The Schrödinger Equation and the Scale Principle

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1 Te Scrödinger Equation and te Scale Princile RODOLFO A. FRINO Jul 014 Electronics Engineer Degree fro te National Universit of Mar del Plata - Argentina rodolfo_frino@aoo.co.ar Earlier tis ear (Ma) I wrote a aer entitled Scale Factors and te Scale Princile. In tis aer I forulated a new law wic describes a nuber of fundaental quantu ecanical laws. Since ten I found tat oter quantu ecanical laws including te Bor ostulate and te De Broglie wavelengt forula also obe te scale rincile. Later I roved tat tis new law also describes te forula for te Scwarscild radius, te equation for Einstein s relativistic energ and Newton s law of universal gravitation. Now I discovered tat te Scrödinger s equation can also be elained in ters of te resent forulation. Kewords: Scrödinger equation, De Broglie wavelengt, wave nuber, wave function, differentiation, first order derivative, second order derivative, Lalacian oerator. 1. Introduction In 01 I forulated te Scale Princile or Scale Law. I ublised te first version of tis aer in Ma tis ear (014). In te first version tis law was called te Quantu Scale Princile. However after finding tat Einstein s relativistic energ also obes tis law, I canged its nae to te scale rincile or scale law. Since te first version te rincile as evolved to te resent for given b te following relationsi: (1) Meta Law: Scale Princile or Scale Law (1a) Meta For (Meta Quantities) (1b) Elicit For (ratio, eonents and scale factor) M1 R S M M 1 diensionless Meta Quantit 1 M diensionless Meta Quantit S diensionless Meta scale factor R Meta Relationsi Te Q 1 Q n [ < > ] (See details below) Q 3 S Q 4 Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 1

2 Te above sbols stand for a) Quantities: (i) Q 1, Q, Q 3 and Q4 are sical quantities of identical diension (suc as Lengt, Tie, Mass, Teerature, etc), or ii) Q 1 and Q are sical quantities of diension 1 or diensionless constants wile Q 3 and Q 4 are sical quantities of diension or diensionless constants. However, if Q 1 and Q are diensionless constants ten Q 3 and Q 4 ust ave diensions and viceversa.(e.g.: Q 1 and Q could be quantities of Mass wile Q3 and Q 4 could be quantities of Lengt). Te sical quantities can be variables (including differentials, derivatives, Lalacians, divergence, integrals, etc.), constants, diensionless constants, an ateatical oeration between te revious quantities, etc. b) Relationsi te: Te relationsi is one of five ossibilities: less tan or equal to inequation ( ), or less tan inequation (<), or equal to - equation (), or a greater tan or equal to inequation ( ), or a greater tan inequation (>). c) Scale factor: S is a diensionless scale factor. Tis factor could be a real nuber, a cole nuber, a real function or a cole function (strictl seaking real nubers are a articular case of cole nubers). Te scale factor could ave ore tan one value for te sae relationsi. In oter words a scale factor can be a quantu nuber. Tere ust be one and onl one scale factor er equation. d) Eonents: n and are integer eonents: 0, 1,, 3, Soe eales are: eale 1: n 0 and 1; eale : n 0 and ; eale 3: n 1 and 0; eale 4: n 1 and 1; (canonical for) eale 5: n 1 and ; eale 6: n and 0; eale 7: n and 1; It is wort to reark tat: i) Te eonents, n and, cannot be bot ero in te sae relationsi. ii) Te nuber n is te eonent of bot Q 1 and Q wile te nuber is te eonent of bot Q 3 and Q 4 regardless on ow we eress te equation or inequation (1). Tis eans tat te eonents will not cange wen we eress te relationsi in a ateaticall equivalent for suc as Q 4 Q 3 [ < > ] Q S Q 1 n Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts

3 iii) So far tese integers are less tan 3. However we leave te otions oen as we don t know weter we sall find iger eonents in te future. iv) Wen bot eonents, n and, are equal to one, ten we sa tat te equation is in its canonical for. Wenever we eress a articular law of sics in te for of te Scale Law, we sould use its canonical for, if ossible, rovided we don t i u te variables. Q1 Q4 < > S Q Q3 n n Te scale law (1) can also be written as [ ]. Derivation of te Scrödinger Equation How did Scrödinger derive is faous equation []? It ust ave coe fro at least te following knowledge: (a) te De Broglie s relationsi between te wavelengt and te oentu of a article, (b) te classical ecanical energ of a article, (c) te classical wave equation (wic uses te second order derivative wit resect to tie as oosed to te first order derivative in Scrödinger s equation), and (c) differentiation. But te question still reains: were did te idea of using a cole wave function coe fro? To derive te Scrödinger s equation we sall ostulate te eistence of a quantu field or cole wave function (,,, t) tat reresent all tere is to know about a given article. Ten we sall use differentiation. Te derivation is a sile and straigtforward ateatical rocess. Before starting te derivation, it is wort to reark tat te units of te alitude of te 3/ 3/ wave function are 1 ( stands for eters). Now we sall derive bot te tie indeendent and tie deendent Scrödinger s equations. Te tie deendent equation is: U i t () In general te wave function will deend on te tree satial variables,, and on te teoral variable t, ateaticall f (,,, t) (3) Let us ostulate tat tere is cole quantu field or a cole wave function suc as i ( k k k wt) (,,, t) Ae (4) And tat tis function as a robabilistic interretation as te one given b Ma Born [3]. Tis is just an eale of te wave function to illustrate te derivation rocess. We could ave used oter aroriate wave functions suc as: Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 3

4 ) iwt (,,, t) Asin( k )sin( k )sin( k e, or ) iwt (,,, t) Acos( k ) cos( k ) cos( k e, etc. and we would ave got te sae concetual results. Te wave nubers of equation (4) are defined as k k k π (5) π (6) π (7) Considering te De Broglie relationsi we can write (8) (9) (10) were,, coonents of te oentu of te article along te ais, ais and ais resectivel,, De Broglie wavelengts associated wit a article of ass. (Altoug te article as onl one wavelengt () and since te oentu of te article as tree coonents suc as te ones tat arise fro using a rectangular coordinate sste, we can associate a different wavelengt (,, ) wit eac coonent (,, ) of te oentu.) Planck s constant Ten we can eress te wave nubers in ters of te coonents of te oentu. Tis gives k (11) k (1) k (13) Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 4

5 Now we eress te angular frequenc ω in ters of te classical ecanical energ E of te article. (For assless article suc as otons te total relativistic energ equals its kinetic energ. However ere we are dealing wit te general case of assive articles and terefore we use te classical ecanical energ). Tis gives E ω (14) Substituting k, k, k and ω in equation (4) wit equations (11), (1), (13) and (14) resectivel ields (,,, t) Ae i E t (15) i ( Et ) (,,, t) Ae (16) Now we define te variable θ (,,, t ) θ to silif te differentiation rocess Et θ (17) Introducing θ in equation (16) we get ) iθ (,,, t A e (18) According to Euler equation (18) can be rewritten as ( cosθ sin θ ) (,,, t) A i (19) We now calculate te first order artial derivative of variable wit resect to te satial A θ θ sin θ i cosθ (0) θ (1) A ( sin θ i cosθ ) () Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 5

6 We continue b calculating te second order artial derivative of wit resect to te satial variable i A θ θ θ cosθ sin (3) ( ) cosθ sin θ i A (4) Coaring equations (4) wit (19) we can write te following relationsi (5) A siilar rocess leads to (6) (7) Adding te tree revious equations ields (8) (9) (30) 0 (31) Now we consider te classical ecanical energ E of te article U K E (3) Were E classical ecanical energ of te article Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 6

