INCURSION OF THE GOLDEN RATIO Φ INTO THE SCHRÖDINGER WAVE FUNCTION USING THE Φ RECURSIVE HETERODYNING SET

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1 INCURSION OF THE GOLDEN RATIO Φ INTO THE SCHRÖDINGER WAVE FUNCTION USING THE Φ RECURSIVE HETERODYNING SET S. Giadioto *, R. L. Amoroso & E.A. Rauscher * Advaced Laser Quatum Dyamics Research Istitute (ALQDRI) 3 Briar Hollow Drive, St. Louis, MO 6346 USA Noetic Advaced Studies Istitute, 68 Jea St, Oaklad, CA USA oetic.advaced.studies@midsprig.com Abstract. Icursio is suggested to be a fudametal physical priciple of the uiverse []. We take steps i this directio by aalyzig recursio i the Schrödiger equatio. The Golde Ratio Phi (Φ ) is a extraordiary ad ubiquitous irratioal umber of value.6833 Phi s presece may be see i both biological ad astroomical realms ad recetly i the quatum mechaical ad physical realms. I the biological realm, the umber Phi ca be see i both Phyllotaxis ad DNA. I the astroomical realm, its presece is foud i the spiral structures of galaxies. I physics, Phi ca ow be related to the g-factors of the electro, proto ad eutro. This paper will demostrate that Phi is also itimately related to the quatum realm by virtue of its presece i the quatum mechaical wave fuctio?(x, y, z, t). The basis for the compact icorporatio of Φ ito the wavefuctio will be derived by solvig the Schrödiger Wave Equatio ad the use of the Phi recursive heterodyig set of wavelegths?. Solutios to the Schrödiger Wave Equatio based o these recursive wavelegths ad Φ will be derived i both Cartesia ad Polar coordiates. Keywords: Schrödiger wave equatio, Phi, heterodyig set, quatum mechaics, state fuctio, B4C. Itroductio The state fuctio derived from solvig the Schrödiger Equatio is a compact relatioship that icludes the Four Basic Costats (B4C), e, p ad Φ origially proposed by Michael Heleus [] whereby he had show a iterestig relatioship betwee the B4C i relatio to the buildig of both the great pyramid of Giza, Egypt ad the Partheo i Athes, Greece. Heleus has postulated that these aciet builders erected structures based o two orthogoal axes. The umbers Phi ad two for the North-South axis ad the umbers e ad p for the East-West axis. Heleus also foud that the B4C are coordiated by a rule of expoets such that a ew costat is created which is the least-mea-square error optimized value of the umber which is simultaeously a root of each of the 4 costats whereby the idex of the root is very close to a iteger of 3 digits or less. This optimized value Heleus desigated as HC (Heleus costat) equal to.647, which is simultaeously approximately the 8 th root of phi, the 5 th root of, the 66 th root of e, ad the 9 th root of pi. If all of these root idices are added up ad the divided by four, the umber is obtaied. The fie-structure costat a is equal to the reciprocal of The differece betwee the fie-structure costat ad the reciprocal of oly amouts to about.4% ad is therefore well withi the bouds of beig scietifically sigificat. The Golde Ratio Phi (Φ ) has bee termed The World s Most Astoishig Number by Mario Livio [3]. Phi ca easily be derived by solvig the simple quadratic equatio, x x =. The two roots of this equatio are Φ = ad φ = or. x = ( ± 5) / There are may hudreds, if ot thousads of relatioships ivolvig Phi. The irratioal umber Φ arises from the well kow Fiboacci Series whereby each successive umber of the series is equal to the sum of the proceedig two (i.e.,,,,, 3, 5, 8, 3,, 34 ). Zero represets the th Fiboacci umber F(), oe represets the first Fiboacci umber F() ad 34 represets the ith Fiboacci umber F(9). It is iterestig to ote that the first ad secod Fiboacci umbers are both equal to oe. The series rather quickly coverges towards the umerical value of Φ whe a umber of the series is divided by the previous umber of the series (i.e., 34/ =.695). Obviously, the farther out the series is take, the closer will be the value of Φ whe the quotiet F()/F(-) is calculated. Some other iterestig exact relatioships to Phi are the followig: (Φ + ) = Φ ;

