The Fine Structure Constant and Interpretations of Quantum Mechanics

EJTP 9, No. 26 (2012) 135–146 Electronic Journal of Theoretical Physics

Ke Xiao∗ P.O. Box 961, Manhattan Beach, CA 90267, USA

Received 10 October 2011, Accepted 7 December 2011, Published 17 January 2012

Abstract: The ﬁne structure constant give a simple derivation of the localized wavefunction, Schrödinger equation, double-slit and the uncertainty principle in Quantum Mechanics. The wave-particle duality link to the space-time property of matter by Planck constant. The 2 2 double-slit formula is |ψX | =2|sinc(a kX)| [1 + cos(akX − ωδT + ϕ)], where the cross-linked T = T cos (θ ) angle for double-slit, 1 c 2 and vice versa. c Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Fine structure constant; Wavefunction; Uncertainty Principle; Quantum Mechanics PACS (2010): 03.65.-w; 42.25.Fx; 61.05.fg

1. Introduction

The fine-structure constant α is deeply involved in the Quantum theory.[1,2] Pauli consid- ered quantum mechanics to be inconclusive without understanding of the fine structure constant.[2] Feynman also said that nobody understands quantum mechanics.[3] There are many controversies surrounding its reality, determinative, causality, localization and completeness. Beyond the Copenhagen interpretation, other interpretations include Many- World, Hidden Variable, de Broglie-Bohm, Ensemble, von Neumann/Wigner, Quantum logic, etc, each with its own compromises. As a new approach, here we discuss a fine structure constant interpretation of Quantum Mechanics.[4] (1) Wavefunction and Wave-Particle Duality One basic problem in the quantum interpretation is the wavefunction. From 1926 to Hˆ Ψ=i ∂ Ψ 1928, there were some proposed quantum wave equations ∂t ; for example

∗ Email: [email protected] 136 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146

[ −2 ∇2 − (r)]Ψ(r )= ∂ Ψ(r )( ) ¨ 2μ V ,t i ∂t ,t a Schrodinger (1) → 1 (→ · (→ − ))2 + ]Ψ = ∂ Ψ ( ) 2m σ p qA qφ 1,2 i ∂t 1,2 b Pauli 2 mc 2 1 ∂ [∇ − ( ) ]Ψ(r )= Ψ(r )() − ,t c2 ∂t ,t c Klein Gordon 2 3 ∂ [βmc + αkPkc]Ψ1,2,3,4 = i Ψ1,2,3,4 (d) Dirac k=1 ∂t where the non-relativistic equations are (a) for the atomic electron conﬁguration and (b) for the Dirac limitation of spin-1/2 particles, and the other two relativistic quantum equations are (c) for the spin-0 free particle and (d) for the spin-1/2 particle-antiparticles. The diﬀerent Hamiltonians Hˆ on the left-side were proposed based on the imitations of various classical physical processes, such as Schrodinger’s Hamiltonian imitating the optical wave equation. The eigenvalues of a negatively charged electron orbiting the positively charged nucleus are determined by the time-independent Schrödinger equation Hˆ Ψ(r)=EΨ(r).[5]

Wavefunction Ψnlm =Rnl (r)Ylm (θ, φ) (n, l, m are quantum numbers) is geometri- cally quantized in a 3D cavity. According to Born, Ψ is the probability amplitude and ΨΨ∗ is the probability density.[6] Each state only allows two counter-spin electrons (m = ± 1 ) e2 |Ψ |2 = |Ψ |2 s 2 per the Pauli principle, nlm α c nlm is the probability density of the paired charge. The Schrödinger wavefunction Ψnlm indeed describes the twins in a box. Dirac’s√ relativistic quantum equation solves the spin-1/2 problem. Then, e |Ψ |2 = |Ψ |2 nlmms α c nlmms is the probability density of the charge. The ﬁne structure constant can be deﬁned as the conservation of angular momentum

e2 = ±α (2) c The wavefunction had an entropy format S = klnΨ for Hˆ Ψ=EΨ in Schrödinger’s ﬁrst paper in 1926.[5] The Boltzmann constant k is linked to α by the dimensionless black- 2 = e2( 4σ )1/3 =( p )1/3 =086976680 = body radiation constant αR and primes αR ck4 p2+1 α . α 1 Ψ(r ) |Ψ(r )|2 157.555 .[7] A plane wave function ,t and the Born probability density ,t are