7 K classical kinetic energ of te article U otential energ of te article In general te ecanical energ will be a function of tie. Te classical kinetic energ is 1 K v (33) wic can be written as K (34) were is te oentu of te article and is given b v (35) Were is te ass of te article and v is te seed Te ecanical energ can be eressed in ters of te kinetic energ as follows E U (36) Solving for te square of te oentu ields ( E U ) (37) Substituting in equation (31) wit te second side of equation (37) ields ( E U ) 0 (38) Considering tat (39) π we finall find te tie indeendent (TI) Scrödinger s artial differential equation 8π ( E U ) 0 (TI Scrödinger s equation) (40) Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 7

8 If we introduce te Lalacian oerator wic is defined as (Lalacian oerator) we can eress equation (40) in ters of te Lalacian 8π ( E U ) 0 (TI Scrödinger s equation - Lalacian for) (41) We now calculate te first order artial derivative of variable t wit resect to te teoral t A θ θ sin θ i cosθ (4) t t θ t E (43) t AE ( sin θ i cosθ ) (44) Multiling b i ields t AE ( i sin θ i cosθ ) (45) i AE ( cosθ i sin θ ) t (46) Coaring equation (46) wit (19) we get i E (47) t Now we sall substitute E in equation (41) wit te first side of equation (47) to get te tie-deendent (TD) Scrödinger s artial differential equation. A sile ateatical work leads to te result we are looking for: te tie deendent Scrödinger s equation U i t (Tie deendent Scrödinger s equation) (48) 3. Te Scrödinger Equation as a secial case of te Scale Princile Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 8

9 We sall sow tat te tie-indeendent Scrödinger equations (given b equation (41)) obes te scale law. Te scale factor will be deduced fro coarison wit te revious section. Tis articular case requires a ore elaborated scale table tan te ones we used before. Wile a scale table wit a air of balanced coluns was enoug for te revious cases [4, 5, 6, 7, 8] now it isn t. Wat we need is to create a balanced table wit unbalanced coluns. Tis can be acieved if we consider te De Broglie equation (49) Tis can be re-written as (50) v If inside te square root we ultil and dividing b te above equation takes te following for v 1 v (51) We recognie 1 v as te classical kinetic energ K of te article. Ten we write (5) K According to equation (3) we substitute K wit E U. Tis gives (53) ( E U ) Squaring bot sides (54) ( E U ) Te idea is to lace in te first alf of te scale table and 1/ in te oter alf so tat wen we ultil te we sall get. Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 9

10 To balance te table we erfor a sile diensional analsis: we lace bot 1 ( LENGTH 7 / ) and ( LENGTH ) on one alf of te table; and ( 1 LENGTH ) 3/ and ( LENGTH ) on te oter one. Wit tese considerations we can draw te following scale table Lengt Lengt Lengt Lengt Lalacian of te Wave function Wavelengt 1/Wavelengt Wave function 7 / 1 3/ 1 TABLE 1: Tis sile scale table is used to sow tat te Scrödinger equation obes te scale law. Te table ust be balanced as a wole. Now we write te following relationsi 1 S (55) 1 S (56) Fro equations (54) and (56) we get ( E U S (57) ) If we adot S 4π (58) Equation (57) transfors into equation (41) 8π ( E U ) 0 (Tie indeendent Scrödinger equation) (59) Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 10

11 wic is te tie indeendent Scrödinger equation. Now we consider equation (55). Tis equation can be written in te for of te scale law as follows S (60) were n Q 1 Q 0 Q 1 ( doesn' t atter because 3 Q4 1 ( doesn ' t atter because S 4π 0) 0) We sould eress equation (60) in te canonical for (n 1). Tus we write 4 π (Scale Law for te Scrödinger s equation) (61) were n 1 Q 1 Q 1 Q 3 Q 4 S 4π Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 11

