υ lm 5. u. thee ae thee possibilities: a finite u ( 0) must be excluded, because afte etuning to ψ, thee appeas undesiable / tem, which is not a solution of Laplace equation. The second possibility u ( 0) = must also be ejected, as the pesence of infinite tem in 3
4 ( 0) 0 equation is nonsense. So, thee emains only possibility, u = and if at the same time we take this function to behave as u ε ( ) + 0, ( ε > 0) (7) then this tem kills the delta function and we etun to the standad equation. This consideation shows, that the standad educed adial equation is valid only in cases when the estiction u ( 0 ) = 0 is fulfilled 3. In this case standad equation looks like d d u = 0 (8) and has a solution () = a b u + (9) Taking into account the above estiction, we obtain that b = 0 and, theefoe u ψ = = a = const. While R.Feynman mentioned: found that the following ψ is a solution fo the electostatic potential in fee space b ψ = a + (0) Then R.Feynman continued: Something is evidently wong. In the egion whee thee ae no electic chages, we know the solution fo the electostatic potential: the potential is eveywhee constant. That coesponds to the fist tem in ou solution. But we have also the second tem, which says that thee is a contibution to the potential that vaies as one ove the distance fom the oigin. We know, howeve, that such a potential 4
5 coesponds to a point chage at the oigin. So, although we thought we wee solving fo the potential in fee space, ou solution also gives the field fo a point souce at the oigin. We see that igoous consideation of adial Laplace equation puts all things on thei own places. The estiction u( 0 ) = 0 has a decisive meaning. The natual question aises: Why does the bounday condition-like estiction aises in the fee equation? This happens because tansfomation to spheical coodinates does not involve = point and the tansition to u () function feels this, because of facto. 0 The second tem in Feynman consideation b / is not a solution at all. Indeed, afte its substitution into Laplace equation we obtain a delta function, instead of zeo. Anothe way to avoid this solution is a compaison to Catesian solution, whee the wave function at the oigin is constant, as it is clea fom the solution of the in this coodinates 6 ψ = 0 equation ψ = e ± iαx e ± iβy e ± α + β z () The same can be demonstated by consideing chaacteistic equation fo (), substituting thee ψ s, it follows s s ( s + ) = 0 integating this equation by the spheical element s s + ( s + ) ( ) = 0 s+ b a d in abitay bounds, we obtain () (3) it follows that we have only one solution s = 0, which coesponds to ψ = const accodance with Catesian behavio., in 5
6 III. YUKAWA POTENTIAL Anothe place whee R.Feynman made use the elation () is the Yukawa potential (ibid. Chapte 8). It is commonly believed that the Yukawa potential is a spheically symmetic solution of well-known wave equation φ μ φ = 0 (4) Aming with the pevious consideation, R.Feynman wote this equation in the following fom ( φ) μ φ = 0 (5) solution of which is φ = ± Ke μ and theefoe afte a suitable bounday condition at infinity, it follows the Yukawa potential μ e φ = K (6) But we know, that a igoous application of coect elation (3) gives μ e μ e = μ ( 3 ) μ () 4 πδ e, (7) Inteesting enough that this fom was given in the ealie book 7. It follows that the Yukawa potential is not a spheically symmetic solution of Eq. (4) eveywhee, but it is such only ahead of oigin. We see fom Eq. (7) that the Yukawa potential is a solution of nonhomogenious wave equation with a souse tem on the ight-hand side φ μ φ = 4 πk δ 3 ( ) ( ) (8) 6
7 IV. CONCLUSIONS In conclusion, we have demonstated that caeful consideation of Laplacian opeato nea the oigin emoves all the inconsistencies elated to Feynman s analysis. The only point which must be claified is a igoous detemination of a chaacte of aspiation to zeo of u() function in ode to povide u ( ) δ ( ) = 0, which is a business of the theoy of distibutions. Moeove we note that things will be changed in many applications whee the Laplace opeato is used in spheical coodinates. ACKNOWLEDGEMENTS Authos acknowledges financial suppot of the Shota Rustaveli National Science Foundation (Pojects DI/3/0 and FR//4) a) Electonic mail: anzo.khelashvili@tsu.ge b) Electonic mail: teimuaz.nadaeishvili@tsu.ge 7
8 R.Feynman, R.Leighton and M.Sands, The Feynman Lectues on Physics (Addison- Wesley Publishing Company,Inc.Reading,Massachusetts, Pablo Alto, London,964) Vol.II, Electomagnetism. Chapte 0. Anzo A.Khelashvili and Teimuaz P. Nadaeishvili, Unexpected delta-function tem in the adial Schodinge equation, e-pint axiv: Anzo A.Khelashvili and Teimuaz P. Nadaeishvili, What is the bounday condition fo adial wave function of the Schödinge equation?, Am. J. Phys. 79 (6), (0). 4 Anzo A.Khelashvili and Teimuaz P. Nadaeishvili, Delta-like singulaity in the Radial Laplace Opeato and the Status of the Radial Schodinge Equation, Bulletin of the Geogian National Academy of Sciences (Moambe) 6 (), (0). axiv: See, any book on quantum mechanics. 6 J. D. Jackson, Classical Electodynamics, 3d ed. (John Wiley & Sons, New Yok, 999). 7 H.Bethe and F. Hoffmann, Mesons and Fields, (Row Peteson and Company, New Yok, 955). Vol, p
Keywords: Schrödinger equation, radial equation, boundary condition, singular potentials.
What is the bounday condition fo the adial wave function of the Schödinge equation? Anzo A. Khelashvili a) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9, 19,
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