− i p·r−Et − i Ψ(r,t)=Ae ( ) = Ae Etψ(r)=f(t)ψ(r) (3) ∗ − i − i ∗ |Ψ(r,t)|2 =ΨΨ =[Ae Etψ(r)][Ae Etψ(r)] − i i ∗ = Ae Etψ(r) · Ae Etψ (r)=A2|ψ(r)|2 The Einstein/de Broglie wave-particle duality is linked to the reciprocal space- time properties of matter.[8,9] Note that period T =1/ν [T] and wavelength λ = 1/k [L], the property of particle-wave is deﬁned by the spin over time-space. E = ω i.e.E= hν = h/T (spin/time) (4) p = k i.e. p = hk = h/λ (spin/space) where the conservation of angular momentum in the reduced Planck (Dirac) constant h = ET = pλ (i.e., = E/ω = p/k), and the electron spin /2 can only be interpreted by the 4-dimensional spacetime of the relativistic Dirac equation.[10,11] Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146 137

(2) Schrödinger Equation Derivation From α We notice that e2/c in (2) has the same dimension with , Et,andp · r. Let’s use i = −1/i,ande−iπ = −1=i2 =lne−1 to rewrite (2) in the four-dimensional space-time {r,t} = {x, y, z, t}, i.e., link the charge to space, time, and spin

e2 = p · r − Et = ±α = ±i2 · α · lne−1 = i ln e∓iα (5) c where (p · r − Et)/ =(k · r−ωt)=k(r−ct) link to the relativity. (5) may also work for Et or p · r separately (e.g., let t ≡ 0 for the time-independent wavefunction). From (5),

´ ln ∓iα = − i (p · r−Et)=− (k · r− t)= Ψ du =lnΨ e i ω A u A (6)

i iHˆ t/ iEt/ − In Hilbert space (e = e ), a nonlocal plane wavefunction Ψ(r,t)=Ae Etψ(r), i.e., Ψ(r,t)=Ae−i(p·r−Et)/ is deﬁned as the exponential (6), and the wavefunction can be locally quantized by the ﬁne structure constant as (5)

− i p·r−Et Ψ(r,t)=Ae ( ) = Ae∓iα = A cos(∓α) ∓ iA sin(∓α)=a ∓ bi (7) ´ Ψ(r )= (r)= −iEt/ (r) |Ψ|2 =1 In (7), obviously, ,t anψn ane ψn and V dV .Itisa Hermitian function, where the real part is an even function and the imaginary part is an − i p·r−Et − i · · · −Et odd function. From Ψ(r,t)=Ae ( ) = Ae (Px x+Py y+Pz z ),

∂ Ψ(r )=− i Ψ(r ) ∂t ,t E ,t (8) ∂2 Ψ( )=− 1 2Ψ( ) ∂x 2 x, t 2 Px x, t i.e., the operators can also be derived mathematically from (8)

Eˆ = ∂ ; Pˆ = ∇; Pˆ2 = −2∇2 i ∂t i (9)

The Hamiltonian Hˆ = Tˆ + Vˆ = E is the sum of kinetic and potential energy, and T = 1 P2 2μ . In this way, the Schrödinger equation can be derived from the ﬁne structure constant (2)

[ −2 ∇2 − (r)]Ψ(r )= ∂ Ψ(r ) 2μ V ,t i ∂t ,t (10) −2 ∇2 − (r)]Ψ(r)= Ψ(r) 2μ V E

To solve the Schrödinger equation for the atomic electron configuration, we need to convert Ψ(r)=Ψ(x, y, z) to Ψ(r)=Ψ(r, θ, φ). However, the quantum principle is independent of the coordinate system. The Schrödinger equation can also be mathematically transformed into Heisenberg matrix mechanics or Feynman path integral formulation. Therefore, the quantum principle is also independent of the mathematical expression (some confusion may be caused by usage of different symbols and terms). Although there are different ways to describe Quantum theory, all theoretical physicists agree that 138 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146

the ﬁne structure constant is the magic number , and it can not be derived from the − i p·r − Quantum Mechanics.[1,2] From (7), Ψ(r)=e ( ) = e iα is for a negatively charged electron orbiting a positively charged nucleus, which has the complex number conjugates Ψ(r)=a − bi and Ψ∗(r)=a + bi,thenΨΨ∗ is a real number for the probability density

|Ψ|2 =ΨΨ∗ =(a − bi)(a + bi)=a2 + b2 =1 (11) √ Speciﬁcally, if Ψ=a(1 − i), then the normalization from (11) yields a =1/ 2. After localization, the variable wavefunctions become complex numbers (vectors in