12 Tus we ave roved tat te tie indeendent Scrödinger s equation obes te scale law. Let us define te function Units(Q) tat will give us te units of te quantit Q and let us introduce a new concet to quantu ecanics: te Psi radius R wic we sall define as R (Psi Radius) (6) Ten let us verif te units of 3/ 3 Units 7 / 7 R ( R ) Units Tus te units of R are eters as it sould be. Now we eress te Scrödinger s equation in ters of te Psi radius R R 4 π (63) R 4π (64) or R i π (65) Finall i π R (Scrödinger s equation in ters of te Psi radius) (66) Now we see tat te wavelengt of te so called aterial waves is te lengt of a circle of radius R ultilied b te iaginar nuber: i 1. For tis reason equation (6) was defined as a radius and not as a wavelengt. Tus we see tat is real if and onl if R is iaginar. To obtain te tie deendent Scrödinger equation we sil substitute E in equation (53) wit te second side of te following equation Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 1

13 E 1 i (67) t Tis gives 1 i t U (68) Now after sile algebraic stes we get te tie deendent Scrödinger s equation U i t (Tie deendent Scrödinger s equation) (69) 3. Conclusions In suar, we ave derived bot te tie deendent and te tie indeendent Scrödinger equations fro te following assutions: (a) Tere is a cole quantu field or cole wave function wic cannot be easured, (b) Te article as an associated wavelengt tat obes te De Broglie relationsi between te wavelengt and te oentu, (c) Te kinetic energ of te article is te classical kinetic energ (d) Te article obes te rincile of conservation of energ. We ave also defined a new quantu ecanical quantit we called te Psi radius wose full sical eaning is unknown. Altoug in order to obtain te value of te scale factor we ad to coare two equations, we ave roved tat te Scrödinger equation is a secial case of te Scale Law. Tus te Scale Law describes quantu ecanics ost owerful forulation te Scrödinger equation. Taking into consideration tat te Scale Law describes several noral laws as I ave sown bot on revious aers [4, 5, 6, 7, 8] and on tis aer, we can consider te scale law as a ore fundaental law tan te secific laws it describes because te scale law ust ave been conceived before te Big Bang. Tis eans tat te Scale Law wouldn t be a noral law of sics but a Meta Law: a law tat would ave sawned oter laws of sics. But w would tere be onl one Meta Law? Coon sense indicates tat tere ust be oter Meta Laws wic we aven t been able to discover et. Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 13

14 Tus eac noral or secific law of sics ust obe one or ore Meta Laws and one Meta Law governs a nuber of natural laws. Te answer to te question Were do te laws of sics coe fro? is: te coe fro Meta Laws [9]. REFERENCES [1] R. A. Frino, Scale Factors and te Scale Princile (version 7), vixra: , (014). [] E. Scrödinger, Quantiation as a Proble of Proer Values; arts I-IV, Four Lectures on Wave Mecanics, Blackie & Son Liited London and Glasgow, (198). Tese four lectures are included in te Book Te Dreas te Stuff is ade of, Introduction b S. W. Hawking, Running Press, (011). [3] M. Born, Te Statistical Interretation of Quantu Mecanics, Nobel Lecture, (1954) [4] R. A. Frino, Te Secial Teor of Relativit and te Scale Princile, vixra: , (014). [5] R. A. Frino, Te Scwarcild Radius and te Scale Princile, vixra: , (014). [6] R. A. Frino, Te Fine Structure Constant and te Scale Princile, vixra: , (014). [7] R. A. Frino, Te Bor Postulate, te De Broglie Condition and te Scale Princile, vixra: , (014). [8] R. A. Frino, Te Universal Law of Gravitation and te Scale Princile, vixra: , (014). [9] R. A. Frino, Were Do te Laws of Psics Coe Fro, vixra: , (014). Te Scrödinger Equation and te Scale Princile v3. Corigt 014 Rodolfo A. Frino. All rigts 14

Derivation Of The Schwarzschild Radius Without General Relativity

Derivation Of The Schwarzschild Radius Without General Relativity Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas: