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Journal of Physics

Volume1|Issue18

Research Article Open Access

The Difficulties and Contradictions of Quantum Mechanics and Their Throughout Eliminations

Pang Xiao Feng

Corresponding author: Pang Xiao Feng, Institute of physical electron, University of Electronic Science and

Technology of China, Chengdu 610054,; Email: [email protected]

Citation: Pang Xiao Feng, (2018), The Difficulties and Contradictions of Quantum Mechanics and Their Throughout

Eliminations: Nessa J Physics

Received: 30th August 2018; Accepted: 14th September 2018; Published: 19th October 2018

Copyright: © 2018 Pang Xiao Feng. This is an open-access article distributed under the terms of the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original author and source are credited.

Abstract

The difficulties and contradictions of quantum mechanics, which is the foundations of model science, are first

collected, commented and elucidated in detail and systematically using a large number of experimental facts and

theoretical results obtained from the solutions of its basic Schrödinger equation in virtue of their comparisons with the

objective and intrinsic features of the microscopic particles having the duality of wave and corpuscle. These existed

difficulties and contradictions focus mainly the properties of basic Schrödinger equation, of which the solutions have

only a wave feature, not corpuscle feature due to the fact that no other interactions can be used to restrain the

dispersed effect of microscopic particle arising from the kinetic energy in dynamic equation and Hamiltonian of the

systems, thus quantum mechanism describes only the wave feature, cannot represent the wave-corpuscle duality of

microscopic particles. This is also fully not consistent with people’s traditional knowledge. These results lead to a

series of difficulties and contradictions in quantum mechanics, which cannot be always overcome in its framework up

to now, so, it is only an approximate theory. In such a case it is very necessary to eliminate and solvate these

difficulties and contradictions for promoting the development of quantum mechanics toward. Our research discovered

that these difficulties and contradictions are due to neglect the real nonlinear interactions existed among the

microscopic particles. In order to eliminate these difficulties and contradictions, the nonlinear interaction, such as

2

b , that is closely related the states of microscopic particles, should be added in original Schrödinger equation

and Hamiltonian. From the solutions obtained from different Schrödinger equation containing 2

b we affirmed that

they have a evident and clear wave-corpuscle duality because the nonlinear interaction restrained now the dispersed

effect of microscopic particle arising from the kinetic energy, then the microscopic particles are localized. This means

that the microscopic particles described by the Schrödinger equation including the nonlinear interaction possessed


Journal of Physics Volume 1| Issue 18

certainly a wave-corpuscle duality, which corresponds perfectly and completely with their intrinsic features and

experimental results. Then the difficulties and contradictions of quantum mechanics were through eliminated, thus we

affirmed that the microscopic particles should be described by nonlinear Schrödinger equation. In such a case we

elucidated and confirmed further the real and wide existences of the nonlinear interaction in all physical systems, they

are formed and produced by means of four mechanisms of self-interaction, self-trapping, self-focusing and self-

localized in the physical systems. Therefore we affirmed and concluded that the dynamic properties of microscopic

particles should be described by the nonlinear Schrödinger equation, instead of linear Schrödinger equation in original

quantum mechanics.

Keywords: quantum mechanics, difficult, contradiction, nonlinear interaction, wave-corpuscle duality, Schrodinger

equation, elimination, nonlinear equation, wave feature, corpuscle feature.

Introduction

As are known, the quantum mechanics is only fundamental theory of model science, a lot of new natural and

applicable sciences were built based on it, such as quantum electrodynamics, quantum field theory, quantum physics,

quantum chemistry, quantum biology, and so on. Hence, Its contributions on developments of science and technology

cannot be denied never. Thus, we learnt and used it always.

It is quite clear that quantum mechanics was established by several great scientists such as Bohr, Born, Broglie,

Schrödinger, Heisenberg, Born, Born, and Dirac et al. in the early 1900s [1-18]. It is mainly used to research, describe

and elucidated the properties of motion of microscopic particles, involved electron, proton, phonon, photon, exciton as

well atoms, molecules and other particles.

In quantum mechanics, the state of microscopic particles is represented a wave function ),( tr , it states of

movement and properties can be obtained from only dynamic equation, or following Schrödinger equation (here

called linear Schrödinger equation)

(1)

Where is the kinetic energy operator, V( ,t) is the externally applied potential operator, m is the mass of

the particles, is the coordinate or position of the particle, and t is the time. This theory tells us that once the

externally applied potential field and initial states of the microscopic particle are first known or given, then the states

and properties of the microscopic particles at any time later and any position can be determined by the Schrödinger

equation (1). Therefore, this theory simplifies greatly to research and find the properties and rules of movement of

microscopic parties in different matters and materials. This is a great creation and progress of science, which

promotes the development of science and technology.

( )( ) ( ) ( )

22

,, , ,

2

r ti r t V r t r t

t m

= − +

2 2 / 2m r→

r→



Quantum mechanics give only the wave solution, cannot give the solution of wave-corpuscle daulity, it’s a wave

theory

Just the importance of quantum mechanics, we should completely, deeply and systematically research itself features

and essences. From these investigations we can find its many difficulties, which are described as follows [9-16].

For example, at 0),( =trV , the solution of Eq.(1) is a plane wave, which is represented by

(2)

Obviously, this solution is a plain wave, where k, A’ and are its wave vector, frequency, and amplitude of the

wave, respectively, if P= kp = , then its energy is

(3)

This is a continuous spectrum of the energy of the particle. These results manisted that the particle moves in a wave

having a content energy and speed in total space- time region.

If the free particle can be confined in a small space, such as, a rectangular box of dimensions a, b and c, the solution

of Eq. (1) is a standing wave, which is denoted by

(4)

Where n1, n2 and n3 are three integers. This result manifests that the microscopic particle is stil a wave and cannot be

localized, although a, b and c are very small.

If the potential field is further varied, for example, the microscopic particle is subject to a conservative time-

independent field, 0),( trV , then they satisfies the time-independent linear Schrödinger equation:

(5)

where ]/exp[)('),( iErtr −= .If rFrV .)( = ,where F is a constant field force, such as the microscopic particle

is in an one- dimensional electric-field having uniform strength E’, then xeExV ')( −= , then its solution is

(6)

( ), 'exp[ ( )]r t A i k r t → →

= −

22 2 21

( ), ( , , )2 2

x y z x y y

pE p p p p p p

m m= = + + −

( ) 31 2, , , sin sin sin iEtn zn x n yx y z t A e

a b c

−

=

( )2

2 ' ' '2

V r Em

− + =

3 2(1)

1 2

2' ,

3

xA H

l

= = +



where H(1)(x) is the first kind of Hankel function, A is a normalized constant, l is the characteristic length, and is a

dimensionless quantity. In this case the microscopic particle is denoted by a dispersed wave because → , it

approaches

3/24/1 2/3

')(' −−= eA , (7)

This result affirm still that the microscopic particle is also a damped wave.

If 2)( axxV = , which denoted that the microscopic particle is acted by an elastic force. In the case, its eigen wave

function and eigenenergy are

(8)

and

....)2,1,0..()2/1( =+= nnEn (n=0,1,2,…) (9)

respectively, where )( xH n is the Hermite polynomial. In such a case the microscopic particle is still a decaying wave

and cannot be also localized.

If Eq.(1)is used to investigate the quantum features of the electron in the hydrogen atom, in this case its potential is, in

general, represented by a Coulomb interaction rerV /)( 2−= , which denoted the electric interaction between the

electron and nucleon. Thus the dynamic equation of the electron is following Schrödinger equation:

(10)

Its corresponding solution is denoted in a product of associated Legendre polynomial )(rR and

the associated Laguedre polynomial, ),( Y [9-12], it is denoted by

(11)

where ]'exp[)(cos),( ' mPY m

l −= , 'mN and )(cos' m

lP are the normalized coefficient, and are the

angles in a spherical coordinate, n, l and m’ are main, trajectory and magnetic quantum numbers of the electron,

respectively. Its eigenenergy is denoted by

,.......)3,2`,(,2/ 224 =−= nnmeE (12)

2 2 2'( ) ( )a x

n nx N e H x −=

2 22 ' ' '

2

eE

m r − + =

' '

' ''( , , ) ( ) ( , ) ( ) (cos )m im

m l lm nl lr R r Y N R r P e = =



Because )(rRnl is a decaying function involving ]2/exp[ oar− and )(cos' m

lP is composed esin and cos . This

means that the electron has only decaying feature, but not corpuscle nature in the hydrogen atom.

If the potential V ( ,t) of microscopic particles in Eq.(1) is changed further and continuously we may infer and

suppose that the solutions obtained from Eq. (1) possess only the wave features, a localized solution cannot obtain be

obtained always. This implies that the microscopic particles have only a wave feature and have not corpuscle feature,

if they are described by the linear Schrodinger equation in Eq.(1) no matter what of the potential 0),( trV . and

used in Eq. (1) or Eq.(5), then we cannot still obtain a localized solution. This signifies that Schrödinger equation in

Eq. (1) give only the solution of wave no matter what forms of the potentials. Then we can only conclude that the

microscopic particles described by the quantum mechanics have only a wave feature, but corpuscle feature, i.e., they

cannot be localized always. This is just the essential and intrinsic features of quantum mechanics, which cannot be

varied. These are just the layer upon layer difficulties of quantum mechanics. These make us believe the quantum

mechanics cannot describe completely and correctly the wave –corpuscle feature of microscopic particles, which were

obtained and verified from plenty of experimental results.