Hilbert space). The complex addition (vector sum) Ψ12 of Ψ1 = a + bi and Ψ2 = c + di is

Ψ12 =Ψ1 +Ψ2 =(a + bi)+(c + di)=(a + c)+(b + d)i (12)

A superposition f(x1 + x2 + ···)=f(x1)+f(x2)+··· is interpreted as a vector sum as (12). There is an interference term when calculating the probability density from (12) Ψ Ψ∗ =[( + )+( + ) ][( + ) − ( + ) ] 12 12 a c b d i a c b d i (13) =(a + c)2 +(b + d)2 = a2 + b2 + c2 + d2 +2(ac + bd)

Ψ (r )Ψ∗ (r )= 2 + 2 + 2 + 2 +2( + ) iEδt/ or 12 ,t 12 ,t a b c d ac bd e . This enlightens the physical interpretation of wavefunction. There is a hidden fine-structure constant α in the = − ( α )2 localized wavefunction (7). It naturally yields the eigenvalues En Ee n from the = 1 2 = ± ( α )4[ 3 − n ] Schrödinger equation, where Ee 2 mec ; the fine-structures EF Ee n 4 j+1/2 are from Dirac’s relativistic equation, where j = l+ms is the total angular momentum.[4] α5 1 In the magnetic field, QED yields the Lamb shift E = E 3 [k(n, l)± ] and L e 2n π(j+1/2)(l+1/2) = ± gp α4 F (F +1)−j(j+1)−I(I+1) the hyperfine structures due to the “nuclear spin” EH Ee β n3 j(j+1)(2l+1) , where the total angular momentum F is equal to the total nuclear spin I plus the total

orbital angular momentum J; the proton g-factor gp =5.585. This gives experimental confirmation of a connection between α ∼= 1/137 and β ∼= 1836 . In fact, the reduced · μ = me mp = β·me ≈ m β masses me+mp β+1 e always involve in the Schrödinger equation (10). Quantum theory not only describes the electron configurations around the nucleus, but also considers the effect of nuclear mass. Neglecting interaction between electrons, the = 1 2 =025536[MeV] n above eigenvalues are linked to Eμ 2 μc . and α .

α 2 −5 =− ( )2 ∝ =53×10 En Eμ n α . (14) α 3 n 4 −9 =± ( )4[ − ] ∝ =28×10 EF Eμ n 4 j+1/2 α . α5 1 5 −11 = [k(n,l)± ] ∝ =21 × 10 EL Eμ 2n3 π(j+1/2)(l+1/2) α . gp α4 F (F +1)−j(j+1)−I(I+1) 4 −12 =± ∝ =15×10 EH Eμ β n3 j(j+1)(2l+1) α /β .

2 4 5 4 −1 In (14), we have En > EF > EL > EH since α >α >α >αβ . The interpretation of entropy S = klnΨ in the statistical mechanics is the measure of uncertainty,ormixedupness to paraphrase Gibbs.[12] The wave-function Ψ is the number of micro-states as the statistical information or probability amplitude. Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146 139

(3) Double-slit Interference Derivation with α The interference of Young’s double-slit experiments is the “central paradox” of Quan- tum Mechanics.[3] The quanta exhibit strange behavior after passing through the double- slits: (a) There is a deﬁnite cosine-type symmetric interference pattern for the multi- quanta (photon or electron), regardless of whether they all come together or one at a time; (b) The individual quanta has an unpredictable path and target point; (c) There is a white background noise, a photon can be found even in the forbidden-zone; and (d) No interference for single-slit. In Fig. 1 (right), the 2D double-slit plane is illustrated so that the slits 1 and 2 are at x and −x, a moving target receiver at the point X, with distance 2 2 1/2 2 2 1/2 Y between the double-slit and target, L1 =[Y +(X +x) ] and L2 =[Y +(X −x) ] . Note that the experimental condition requests Y X |x| λ,soL ∼= aX is linear.

