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На 6a3i методу V-функцп здтснюеться продовження оптико-мехашчног аналоги. Розглядаеться траектор-но-хвильовий рух частинки. Вказуеться на наявтсть квантування енерги частинки, ршення без частинки в разi прямолшшного рiвномiрного руху з постшною швид-тстю). Дослгджуються властивостi хвильовог природи руху електрона в водородоподiбному атомi як ршення прямог задачи Показуеться споыб знаходження ктцевого ршення стащонарного хвильового рiвняння
Ключовi слова: варiацiйний принцип, пряма задача динамти, зворотна задача динамти, оптико-мехашч-на аналогiя, хвильовий рух, траекторний рух, хвильова
функщя, хвильове рiвняння
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На базе метода V-функции осуществляется продол-жениеоптико-механическойаналогии.Рассматривается траекторно-волновое движение частицы. Указывается на наличие квантования энергии частицы, решения без частицы в случае прямолинейного равномерного движения с постоянной скоростью. Исследуются свойства волновой природы движения электрона в водородоподоб-ном атоме как решение прямой задачи. Показывается способ нахождения конечного решения стационарного волнового уравнения
Ключевые слова: вариационный принцип, прямая задача динамики, обратная задача динамики, оптико-механическая аналогия, волновое движение, траек-
торное движение, волновая функция, волновое уравнение -□ □-
UDC 51-73
[DOI: 10.15587/1729-4061.2017.1088311
A METHOD OF V-FUNCTION: ULTIMATE SOLUTION TO THE DIRECT AND INVERSE PROBLEMS OF DYNAMICS FOR A HYDROGEN-LIKE
ATOM
N. Valishin
PhD, Associate Professor Department of Special Mathematics* E-mail: [email protected] S. Moiseev Doctor of Physical and Mathematical Sciences, Professor Kazan Quantum Center KNITU-KAI* *Kazan National Research Technical University named after A. N. Tupolev-KAI K. Marksa str., 10, Kazan, Russia, 4201 11
1. Introduction
Variational principles and optical-mechanical analogy were essential for the development of quantum mechanics. The analogy of motion of the mechanical conservative systems and the propagation of light rays in an optically heterogeneous medium was first paid attention to in paper [1]. Thus, from the stationary Hamilton-Jacobi equation
L. de Broglie shed a new light on the optical-mechanical analogy [2-4]. He considered the correspondence between a wave and a particle based on equations (1) and (2), and on the basis of variational principles by Maupertuis and Fermat. It is the very optical-mechanical analogy at the level of geometrical optics that allowed L. de Broglie to establish wave properties of the particle. Thus, if one puts 9 = W /h, in (2), then we obtain from (1) and (2)
2m(E-U) = (3)
x2 _ h ~ h2' (3)
recorded for a single particle:
,\2
dW
dqt
= 2m(E - U ),
(1)
where W(q) is the Hamilton's characteristic function, U(q) is the potential energy of a particle, E is the total energy of a particle, m is the mass of a particle; and the eikonal equation that describes the propagation of a light ray
Эф
\ 2
dqt
1
X2'
(2)
where ty(q) is a function of the eikonal - a light wave phase, X is the light wavelength, it follows that these equations are similar in the general form.
where h is the Planck's constant.
Optical-mechanical analogy and the ratio of Louis de Broglie (3) were consequently employed by Schroding-er to formulate a wave equation [5].
Experimental achievements in the study of the behavior of separate microscopic systems revive in turn sustainable interest in verifying the basic provisions of quantum theory and stimulate a deeper rethinking of its physical principles, a role of information in the theoretical description of the micro-particle behavior [6, 7].
2. Literature review and problem statement
Optical-mechanical analogy is, first of all, a view of the nature of light. Optical-mechanical analogy has remained
relevant up to now [8-10]. Study [8] shows the existence of connection between the trajectories of particles under the action of nonholonomic constraint, and the trajectories of light rays with a variable refractive index. Article [9] provides a proof of the existence of a new optical-mechanical analogy between the equation of rotational motion of the body in mechanics (taking into account the principle of relativity) and the first pair of Maxwell's equations. The Hamilton's optical-mechanical analogy between a material's particle trajectory in potential fields and the trajectory of light rays in media with a continuously changeable refractive index has played an important role in the substantiation of Schroding-er's wave mechanics [10]. In this case, based on the existing variational principles, this analogy is drawn only at the level of geometrical optics. In the given paper, the motion of an object is explored using a V-function method, which consists of a local variational principle (LVP), new statement of the direct and inverse problems of dynamics [11, 12].