2 π 5 Si (ilφ ) = i/; Si il Φ = ; Φ = Cos(p/5) ad Si (l i Φ ) =. Additioally, o other 4 umber besides Phi is kow to have the property that whe it is added to oe it exactly equals the square of itself.. The Heterodyig Set of Wavelegths The heterodyig set of wavelegths may be expressed i the followig form [4]: () = Expressio () simply defies the heterodyig set as a ifiite summatio of wavelegths. It states that a sigle wavelegth is actually the sum of a ifiite umber of wavelegths. Now, if we itroduce the mathematical cocept of scale ivariace we may create ay particular scale factor we wish accordig to the followig [5]: () f (scale factor) = = The scale factor f is arbitrary ad could be set to ay umber whatsoever. Now, we ca allow each idividual wavelegth to be equal to a ifiite sum of wavelegths as show i equatio (3) below: (3) Let = i+ ad + = + ad + = i+ 3 for all = Likewise, the followig relatioship of the scale factor f therefore follows: i (4) f = = + + i + i+ 3 for = Additioally, the followig relatioship logically ca be obtaied usig equatio (): = + = + (5) i+ i We ow may set up a equality ratio, cross-multiply ad solve for [6]: (6) + = ad therefore + + = + + Now, if we substitute the above equatio (6) ito equatio () we obtai the followig quadratic equatio [7]: (7) + =. Usig the quadratic formula ± b b 4ac a

3 to solve for + we obtai the followig roots: + where Φ = ad φ = ± + 4 = or + =Φ + ad + = φ+ The ifiitely recursive wavelegths may ow be expressed i terms of Φ i the followig maer: (8) i+ =Φ i+ 3 or + =Φ + Also, substitutio ito equatio () gives: Φ = or + + = ; Φ+ + = (9) i+ i+ i+ 3 ( Φ+ ) = ad sice + I summatio form we obtai the followig: + Φ+ =Φ, we therefore obtai Φ + =. () i+ i+ Φ = I itegral form, the above expressio () becomes: () ( i+ d i+ d) i Φ = I double itegral form the above expressio () reduces to the double itegral: () ( + + ) i Φ i ddi = 3. The Time-Depedet Schrödiger Wave Equatio [6]: The Time-depedet Schrödiger wave equatio for a oe particle system i oe dimesio is: (3) h Ψ h + Ψ V Ψ = i t m x where Ψ system ad V(, ) where the potetial eergy fuctio V(, ) is the time-depedet state fuctio of a particle movig i a oe dimesioal Cartesia Coordiate xt is the potetial eergy fuctio of the particle. Now, restrig ourselves to the special case xt is oly a fuctio of x, we obtai the followig equatio:

4 (4) h Ψ h + Ψ V Ψ = i t m x Usig the separatio of variables techique ad lettig Ψ = f ( t) ψ where ψ is the timeidepedet state fuctio that is oly depedet ox, we ca solve the above expressio (4) as follows: (5) Ψ df ( t) t = dt ψ ad (, ) Ψ xt d ψ x = f ( t) x dx Substitutio of (5) ito the origial equatio (4) gives: (6) h df t h d ψ x ψ + f ( t) V f ( t) ψ = i dt m dx Dividig equatio (6) by f ( t) ψ we obtai the followig simplified differetial equatio: (7) df t d ψ x h h = + V x i f t dt mψ x dx Now, if we equate the left-had side of equatio (7) to the costat E (total eergy of the system), we obtai the followig equatio: (8) df t f t ie = dt h Upo itegratio of both sides of equatio (8) we obtai the followig relatioship: iet (9) l f ( t) = + C h form we obtai: where C is a arbitrary costat of itegratio. Covertig this equatio i expoetial iet c iet / h () f ( t) = ee = Ae factor i the fuctio ψ ad ca be omitted from / () f ( t) = e iet h h Sice A is a arbitrary costat of itegratio, it ca thus be icluded as a f t. Thus equatio () may be simplified to: 4. Time-Idepedet Schrödiger Wave Equatio [7]: Now, if we equate the right side of equatio (7) to E we obtai the Time-idepedet Schrödiger wave equatio:

5 () h m d ψ dx ψ ψ + V x x = E x The above equatio () describes the motio of a sigle subatomic particle of mass m movig i oe dimesio x. Thus, for cases where the potetial eergy is a fuctio of x oly, there exist wave fuctios of the form: hc h. Sice E = ad h = equatio (3) ca be re-writte as π π / Ψ xt, = e ψ x. Normalizatio of the wave-fuctios, of course, requires that multiplicatio of iet / h (3) Ψ = e ψ (4) * * each wave-fuctio by its complex cojugate ( Ψ adψ ) satisfy the followig relatioship: ψ Ψ =. We may ow therefore create a series of wave-fuctios based o the recursive wavelegths: π / π i (5) Ψ = e ψ i i where aturally the l Ψ i = + lψ i. i Or, however, i terms of -recursive wave-fuctios where =,, 3, 8. π / (6) Ψ = e ψ ad l lψ Multiplyig equatio (6) by (7) π Ψ = +. we obtai ( ) l Ψ xt, lψ x + π = or Ψ l + π =. From the previously derived recursive wavelegth equatios we obtai the followig relatioship: terms leads us to the followig: Ψ + l + π = + +. Expasio of ( xt) ( xt) Ψ,, l ( Ψ ) l π =. Sice =Φ we may re-write the above equatio as: + + ( xt) ( xt) Ψ,, l ( Ψ Φ ) l π =. Factorig out + Ψ l we obtai + Ψ l ( Φ+ ) + π = or sice ( Φ+ ) =Φ we obtai Ψ ( Φ + )l = ψ π Φ we get: +. Dividig by

6 (8) (9) l Ψ ψ Ψ π = ψ Φ + π / Φ + = e or,. I expoetial form this expressio becomes: / xt, ψ xe π Φ + Ψ =. We have ow related the three most importat ad ubiquitous irratioal umbers e, π ad Φ as well as the speed of light c, time ad the recursive wavelegths to the state fuctios of the time-idepedet Schrödiger Wave Equatio. Phi is the most irratioal umber kow sice it approaches itself via the famous Fiboacci Series slower tha ay other irratioal umber. The Fiboacci series is as follows:,,,,3,5,8,3,,34,55,89 Each umber i the series is equal to the sum of the two umbers precedig it. If you take ay umber of the series (preferably as far out as possible) ad divide it by the previous umber of the series, you will obtai a value that is very close to Phi. Obviously, the further out you take the series, the closer it will get to Phi. Sice Phi is a irratioal umber, it cotais a ifiite umber of digits ad therefore you would eed to take the Fiboacci Series out to a ifiite umber of terms (i.e., F () (8)) i order to obtai the exact value of Phi. 5. Eigevalue Solutios To The Schrodiger Wavefuctio Based O Φ Usig The Hamiltoia Operator The Hamiltoia operator for a wavefuctio for a oe particle system i a Cartesia coordiate four-dimesioal space-time is defied as follows [9]: ^ h i. i = m (3) H = + V( x,..., z ) Usig the above Hamiltoia (3) to operate o the wavefuctio we obtai: h + V x z Ψ x y z t = Eψ i = m (3) (,..., ) (,,, ) i i i i The solutio to equatio (3) usig Mathematica 5. is as follows: π i i i h i i i i m (3) V( x, y, z ) Φ + e ψ ( x, y, z ) The solutio to the Phi-Based Time-Idepedet Schrödiger Wave Equatio i polar coordiates usig Mathematica 5. is show below i equatio (33): (33) π Φ + e ψ ψ ψ ψ ψ h V r, θϕψ, r, θϕ, r r csc cot + + θ + θ + mr r r ϕ θ θ 6. Three-Dimesioal Plots of the Pre-Expoetial Factors for the Φ -Recursive Wavefuctio Below are show the 3D-plots of the pre-expoetial factors for the Phi-Recursive wavefuctios geerated usig Mathematica 5.:

7 Figure. Plot A: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 8},{ +,^,^ 9}] Plot B: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 7},{ +,^,^ 9}] Figure. Plot A: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 6},{ +,^,^ 9}] Plot B: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 5},{ +,^,^ 9}] Figure 3. Plot A: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 4},{ +,^,^ 9}] Plot B: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 8},{ +,^ 8,^ 3}]

8 Figure 4. Plot A: Plot 3D [.788^ {( * ) *t}/( + ),{ t,,^ 8},{ +,^ 8, }] Plot B: Plot B: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 4},{ +,^,^ 9}] Figure 5. Plot A: Plot 3D [.788^ {( * ) *t}/( + ),{ t,, ^ 4},{ +,^,^ 6}] Plot B: Plot B: Plot 3D [.788^ {( * ) * t}/ ( + ),{ t,,^ 6},{ +,^,^ 8}] Figure 6. Plot A: Plot 3D [.788^ {( * ) *t}/( + ),{ t,, ^ 8},{ +,, }]

9 7. Coclusios I coclusio, it may be surmised that the state fuctio? (x, y, z, t) of the Time-idepedet Schrödiger wave equatio is directly proportioal to a pre-expoetial factor cotaiig the four basic costats (B4C), amely, e, p ad Φ ad the time-idepedet wavefuctio?(x, y, z). This coclusio arises as a direct result of the icorporatio of the heterodyig set of wavelegths ito the actual classical Schrödiger wave equatio. Also, it appears that by so doig, this paper has demostrated that idividual wavelegths ad/or frequecies are actually a summatio of a ifiite umber of wavelegths or frequecies. This cocept, brought forth i this paper teds to support the so-called May Worlds Iterpretatio (MWI) of quatum theory as opposed to the Copehage Iterpretatio whereby the collapse of the quatum mechaical wavefuctio? occurs as a result of the mere observatio of a subatomic particle. Additioally, perhaps the most importat cocept resultig from this paper is that the ubiquitous irratioal umber Phi (Φ ) is both a itegral ad essetial costat i the quatum mechaical realm of reality. Refereces [] Dubois, D.M., 998, Computig Aticipatory Systems with Icursio ad Hypericursio, I Computig Aticipatory Systems: CASYS st Iteratioal Coferece, D. M. Dubois (ed). America Istitute of Physics, AIP Coferece Proceedigs 437, pp [] Heleus, M. refereces authors Rocky McCollum ad Peter Tompkis who foud that the compass poit axes of the Great Pyramid at Giza, Egypt, ad the Partheo i Athes, Greece related to pairs of the B4C. [3] Livio, M., The Golde Ratio, Broadway Books, New York (). [4] Va de Bovekamp, F., Witer, D., Private Commuicatio. [5] Va de Bovekamp, F., Witer, D., Private Commuicatio. [6] Va de Bovekamp, F., Witer, D., Private Commuicatio. [7] Levie, I.N., Quatum Chemistry, 4th Editio, Pretice Hall, Eglewood Cliffs, (99), p-3. [8] Levie, I.N., Quatum Chemistry, 4th Editio, Pretice Hall, Eglewood Cliffs, (99), p.3. [9] Levie, I.N., Quatum Chemistry, 4 th Editio, Pretice Hall, Eglewood Cliffs, (99), p. 47.