Figure 1 The cross-linked photon scattering (left) and 2D illustration of Double-slit (right)

Since the phase velocity c = νλ = ωň = ω/k and T = ω−1, from Compton scattering c c c ň − ň = ň (1 − cos θ) − = (1 − cos θ) T − T = T (1 − cos θ) c (i.e., ω ω ωc ), we have c . [13] For the each slit in Fig. 1 (left)

− = [1 − cos( )] (1) T1 T Tc θ1 (15) − = [1 − cos( )] (2) T2 T Tc θ2

2 −21 Let (15-2) subtract (15-1), and Tc = ňc/c = /mec =1.288 × 10 [sec], then

= − = [cos( ) − cos( )] T T2 T1 Tc θ1 θ2 (16) (16) establishes the cross-linked angle for the double-slit

If = cos( )then = cos( ) T1 Tc θ2 T2 Tc θ1 (17)

where the T1 on the slit-1 is related to the scattering angle θ2 on the slit-2, and vice versa. It is a random variable for each particle noted as T = δT. −iωT1 −iωT2 Assuming ψ1 = A1e and ψ2 = A2e after the photon scattering for slits 1 and 2, where |A| |A1| |A2| is real, T = Tc cos(θ2) and T = Tc cos(θ1); The target 1 2 −iωT1 ikL1 −iωT2 ikL2 i2πL1/λ i2πL2/λ wavefunction at X is ψX = Ae e + Ae e = ψ1e + ψ2e , where λ is the wavelength of the quanta (photon or electron). The probability given for the quanta at the target point X is the same as (13) but in the exponential form 140 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146

2 −iωT1 ikL1 −iωT2 ikL2 iωT1 −ikL1 iωT2 −ikL2 |ψX | =(Ae e + Ae e )(Ae e + Ae e ) (18) − − − − − = |A|2 · [1 + 1 + ei(kL1 ωT1 kL2+ωT2) + e i(kL1 ωT1 kL2+ωT2)] = | |2{ 2+2cos[( − ) − ( − )]} A k L1 L2 ω T1 T2 White Wave Particle noise interference scattering =2|A|2[1 + cos(kL − ωδT )]

space time

1 (p · r−Et)=(k · r− t)=k(r−ct) The angle term in (18) is same as ω in (6), i.e., related 2 ň = αa = α to the ﬁne structure constant in (5). It is c 0 4πR∞ for the free electron and 2 = mp α a = mp α 1 mic mi β 0 mi β 4πR∞ for the atomic bonding electron in the Compton scattering. (18) clearly show that the wave-particle duality by the wave-vector k and the photon- frequency ω linked separately to the space L and time δT . The visible photon-frequency ω ∼ 1014[Hz]. The term of ωδT are random and small

eﬀect on the visible-light scattering by the tightly bound electron (mi me). Therefore, 2π ( − )= the term λ L1 L2 akX is the major-variable in control. The cosine term becomes {−1 0 1} 2π ( − )− ( − )= − = {(2 ±1) ( + 1 ) 2 } , , when λ L1 L2 ω T1 T2 akX ωδT n π, n 2 π, nπ where = 2π =0±1 ±2 ··· | |2 = {0 2 4)}| |2 k λ , n , , , , and let ψX , , A . The continued distribution is changes into a quantum interference pattern during coherence and cross-correlation.[14]

Assuming A = sinc[kL] and A1,2 = sinc[k(L±x)], the graph of (18) matches perfectly with experimental data from MIT for the far target using lasers (Fig. 2), 2 and Teachspin for the near target with “one photon at a time” (Fig. 3).3 The scatter graph in Fig. 2 and Fig. 3 clearly show that the interference pattern with the random scattering eﬀect as the individual particle of “one photon at a time.”

2 Figure 2 The scatter graph of the probability density |ψX |DS in (18) (top); and the Laser double-slit experimental data for the far target from MIT (bottom)

2 http://scripts.mit.edu/~tsg/www/demo.php?letnum=P%2010 (2011) 3 http://www.teachspin.com/instruments/two_slit/experiments.shtml (2011) Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146 141

2 2 2 Figure 3 The scatter graph of |ψX |DS, |ψ1| and |ψ2| in (18) (left); and the “one photon at a time” double-slit experimental data for the near target from Teachspin (right)

(18) indicates that the double-slit experiment has the interference pattern (a) with a random quantum scatting (b) and a background white noise (c). Notice that the cosine function is an even function responsible for making the symmetric interference pattern appearing on the target plate. It also shows that there is no interference pattern for i2πLk/λ 2 −iωtk ikLk iωtk −ikLk 2 the narrow single-slit (d), since |ψke | =(Ake e )(Ake e )=|Ak| where k =1, 2 (Fig. 3). As an exception, a slit which is wider than a wavelength produces Fraunhofer diffraction similar to the weak interference, due to the slit-edge electron-photon scattering effect. The probability for the diffraction of a wide single-slit at the target point X can be derived as a Taylor-Maclaurin series or Euler-Viete infinite product,

∞ | |2 = | |2 = |sinc( )|2 =[ (−1)n (akX)2n ]2 ψX A akX (2n+1)! (19) SS n=1 = { ∞ [1 − ( akX )2]}2 =[∞ cos( akX )]2 nπ 2n n=1 n=1

= πd sin =2 where akX λ θ and slit width d x. Unlike θ in the Fraunhofer approximation for the far ﬁeld limitation, X can be measured directly regardless far or near target. (19) looks like but is not Gaussian distribution. Fig. 4 shows the graph of (19) compared with the single-slit diﬀraction experimental data using laser and adjustable slit-width.