In some problems, light manifests itself as a particle. In other problems - as a wave. In other words, a dualism of the wave and the particle is detected. The same dualism manifests itself also for the particles of matter. Continuing attempts to comprehend paradoxical manifestations of a corpuscular-wave dualism in the motion of the electron (as well as other micro-particles) do encourage undertaking new research [13-16].
This makes it possible to argue about theoretical interest in the approach based on corpuscular-wave monism to explaining the nature of the particles (the object). In particular, a theory being developed can apply the description of physical reality where the existence of the particle trajectory is taken into account, which reflects the fact of the existence of the particle, while it is also accepted that the motion of a particle is determined by a physical wave V(x,t).
Along with a system of equations (4), we shall also introduce a wave function V(x,t). The rate of its change for the system being studied (4) will be determined by expression
dv = -d V + #- V T f.
dt dt dx
Consider an isochronous variation of the rate in the wave function change
d( dV )=l+4-T f+îxV Tf,
here
dV = dXV TSx, df = dXf dx.
We accept that at a variation in the rate of change in
wave function ), an object from a certain initial state
passes into the state that is different by a new spatial coordinate x + dx. Such a transition will be referred to as a wave transition of the object, at which a magnitude dV assigns the possible wave transition from the initial state to the new state, while dx determines trajectory variations. When implementing a wave transition, the spatial variation takes the form of displacement dx = dx = Xdt, implemented in space.
Let us formulate an LV principle: out of all the possible transitions to a new state, the only one, which is actually carried out, is the one at which in each moment a rate of change in the wave function V(x,t) takes a stationary value
S( dV > = 0
(5)
3. Research goal and objectives
The goal of present work is to demonstrate the capabilities of a V-function method to study the motion of an object.
To accomplish the goal, the following tasks have been set:
- to perform optical-mechanical analogy at the level of wave optics based on the local variational principle, taking into account the wave and the trajectory motion of the object;
- to conduct research into the properties of wave nature of the electron motion in the hydrogen-like atom as a solution to the direct problem of dynamics;
- to devise a technique for finding the ultimate solution to the stationary wave equation for a hydrogen-like atom.
By assuming the feasibility of (5), we also accept that a wave function satisfies additional condition for a full variation in the change rate of wave function V(x,t):
(6)
A( dtV )=°
where A(.) = S(.) + d- (.)At.
dt
Once we have classical equations (4) and conditions (5), (6), we shall derive a wave equation for V(x,t), taking into account the implementation of wave transition (8x = dx = xdt ) in (5) and (6):
4. Research materials and methods
We shall define the content of a V-function method. Let the trajectory motion of an object is assigned by a system of differential equations from classical physics:
ÎX = f( X >
(4)
where the vector of phase particle coordinates x(t) = = (xv x2,..., xn )T is assigned in n-dimensional Euclidean space (x e Rn), t is the time.
I dt
aÎ dt m+3 If
dt2 dtdx
f + 2 fTWf + 2 — f \dt =
dx dt
=K f
a2v
'dV - fTwf-dyTdf
dt2
dx dt
dt
w -fwf -
dVT df dx dt
= 0,
(7)
where V(x,t) is the piecewise continuous, finite, single-valued function, W = [d2xxV(x, t)] is the function matrix. Equation (7) is a necessary and sufficient condition for the feasibility of (6). We shall demonstrate that there is equality
dVT d . „ — —x = 0. dx dt
According to the method of a V-function, particle motion occurs in such a manner that at each point in time, the particle's velocity is co-directed with the wave function gradient, that is
^ F1 = dx
dx^
x .
Hence, we obtain dV/dx _ k2(x)x. Further, we assume that a velocity field in a three-dimensional space coincides with the field gradient corresponding to it, which occurs at k2(x) _ k2, and, accordingly, we obtain equality
Condition 1.
We obtain from equality (9) a boundary condition for wave in point x = xM of the trajectory of a particle
/ x=x&,! 2 x= Xi*
(13)
Condition 2.
Assuming the implementation of wave transition in (5), we shall obtain
-J^- V T x = const. dx
(14)
Employing condition (14), for a full variation (6), we in turn obtain equality
dV / dx = k2 X.
