THE LEAST-ACTION PRINCIPLE & QUANTUM MECHANICS Vu Huy Toan (Vietnam) email@example.com LEAST ACTION PRINCIPLE AND QUANTUM MECHANICS I. INTRODUCTION. II. BASIS CONCEPTS. 1. Effect of and action among physical objects 2. Least effect and least action. III. LEAST ACTION PRINCIPLE (LAP). IV. CONSIDERATION OF SOME PHENOMENA ON THE BASIS OF LAP. 1. The particle movement under the action of a force. 2. Wave property of fundamental particles. 3. Movement orbit quantization of electron in atom. V. CONCLUSION. I. INTRODUCTION Return to the classical mechanics… ε = hf (1) h = 6,63×10-34 Js – Planck constant + Hamilton - Ostrogratsky: t1 H = L dt t0 +Maupertuis – Lagrange: t1 H = 2Edt . t0 (2) (3) The problem is that: According to its definition, function H must be the total energy of the mechanical system gained in the duration of time from t0 to t1. Should we understand this as cause or consequence? Is it action or effect? What if the action doesn’t reach the least action threshold? What could the mechanical energy state of the micro objects under the action of external force be if this least effect is taken into account? Is the so called “wave-particle dualism” the cause of orbit quantization of electrons in atoms or do they both have other similar causes because they are both associated to Planck constant? II. BASIS CONCEPTS. 1. Effect of and action among physical objects. t1 t1 H = mv2dt = m(v0 + at)2dt = t0 t0 t1 = m(v02 + 2v0at + a2t2)dt = t0 t1 t1 = mv02dt + 2 ma(v0t + at2/2)dt = t0 t0 t1 t1 = 2T0dt + 2FS(t)dt = H0 + Ht t0 t0 t1 H0 = 2E0dt, t0 t1 t1 Ht = 2FS(t)dt = 2A(t)dt t0 t0 t1- D = 2E(t) dt t0- A(t) = E(t) Ht = D (4) (5) (6) (7) 2. Least effect and least action. As recognized above, there cannot be any effect smaller than the h least effect value. This also means that H effect function pursuant to (4) is not continuous and it can only be the multiple number of h: H = nh, (n = 1, 2, 3,…) (8) On the other hand, any occurring physical processes are associated with the energy exchange. That exchange occurs on every small portion. Thus, corresponding to the least effect h is the least action d. h = d. (9) III. THE LEAST ACTION PRINCIPLE "For a physical object to change its energy state, the action on it can not be smaller than the least action". t1- D = 2E(t)dt d = h/ (10) t0- If E(t) = E = const and = 1 then: D = 2E(t1- t0) = 2Et h (11) This action time interval ∆t can not be too long and must be limited by some conditions: - In case the living time of the given object is equal to τ1, ∆t τ1; - In case the possible time interval for the two objects to exchange their energy is equal to τ2, ∆t τ2 ; - In case the object under the action is moving in the period of T, ∆t T Effect radius of physical object. A B R Figure 1. Effect between electric charges. “Effect radius ” A d B 2/T R R cT >> d Figure 2. Effect of a physical object (A) consisted from anti-polarity electric charges on an electric charge. (B). “Effect radius” A B 2/TB B A 2/TA R If A > B then R = cTA Figure 3. Effect between the physical objects consisted from anti-polarity charges. IV. CONSIDERATION OF SOME PHENOMENA ON THE BASIS OF LAP 1.The movement of PO under the action of a force a) The action direction of a force coincides with the movement direction of PO b) Particle moves under the action of perpendicular to movement direction force 2.Wave property of fundamental particles 3. Movement orbit quantization of electron in atom a) The action direction of a force coincides with the movement direction of PO F = d(mv)/dt or F = dp/dt mdv/dt = ma = F, (13) (14) a= dv/dt= limto(v/t)= limtoatb (15) matb = m(v/t ) = F. atb = v/t = F/m = const. E1= E1-E0 =mv12/2. (16) (17) atb= v1/t1= F/m _____ ______ v1= 3 hatb/m = 3 hF/m2 ______ t1= 3 hm/F2 (19) (20) (21) atb = vn/tn = F/m (22) (vn)/( tn) = vn/tn (23) vn = atbtn (24) En = En– En-1= m(vn2- vn-12)/2 m(vn2- vn-12)(vn- vn-1)/atb = h. Movement equations of PO 3 vn - 2 2 3 3 vn-1vn - vn-1 vn +vn-1 - v1 = 0 3 2 2 3 3 tn - tn-1tn - tn-1 tn + tn-1 - t1 =0 (27) (28) Sn = Sn = vntn+1 (29) Sn = atbtntn+1= (F/m)tn(tn+1- tn) (30) v(t),S(t),vn,Sn v4 v3 v2 S(t)=v0+at2/2 vn = atbtn v(t)=(F/m)t v1 Sn t0=0 t1 t2 t 3 t 4 tn Figure 4. The discrete of movement parameters of particles. b) Particle movement under the action of perpendicular to movement direction force vΔt1 F α1 A(t0) B(t1) t0 Sy= vΔt1sinα1 t2 Figure 5 E = E – E' = mv2/2 - mv’2/2 = = m(v2-v2cos21)/2=(mv2sin21)/2 (31) A1 = FSv = Fvt1sin1 (32) D = Ht1 = mv2t1sin21 = h (33) t1 = (mvsin1)/2F _______ sin1 = 32hF/m2v3 = v0/v (35) t1 = mv0/2F = v0/2atb1 (36) S1 = vt1 (34) 2. Wave property of fundamental particles e1 Screen1 2 Screen 2 Figure 6. Diffraction of electron beam E(t1) E(t2) E(t3) E(t) f1 f2 α1 α2 Figure 7 Figure 8. The Defocus Lens model of the one-slit potential field c3 c b a I2 b2 c' b' a' a I1 I0 Figure 9. The movement deflection of electron. C’ C B’ I2 α3 α2 I1 B A A’ α1 I0 Figure 10. The electron deflection in potential field + + f1(t) E1(t) + + + + + + + d f2(t) E2(t) + + + + + E2(t)+E2 e- E E1(t)+E1 f1(t)+f1 f1(t)+f1 Figure 11. Distribution of the double slits potential fields. e- ΔE E2(t)+ΔE2 f2(t)+Δf2 E1(t)+ΔE1 f1(t)+Δf1 Figure 12. Distribution of the double slits potential fields vt1sin2a = Sa sin2a = h/mv = h/p (39) D1= Ht1= 2Eb1tb1= mvSb1sin2b1 = h (40) D2= Ht2= 2Eb2tb2= mvSb2sin2b2 = h (41) mv(Sb1sin2b1+Sb2sin2b2) = 2h Sb1sin2b1 + Sb2sin2b2 = 2h/mv = 2h/p Sc1sin2c1+ Sc2sin2c2 + Sc3sin2c3 = 3h/p Sknsin2kn = n(h/mv) = n(h/p) (42) 3. The orbit quantization of electron in atom. 0 on vn vn v =vncoson A B Figure 13. Movement orbit of an electron in atom is only a broken line. + F αon vn vα = vncosαon Figure 14. Movement orbit of the electron in atom only is a broken line The orbit angular momentum of electron in atom kn.on= 2 (43) Htn= kn Hto= kn2Entn= 2EnTn= = 2knmevnrnsin2onsin(on/2) = knh (44) Mn= mevnrn= h/(2sin onsin(on/2)) = 2 =[2/2sin onsin(on/2)].h/2 = mn(45) 2 Discrete polygon orbits of electrons in atom. The stability of the electron polygon orbits in atom. + IV.CONCLUTION 1. According to LAP, the fundamental particle can merely move in “jerky steps” varying its velocity along with each step, but not in regularly increasing or decreasing manner. 2. The “physical wave” does not exist! The particle just has “wave-like manifestation”, but not “wave property”. There is a new invention of the particle’s property, that is: "The particle’s movement can be only deflected at limited and specified angular quantum and that can not be as small as wanted ". 3. In case the movement deflection of an electron in atoms occurs under the action of Coulomb force at uniform angular quantum whose sum is multiplication of 2, the electron’s orbit has a shape of a regular polygon inscribed to a circle with radius rn, from which the orbit quantization condition of electron in atom can be formed.