2 Figure 4 The graph of the probability density |ψX |SS in (19) (top); and the single-slit diﬀraction experimental data from MIT (bottom) 142 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146

2 Figure 5 The graph of |ψX | where cos(φ+π)=− cos(φ) in (20) (top); and the widely-separated very narrow double-slit experimental data (bottom)

Therefore, the double-slit equation is simply the diﬀraction of two narrow single-slits with wave interference and photon scattering. This is supported by the experimental result of very narrow but widely separated double-slit, i.e., moving two single-slits closer, in the double-slit experiment of electron.[15] This creates a symmetric pattern that may need an additional phase-shift of π in (20) (Fig. 5). From (18) and (19), and Fig. 5, with the optional phase shift ϕ

| |2 =2·|sinc( )|2[1+cos( − + )] ψX DS a kX akX ωδT ϕ (20) two diﬀraction interference scattering phase shift =4· sinc2(akX) · cos2[(akX − ωδT + ϕ)/2] = πδ sin δ = πd sin =2 δ where a kX λ θ and is the narrow slit width; akX λ θ and d x (d> for ϕ =0in Fig. 2-3 and d δ for ϕ = π in Fig. 5). The total number of fringes are N =2m − 1 for ϕ =0and N =2m for ϕ = π where m = d/δ = a/a equal to the ratio of the width of two-slits d divides the slit width δ. The double-slit experiment is a combination of the diﬀraction, scattering, and interference processes. The same principle can be applied to N-narrow slit experiments (Fig. 6).[15]

N/2 N/2 ψX = ψ±n = A cos(akX − ωδT ) (21) n=0 n=1 sin[N(akX+ωδT )/2] = A sin[(akX+ωδT )/2] The probability for the diﬀraction grating of N-narrow slits at the target point X is

2 2 sin[N(akX−ωδT )/2] 2 |ψ | = sinc (a kX){ } X NS sin[(akX−ωδT )/2] (22) 1−cos(2akX) 1−cos[N(akX−ωδT )] = · 2(akX)2 1−cos(akX−ωδT )] 2 =2 sin(2φ/2) =[2cos( 2)]2 = 2(1+cos( )) = 4 cos2( 2) If N , sin(φ/2) φ/ φ φ/ , (22) go back the double-slit equation (20). Unfortunately, the wave pattern is so attractive in the classical grating equation, and the particle scattering is ignored. (22) contains a particle scattering term ωδT , and its scatter graph compare with experiment is shown in Fig. 6.4 4 http://electron9.phys.utk.edu/phys136d/modules/m9/diﬀ.htm (2011) Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146 143

2 Figure 6 The scatter graph of the probability density |ψX |1,2,3,4,5,7 in (22) (top); and the N-narrow slits (1, 2, 3, 4, 5, 7) diﬀraction data (bottom)

In this model, the slit-edge electron-photon scattering plays a hidden role (each slit has two edges), which is linked to the fine structure constant.[7] Compton scattering creating redshift has been confirmed by sunlight and the MIT double-slit experiment (Fig. 2). The fine structure constant must also play critical rule in the double-slit experiment of electron, neutron, He-atoms, and C-60 molecules. Fig. 7 shows that there is a classical distribution (or de-coherent) if the quanta-slit interaction is weak (e.g., the double-slit experiment for neutron, C-60 and other heavy atoms). Other types of quantum scattering (e.g., Møller) may be involved. The Neutron double-slit data can be described as[16]

2 2 2 2 Figure 7 The scatter graph of the probability density |ψX |NDS = |ψ1 + ψ2| + |ψ1| + |ψ2| in (23) (left); and the Neutron double-slit data (right)

Fig. 7 shows that many quanta have passed the slit without interaction (leaking), they are not taking place in the interference. This is also true for the light-quanta passing a widely separated double-slit. The experiment for which-way also show the de-coherent by the secondary scattering after the quanta passed slit.[17] It further prove that the quantum scattering is a critical issue behind the geometric parameters for the wave-pattern. In other word, there will be no interference without quanta scattering. The ﬁne structure constant is the magic hand behind the double-slit experiment. This model clearly displays the particle-wave duality by particle scattering and wave-interference. 144 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146