(9)
In the case when a wave transition is implemented, relation (5) takes the form
d_ dt
dx
d_ dt
Xdt
dx
dV 1
= 0 ^- X = const. (10)
dx
d (A V ) = (
j ^ A=o, dt[dt )
applying which we find, respectively, condition 2 for wave behavior on the trajectory of a particle
dtV
= k
(15)
Then, taking into account (9) and (10), the equality (8) should hold, that is,
dVT d . k ■t d .
dx dt k2 d / ~2dt(
dt
-k2—( X T x ) = -
1 d_ 2 dt
d_V dx
T
= 0.
Consequently, equation (7), considering (8), takes the form
d2ttV - x TWx = 0. (11)
Moreover, if the following condition is satisfied
• i dxx- = X—1 (i, j = 1,n),
' 1 dX • V J '
equation (11), considering (9), takes the form:
^-tf2V2V = 0, dt2
(12)
where
n
f>2 = 2 = :
and V2 =
tT dxf
where kl2 are some constants. Condition 3.
It follows from the condition of connectedness between a wave and a trajectory (wave amplitude V(x, t) is equal to zero in the point of particle position with coordinate x = xM in time t)
V (x = xM, t ) = V ( x, t = 0) = 0.
(16)
Direct problem of dynamics in the method of V-function is stated as follows:
The differential equations are given that describe the motion of an object (4).
It is required to determine wave function V(x,t) that satisfies equation (12). For the case x _ (n=1), we obtain a solution to equation (12) considering (13)-(16) in the following form:
V ( x, t ) = ±Ae
(17)
Inverse problem of dynamics is stated as follows: For a given wave function V(x,t), which satisfies equation (12), it is required to derive differential equations of the motion of an object (4).
At the given wave function, a solution to the inverse problem of dynamics immediately follows from (9):
Xi = k .
dx-
(18)
Equations (4) and (12) describe trajectory and wave motion of the particle being studied. In order to find a solution to this system of equations, it is required to know the boundary conditions. Note that we shall use as the boundary conditions for (12) the properties of a wave on the trajectory of a particle. The proposed approach to the description of particle behavior includes a system from the trajectory equation (4) and the wave equation (12). Further, we shall find the boundary conditions for wave V(x, t) on the trajectory of a particle.
For the one-dimensional case (n=1), equation (11) takes the form:
d2V(x, t) d2V(x, t) . 2
dt2
dx2
X2 = 0.
(19)
Assume that the wave function is given in the form of a plane wave equation (17), which propagates in the motion direction of the object. Then (17) will satisfy (19) if x = û.
t=0
In this case, it follows from equality (15), where the wave function is given in the form (17), that
dV ( x, t )
—= +iAae u ' = const. dt
Aa = A-^(1 +2 n) = const.
---- = ±i-e VV ' = k2x = k2V.
dx V 22
Hence, considering (20), we obtain -V = k V2 = const.
dV ( x -1
dx
er in the case of Planck oscillator. In this case, we obtain from (24) considering (21)
(20)
ha
— = mV. V
(27)
The constant in the right part (20) is a real number. Therefore, in order to satisfy condition (20), the phase should take the values:
$ = | a x-at = alx-t\ = n+nn, (n=0, 1, 2, 3....). (21)
Since x = V^ —-1 = C, equality (21) takes the form: n n a
aC = ^ + nn ^ra = ^(l+2n) = y(l + 2n)> (22)
that is, in solution (17), natural frequencies can take only certain discrete values. Then (20), considering (21) and (22), will take the form:
By employing the results obtained, it is possible to draw such a correspondence between the wave and the particle [9]
mS2 2E
■ = ■, ra =-= —,
h h
X = —, A = h. mV
(28)
In this case, the wave and trajectory measurements can be described by a single wave function:
±il — x-at :
V(x, t) = Ae > = he
± 4fhrx-hat| ±i1(mVx-Et)
hlV J = he h ). (29)
(23)
This means that equality (22) also takes only discrete values.
From equality (9), considering (17), it follows that
(24)
(25)
Equality (25) is nothing else than the fulfillment of (10) at n=1.
