Download least action principle and quantum mechanics i. introduction

THE LEAST-ACTION
PRINCIPLE & QUANTUM
MECHANICS
Vu Huy Toan (Vietnam)
[email protected]
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) = 2Et  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= limto(v/t)= limtoatb (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 = vntn+1
(29)
Sn = atbtntn+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-v2cos21)/2=(mv2sin21)/2
(31)
A1 = FSv = Fvt1sin1
(32)
D = Ht1 = mv2t1sin21 = h
(33)
t1 = (mvsin1)/2F
_______
sin1 = 32hF/m2v3 = v0/v
(35)
t1 = mv0/2F = v0/2atb1
(36)
S1 = vt1
(34)
2. Wave property of fundamental particles
e1
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
vt1sin2a = Sa sin2a = h/mv = h/p
(39)
D1= Ht1= 2Eb1tb1= mvSb1sin2b1 = h
(40)
D2= Ht2= 2Eb2tb2= mvSb2sin2b2 = h
(41)
mv(Sb1sin2b1+Sb2sin2b2) = 2h
Sb1sin2b1 + Sb2sin2b2 = 2h/mv = 2h/p
Sc1sin2c1+ Sc2sin2c2 + Sc3sin2c3 = 3h/p
 Sknsin2kn = n(h/mv) = n(h/p)
(42)
3. The orbit quantization of electron in atom.
0
on
vn
vn
v =vncoson
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= kn2Entn= 2EnTn=
= 2knmevnrnsin2onsin(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.