The quanta can be counted as one particle at a time, and the multi-quantas are displaced as the cosine-type wave-pattern in space. The electromagnetic wave frequency is in the 0 24 −21 region of ω = kc =10 ∼ 10 [Hz]. Since Tc =1.288 × 10 [sec], the double-slit wave interference can be tested from soft-X-ray (1018[Hz]) to microwave (108[Hz]). The visible light (∼ 1014[Hz]) is a wave-like rather than the particle-like, and the γ-ray (1020−24[Hz]) is a particle-like rather than the wave-like. This physical reality was obscured by the trigonometric identities, however, there are many diﬀerent versions of the classical grating equations. From the discussion in section (2), the quantum physical principle is the same and independent of the mathematical expression and coordinate system. (4) Uncertainty Principle Derivation From α In a thought experiment of the double-slit, Einstein suggested that the particle in- teract with the slit-well should consider the recoil of the wall and conservation of momentum. The Compton scattering obeys the conservation of momentum. Bohr discussed the double-slit and the uncertainty principle as vp ≈ h/t on the debate in 1927.[18] P 1/2 Landau rewrote the uncertainty principle as t c α in 1931. He also pointed out E a weakness in the interpretation of the uncertainty principle, such as, t 2 “does not mean that the energy can not be measured with arbitrary accuracy within a short time.”[18] Here we show, the uncertainty principle can also be derived from the ﬁne structure constant (2). From e2/c which has the same dimension with , Et and p · r, for the spin-1/2 electron

2 2 = 1 e1e2 = 1 e = 1 e t = 1 Et = 2 2 αc 2 ve 2 x 2 (23) 1 e2 = 1 a = 1 v = 1 2 x2 t x 2 me t x 2 me x 2 Px x

where E = e2/r, F = e2/r2 = ma, v = at and P = mv are the simple relationships in Classical physics. The orbital electron velocity is a variable vector that change directions. In (24), if t deceases, x will also decease and make E increase; the same is true for x and Px. This is the most simple and direct physical causality of the uncertainty principle. Experimentally, x and the others can be deﬁned as the standard deviations, e.g.,

1 N 2 1/2 x = σx =[ (xi − x) ] (24) N i=1 → 0 1 If x , then the symmetric standard deviation σPx 2 Px. Therefore, for the spin-1/2 particles

E t 2 Px x 2 (25)

Due to the conservation of angular momentum, the property of e1 must change if e2 is altered. The electrons in an atom are high-speed (αc) charged particles. The distance

between e1 and e2 is constantly adjusting, which causes the measurement uncertainty. The nucleon has a micro-motion and thus the electron orbit could not be a perfect cir- cle. The orthogonal and anti-commute pair [A, B]=±i,thenAB ≥ /2 given Electronic Journal of Theoretical Physics 9, No. 26 (2012) 135–146 145

by Heisenberg-Kennard.[19,20] There are many different mathematical versions of the uncertainty principle, again, the quantum reality is the same. (5) Quantum Mechanics Postulates and α Quantum Mechanics is based on a number of postulates, such as (1) The wavefunction Ψ determines everything that can be known about the system; (2) The operator Qˆ acting on the wavefunction to determine the physical observable q; ˆΨ = Ψ (3) The wave equation Q n qn n determines the eigenfunctions and eigenvalues; Ψ= (4) The superposition of wavefunction anψn is an equivalent expression;´ = |Ψ|2 |Ψ|2 =1 (5) The probability density is given by P and the normalization is V dV . The interpretation of Quantum Mechanics has been of great debate among theoretical physicists, especially, Bohr and Einstein. Einstein was quoted: God does not play dice![21] However, he not only was a pioneer on Brownian motion, but also supported the Born statistic explanation of Psi-function. He had argued for the realism, completeness, determinism, EPR-paradox, etc. What he was really looking for was A DEFINITE RULE. In 1953, he said “That the Lord should play dice, all right; but that He should gamble according to definite rules, that is beyond me.”[22] Here we show, the fine structure constant α is the cornerstone of quantum theory. The wavefunction, Schrödinger equation and other quantum postulates, even the Uncertainty principle can all be derived from it. Just as Pauli and Feynman pointed out: Quantum Theory is inconclusive without understanding the fine structure constant.[2,3] Acknowledgment: The Author thanks Bernard Hsiao for discussion.

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