In relations (28), principal is the equality between the wave phase velocity and the particle speed, while in quantum mechanics, the particle's speed equals the group velocity of waves by L. de Broglie. The energy quantization condition (23) is produced naturally as a result of solving the inverse problem. According to the second relation in (28), energy is transferred by a particle. In turn, according to the third relation in (28), the pulse of a particle determines wavelength X, which coincides with the known formula by Louis de Broglie. In the physical sense, wave V(x, t) characterizes properties of the activity that manifests itself in the motion of a particle. Thus, the wave is connected by its node with the location of the particle and thus guides it, however, the particle (trajectory) generates a wave that propagates with it.
In addition, equation (26) has a solution at k2 = m ^ 0. In this case, we obtain a wave function in the form of a monochromatic flat wave without the particle, which propagates at a given speed and with a given frequency. This can explain the interference pattern when the particle (photon) passes through two slits.
5. Results of studies into particle motion
5. 1. Continuation of the optical-mechanical analogy
Let us consider the trajectory motion of a particle, which dV
satisfies equation (18) x = k—. Trajectory motion of the
dx
particle, as follows from (18), is matched with the wave motion, which satisfies wave equation (19):
d2V(x,t) _( dV_\2 d2V(x,t) at2 I dx ) dx2
= 0.
5. 2. Motion of the electron in a hydrogen-like atom
Let us consider the motion of an object (a particle) in a 3-dimensional potential force field in the Cartesian coordinate system. Let the trajectory equations of the object (the particle) (4) allow the first integral of motion in the form of the law of conservation of energy of a particle, that is,
m( x2 + y2 + i2)
2
U ( x, y, z) = E,
(30)
(26)
n- x-at)
Function (17) V(x,t) = Ae ■ will satisfy equation (26), if equality (21) holds. In this case, we obtain
k ■ \A\ = . rn
Let here k2 = m be the mass of the particle. Then amplitude takes dimensionality of the action. If we accept h
A = — = h, h is the Planck's constant, then the rule of energy quantization follows from (23), similar to that by Schroding-
where m is the mass of a particle, E is the total energy of a particle, U is the potential energy of a particle. Then the object motion (of the particle) is fully described by the following system of equations (30) and (12):
mV2
2
+ U = E,
d2V 2 2 -T-V2V2V = 0, dt2
(31)
where V2 =—2 +--2 +--2 is the Laplace operator, ■2 =
dx dy dz
= x2 + y2 + z2 is the square of the particle velocity. Hence, the second equation, considering the first one, takes the form:
d2V 2( E-U )
dt2
V2V = 0.
(32)
d T (t )
"dtT _ 2(£ - U )V2X(x, y, z) = -a2.
T (t) mX( x, y, z)
Consequently, we obtain the following stationary equation
v2 x + m2 X = 0.
-ß0 + f\kX + m2 X = 0,
where
32=—, a=~
a^ 1 d
-ß• ? H
We shall apply a method of separation of variables in equation (32) (V = X( x, y, z)T(t)),
dr 2_T_
dr
-m2- = 0 = r
, _ 2 a^ 1 d2u 2 u „
H -ß02 +-\—TT + m - = 0 H
r I r dr r
(33)
d2u m2r
---1--
dr2 a-ß2r
= 0.
As is known, potential energy of a hydrogen-like atom is equal to
U(r ) _- Ze2/r. (35)
Then equation (34), considering (35), takes the form
(34) dï where
Represent equation (39) in the following form
a-ß02r
k-
u = 0,
(40)
(36)
,2
k = P2 = 2E ■
The solution to be obtained to the direct problem of dynamics for equation (40) must satisfy natural condition u(r = r0) = 0, (here r0 = a/P02 = -Ze2 /E = Ze2 /|E|), which corresponds to the implementation of boundary conditions (16), at which amplitude of the wave becomes zero at r = r0, where, accordingly, as a solution to the inverse problem, the particle acquires a trajectory. Find the asymptotic solution to equation (40) at r ^^.
In equation (36), we shall proceed to a spherical coordinate system
-ßo2+71*
1 d
r2 dr I dr I r2 sin 8 98 +m2X=0.
1 d ( • ad sin 8 —
1
d8J r2 sin2 8 9^2
X+
where
1 d ( r 2 9
1 9 ( . 9
- I • I ■ ^--I sin 8—
r2 dr I dr ) r2 sin 8 d81 d8
1 d2 .
+—2-2———2 = A
r2 sin2 8 dtf
dr
-- k,2u = 0.
(41)
(37)
We shall record a general solution to equation (41) in the form
u„ = u- (r ) + u+ (r ) = e-k'rf-(r )+ek'rf+ (r ).
Then
u' = ±k2e± k rf±± (r ) + e± k rf±'(r ), u" = e+- k'r ( f'(r ) ± 2k0 f'(r ) + k02f± (r )) and equation (40) takes the form
f±"(r)± 2kf'±(r)+ f±(r) = 0,
(42)
is the Laplace operator in a spherical coordinate system.
We shall search only for the spherically symmetric solutions. Then X = R(r ),
1 d ( 2 d . A=7dTr ( r d =
1
2 d + 2 d2 N d2 + 2 d dr dr2 dr2 r dr
where
ß4 = k20a /ß0 = }Ze m2me /E2, r0 = a/ß2 = -Ze2 /E = Ze2 /|E\.
(43)
Solution to equation (42) will be searched for in the form of the following power series
and equation (37) takes the form
2 a^ 1 d ( 2dR^ 2
-ßo2 +-\—-rI r l + ffl2R = r I r dr V dr I
We shall replace R = — in equation (38) to obtain r
f±±(r) = 11*^ (r - r)m,
(44)
(38)
where a particle trajectory actually becomes localized on the surface of radius r _ r0. Equation (42) after the given substitution of (44) takes the form
m
2
m
z m( m -1) a(m) (ro- r )m-2 ±
±2^0 Z^ (ro-r )m-1 + P1 Z «m1 ) (ro-r )m-1 =0
^¿[(n + 1)na<+> + 2k,nai± + P1«1>](ro - r)n-1 = 0. (45)
n=0
Equality (45) is identically fulfilled only when r = r0, or when all coefficients of the obtained series are equal to zero, that is,
( n +1) nfâ + 2 k,na{± + PX±) = 0.
We shall record a radius of the n-th state of the particles considering (50)
Ze2 2hV
r = — _
,n E Ze2m„
(51)
Next, we shall search for the ultimate solution to equation (40), because solution (48) approaches infinity u(r) at r ^ ". For this purpose, consider a general solution
u = U- (r ) + u+ (r ) = e- K nrf- (r ) + eK f (r ) = Cek"rZ" (r0 -r)m + C2ek"rZn +)(r0 -r)m.
1 m V 0 n ) 2 ^ m=1 m V 0 n )
Hence, it follows that a0 = 0, while coefficients aSz! sat
isfy recurrent relation
a(±) = ±2k0n-Pl a(±>. n+1 ( n +1) n n
(46)
Since, based on the inverse problem of dynamics, we search for the trajectory of a particle that holds provided
Here solution u (r) is considered also in the form of a power series, but a series, as it follows from relation (46), is not discontinued.
It follows from (46) that for sufficiently large values of n, a relation of two coefficients of series (44) takes the form
a(±) ±2k an+\ _ ± 0
a(±) n + \'
P4 = 2k0n (P4 = {Ze2ra2me / E2,
2
k = ' P1 >k > °). (47)
The given condition is satisfied only when series
f+ (r)=z:= «:+) (r. - r ):
But it is the very relationship that exists between two adjacent terms of a series
Ar =. + 2k0r + , (2k0)"rn , (2k0)n+irn e = i +---+... -
n! (n +1)!
(2k0)n+1 n! 2k0
(n +1)!(2 k0)n n +1
is discontinued, that is, a(+) = 0 at m > n +1, which leads to
m '
the following solution
u (r) = Cek°'"r Z " a(+) (r - r) ,
where C is the constant.
Considering equality (47) and
,2 a a m k = pf = -~2Ë~ '
P2
we shall obtain k02 = —2, 4n
2 2 4 4 2
a m Z e4a4m
(48)
2E 16E V
E4a2 2Z 2eim2 E3 Z'e^ 1
E a 16n m a 8 n
E =-
Z 2e'm 1
n 2h2 n2
Therefore, at r ^ , there are asymptotics u±(r)^e °*r, which is why, in order u, ^ 0, the ultimate solution will be sought for in the form of u = u (r)- u+ (r). For this purpose, we shall consider the form that functions u (r) and u (r) take at m=1, 2, 3...n.
Because, at m=1, functions u (r), u+ (r) take the following form:
u+i = ei°',rfli(+) (r0i - r)
ry" <
«- ,1=h"r УI_1«m_) (ro 1- ^ y=
(-1) m-1(2 k0 j)m-1
=e-kojrz^^j^r a-)(-1)m (r-ro,ir=
= -e 01
(49)
(m -1)!
y" (2koj)m-1 -) (r - I =
(m -1)! 11 "<■
0,1;
\m-1
Since, from the results of optical-mechanical analogy (28) we have ra2 = I — , then we shall obtain, from relation
I h )'
(49), the energy value of the n-th state of the particle (a rule of energy quantization)
-k.r t \ (" (2k0,1) I \m-1
= -e 01 (r - roi)fl{ ' y m=1 (m_1)| (r - r01 ) =
= -e 01 (r-r0i)flj >e 01( 01) = «ie0J (r0i-r),
where equality (47) is taken into account, which takes the form P41 = 2k01 and the recurrent relationship (46) for a(m) in the following form
(50)
a(-) = (-1) m-1(2 K)m-1 a(-)
which exactly coincides with the solution obtained in the Bohr model [17], or based on the stationary Schrodinger equation in paper [5].
(m -1)!
Hence, it is clear that if a4(+) = a4 =al(-V2k0,1''0,1, then u+,i(r ) = u-,i(r ).
Let m = 2, then P12 = 4k02,
2k0,2 Pl,2 l + ) -2k0,
2 2-1 1 2-1 1 0,2
u+,2 = ek°' 2ra(+)
1+' ((ro ,2 -r)-ko,2(ro ,2 -r)2)
(-) _ 2k0,2 °2 2 1 Ul 2 1 Ul '
(-) = 2k0,2 - 2 P1,2 (-) = ( 8 - k0,2 ) ( 6 - k0,2 ) (-) = °3 = 3 - 2 °2 = 3 - 2 2 1 Ul =
a\ ',...,
= (-1)2(2 ko,)24 - 3 a(-)
(3 - 2)(2 -1) ai a(-) = (-1)m-1(2ko,2)m-1((m + 1)- m-(m-1) -4-3)^ = am (m (m-1)---3- 2)((m-1)---3-2-1) a1
= (-1) m-1(2 ko ,2 ) m-12m-T)T a1
2- (m -1)!
u = e~k°'2r a
- (-1)m-1(2k,,2)m-1 m +1
1 ym=1 /- 1
(m -1)! 2
r„,-r) =
= e -K 2rai-)
1-) y"
1 m=1
(-1) m-1(2 k02) m-1 m +1
= -e-ko,2'a(-)y" (2k0 2 )m-1 m + 1
1 m=1
(m-1)! 2
(' - '0 ,2 )m =
(-1)m (r - 'o
-e"
(m-1)! 2
^ ((' - r0 ,2)2 e2ko, 2(r-'o,2) )' =
= -e" k0, 2ra(-)e2k0, 2(r-r0, 2)
1(-)e2ko,2(r-r0,2) ((r - 'o2 ) + ko,2(r - ro,2)2 ) =
eVa2 ((ro,2 - r ) - k0,2(r0,2 - r )2 ).
= (-1) m-1(2 m-1(( m + 2)- ( m + 1)-m---5- 4) ^ = %m (m (m-1)- - - 3- 2)((m-1)- - - 3- 2- 1) a
= (-1) m-1(2 ¿0,3) '
-1 ( m + 2)( m +1)
u- ,3 = e
— 3r „ (
C(y" ■
1 ¿-I m=1
3-2-( m -1)! 1 ' ( -1)m-1(2 ko ,3)m-1 ( m+2)( m+1),
( m-1)!
3 - 2
= e c ((r - ro,3)3 e2ko'3(r-®'3)) = e~ k0'3rc(-)e2kj' 3(r-rj' 3)
3!
x(6(r - ro ,3 )+ 6(2ko ,3 )(r - ro ,3 )2 - (2ko ,3 )2(r - ro ,3 )3)=
eh2r a3
3!
(6(ro ,3 - r )-6(2ko ,3 )(ro ,3 - r )2+(2ko ,3 )2(ro ,3 - r )3).
If flj(+) = a3 = a( }e, then u+,3(r) = u-3(r). Let m=n, then P1n = 2nk0n,
(r )=y :=
(ro ,n - r) +
ko nr ( + )
= e 0n a}+)
2ko (1 - n) ,
0 n ( -(ro ,n - r )2 +
(2k )2(1 - n)(2 - n)
1-2
(ro ,n - r )3 +
+ +
(3-2)(1-2)
(2kon)n-1(1 - n)(2 - n) - - - 1
n!( n -1)!
(ro n - r )n
2k„,n (r-r„,n )
(n-1)
/ \ ° If \
u (r) =-a (r-r ) e
-,n\ s n J n 1V 0,n I
If fl,(+) =a =a(f)e'2K", then (r) = u (r). Hence, it follows that the solutions u (r), u+ (r) are linearly dependent. Let us find a second linearly independent solution.
If a(+) =a2 = a1 then u+2(r) = u-2(r).
Let m=3, then P13 = 6k03,
fl( + ) = 2^03 Pl3 fl( + ) = fl( +), 2 2-1 1 2-1 1
a( + ) = 2k0,3 a( + ) = ( 2k0,3 ) ( 4k0,3 ) a(+ )
e 03 a > u+ 3 =-— x
+ ,3 3,
x(6(rn,3 - r)- 6(2kn,3 )(rn,3 - r)2 + (2kn,3 )2 (rn ,3 - r)3)-
-Pi,3 (-) = (-1)1(2ko,3)4 (-), °2 2-1 ai 2-1 ai
u" + p1(r)u' + p2(r)u = /(r). m+ (r ) and (r ) - are the solutions to this equation, then
W =
M+(r) u (r) u'+(r) u'(r)
= u+ (r )u- (r ) - u_ (r )u+(r ).
M+2(r )
u+(r)u-(r)- u (r)u+(r)
u+2(r )
= u+2(r )
/u_ (r vu+ (r
= ^.
(r )
f ua dr = c i J „ (r) J
vu+ (r )/
-f Pi(r )dr
u+(r )
-dr.
Since pt(r) = 0, then
u (r) u+ (r) ~J u+2(r)
= C f—^—dr. J u (r )
a(-) = 2^0,3 - 2 P1,3 a(-) =
3 3 - 2 2
.(-10-ko,3) (-8-ko,3) (-) = (-1)2(2ko,3)25-4 (-)
3 - 2 2-1
(3 - 2)(2-1)
Therefore, the second linearly independent solution will take the form:
1
u_ (r ) = Cu+ (r r
J u+(r)
and, considering solution (48), we shall obtain a solution that falls exponentially with distance u_n(r ~k'°n\
that is,
u-n(r)=
=i :=-«:+> (r„,„ - r )-j
-'2k,, П
n
,=-1®:+ (r0 ,n Г
-dr (52)
Note that wave u_n (r) changes sign during transition r through point ron, which, in accordance with conditions (3) and (4), indicates the existence of a particle's trajectory at u
this point. Since R = , then, in accordance with (52), r
we shall obtain solutions to R-n, (n=l, 2, 3...) (Fig. 1-3). At n = 1, u (r) is equal to:
-2koAr
u-i(r) = e"ir (ro,i - r-a dr.
(roi - r)
Considering that
k =Pi = 1 Ze2a2me = 0,1 2n 2 2nE2 1 Ze24E 2m Ze2m„2
2 2nE2fi2 nh2
and
u2(r)=e
4r / / \
r
1--
(! - k0,2 (r0,2 - Г ))
v v 0 ,2 /
2
1
1 k0,2r0,2
2
1-
— e42 (1 - г)(1 - 4 (1 - 2))x
xj; "
'(1 - 2)2 (1 - 4 (1 - 2))2 Therefore,
rdz.
R_ 2 =
-eiz (1 - z)(3 - 4z) j-e-2 dt
u_ 2(r )_ 1 Д ;J(1 -1 )2 (3 - 4t )2
At n — 3 , u-3(r) is equal to: u_,3(r) = eh''r x
X (6(r0 ,3 - r) - 6(2k0,3 )k0 ,3 (r0 ,3 - r)2 + (2k0,3 )2 (r0 ,3 - r )2 )x
-2ko , 3r
xj-2 dr
(r0 ,3 - r)2 (6 - 12k0 ,3 (r0 ,3 - r)2 + 4k0 ,32 (r0 ,3 - r)2 )
2n hi о о
r01 = k0,1 ■ roi = 2n = 2
Ze m„
u x{r) = e" ■ r„
2r / \
r
1--
V "1/ 2
r,
f \2 1
dr.
We shall replace:
r _
ro,i ' dr — r01dz
Therefore,
-i(r) = e2z ^ -z)j—2dz.
(1- z)
R-,i =
e2 ■(l - z) f——^ dt u_,(r )_ 1 jj(l -1 )2
At n — 2, u 2(r) is equal to: U-,2(r ) =
= гГ l(r0j 2 - r)- k02 к 2 - r)2 )x
xj
(r0, 2 - r)2 (i - k0,2 (r0, 2 - r))'
-dr,
kg 2 ■ Г0 2 — 2n—6
u-,3(r )—^
/ / \ / \ / \ 2\
r/0,3 6 12k0,3r0,3 +4k0,3 r0,3
ч ro,3, ч ro,3, ч ro,3,
2
1-^
d——
6 12kb,3?o;
1-^
/ \A2 r
+4k 2r 2
T4K0,3 '0,3
1-
—e6 z (1-z)(6-72(1-z)+144(1-z)2)x
r e-i2z I-2 dz.
(1-z)2(6-72(1-z)+144(1-z)2) Therefore,
u 3(r) e6(1 -z)(6-72z(1 -z) +144(1 -z)2)
R-,3 =—:-=-:
r r
f eXlt
x I-;-— dt.
(1 -1 )2 (б - 72(1 -1) +144 (1 -1)2 )
We shall construct charts of functions Rn, using the Maple programming complex.
The charts show that starting from second lower state, amplitude of the wave crosses zero more than once, but only at r = r0n the derivative of wave (r, t) changes sign in this point, which according to (l3), indicates the existence of the electron trajectory only on the surface with radius r0 n.
2
j
x
e
Fig. 1. Stationary solution for a wave of the particle (electron) to first lower stationary state (n=1)
I I I I i
1.5 2.0
-0.25 -
Fig. 2. Stationary solution for a wave of the particle (electron) to second lower stationary state (n=2)
Fig. 3. Stationary solution for a wave of the particle (electron) to third lower stationary state (n=3)
The properties of the trajectory and wave Vn(r,t), described above, indicate a different spatial arrangement of the electron in the hydrogen atom compared to the known pattern, described by the Schrodinger's wave function.
6. Discussion of results of the conducted research
The research undertaken indicates that the desire of L. de Broglie to overcome a wave-particle dualism
through the concept of wave-pilot is substantiated here via a continuation of the optical-mechanical analogy, which is solved at the level of wave optics. In this case, the wave function V(x, t) itself is not only connected to the motion of the particle in some way, but directly expresses the motion itself, which is always of a wave nature, be it light, or any other object.
When modeling an electron motion in the Coulomb field, the V-function method makes it possible to establish a rule of energy quantization of a hydrogen-like atom, which fully coincides with the classical results by Schrodinger and Bohr. In this case, discreteness of energy arises from satisfying the conditions following from the V-function method. The trajectory and the electron wave are interconnected, this relation is described by the method of V-function based on the local variational principle and solution to the direct and inverse problems of dynamics. According to the given approach, stationary behavior of the electron on the n-th stable state is described by wave Rn, which subsides exponentially to zero at r ^ ~ In this case, the amplitude of the wave passes zero on the sphere with a Bohr radius ron, which means the existence of the electron trajectory on the sphere of the given radius.
A benefit of the given method is that when simulating the motion of an object, one takes into account its wave motion and the trajectory motion at the same time. Reliability of the results is achieved by confirming the known results of quantum mechanics. In this case, however, the inevitability should be noted of the emergence of difficulties for experimental confirmation of the new results. It should also be noted that the trajectory motion of an object is described by the method of V-function only with a system of stationary differential equations, which can be regarded a constraint of the performed research.
7. Conclusions
1. Based on the method of V-function, we have drawn an optical-mechanical analogy, which thus gained a new continuation. Wave function V(x, t) directly expresses the motion itself, which is always of a wave nature, be it light, or any other object.
2. We obtained a solution to the direct and inverse problems of dynamics in a new statement for a hydrogen-like atom. The method of V-function makes it possible to establish a rule for the energy quantization of a hydrogen-like atom, which fully coincides with the classical results.
3. The ultimate solution to the stationary wave equation for a hydrogen-like atom was obtained. Stationary behavior of the electron is described by wave Rn, subsiding exponentially to zero at unlimited distance from the nucleus, and whose amplitude passes through zero on the sphere with a Bohr radius.
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