Why is this? This can be certified and confirmed from really physical significances of the Hamilton operator of

microscopic systems:

(13)

in quantum mechanics. It is basic relation in this linear theory. Equation (13) indicated clearly that the natures and

features of microscopic particles are determined by the kinetic term because the potential tem , can only change

their states, cannot change their natures and essences because it is not related to the state wave function of the particle.

Thus, there is no other force or the energy to obstruct and suppress the dispersing effect of the particle arising from its

kinetic energy in the system, then the microscopic particle can only disperse and propagate free in this space and

damps gradually. This means that the microscopic particles are unstable and cannot also be localized at all. This is

basic reason that the microscopic particles cannot be localized in quantum mechanics no matter how variations of the

externally applied potentials. This is just the reason and mechanism of layer upon layer difficulties of quantum

mechanics.

r→

2^ ^ ^ ^2 ( , )

2H T V V r t

m= + = +

^

T^

V



The Results of Quantum Mechanics are contradictive with the wave-corpuscle duality of microscopic particles

obtained from the experimental investigations

It is well known that the microscopic particles have all a wave-corpuscle duality, which was confirmed by plenty of

experiments, such as the diffractions of electrons on surfaces of Ni-mono-crystals by Davisson and Germer, in 1927,

and the electron beams with de Broglie wavelength of 0.00645nm, which penetrated over the multi-crystal powders

and Ag-sheet metal by Thomson and Taltakovsky, as well as the diffractions and inflections of the beams of neutrons

and He and H2 atoms, which penetrated or passed LiF crystals at T=2950C by Stern, Gerlach and Easterman [1-18],

and Young’s single and double-slit interference of the wave character of photons by Tuan and Gerjuoy et al., [10-11],

respectively. These experimental results of wave-corpuscle features are all contradicted with the above results

obtained by quantum mechanics, in which microscopic particles have only a wave feature as mentioned above. These

experimental results were described in detail books of quantum mechanics. Based on these experimental results, Bohr,

Born, Schrödinger and Heisenberg, etc., established quantum mechanics in the early 1900s [12-20]. We here exhibited

these experimental results and their analyses, which are described simply as follows.

Figure 1 exhibited the experiment result of the diffraction of the electrons on nickel crystal obtained by Davisson and

Gerner [21], in which they injected the electronic beam, which was launched by an electron gun, to the surface of this

crystal along its vertical direction. In this case, they observed the phenomenon of reflect of the electrons from the

surface of the crystal, where is the reflected angle, is the wavelength of the electron, it is closely related to the

electric-potential V accelerating the electronic beam, their relationship is represented by . Thus this

phenomenon can be described by formula: , where n is the order number of maximum diffraction, d is a

constant of the surface grid of the crystal. This relation is similar to the diffraction effect of X ray. Therefore, it

denoted that the electrons have a wave feature. From this experiment Davisson and Gerner obtained further the De

Broglie relationship of . Thus, the wave-corpuscle duality of the electrons is confirmed by this

experimental result.

Thomson [22-24] observed and measured the diffraction of the electrons on the multi-crystal, in which the electronic

beam was penetrated over the multi-crystal film with the thickness of 10-5 cm. This experimental theorem, which is

shown in Fig.2. From this experiment they obtained the relation of , where n is also order number of

maximum diffraction, d is the space constant of the surface grid of the crystal, and is the angle between the

electronic beam and grid plane of the multi-crystal. On the other hand, they obtained the diffraction rings of the

electrons, which is shown in Fig.3, which are similar with the diffraction of X -ray, where the electronic beam was

accelerated by the electric-potential of 36 x 103e V, which corresponds to the De Broglie wavelengths of 0.00645 nm.

Stern and Fasterman also measured the reflect and phenomena of the He atom and H2 molecule on the surface of the

LiF crystal [6,9], in which the energies and corresponding De Broglie wavelength of the microscopic particles were

changed gradually by modulating the electric-potential. A sensitive barometer recorded their scattering strengths on

the crystal. The result of distribution of reflected strength of He atom beam on crystal at T=2950C was obtained,

0

(150 / )V A =

sinn d =

( / )P h n=

2 sin 'n d =

'



where the result at the reflective angle 00 corresponds to the normal reflectance of the He atom, its strength is

maximal, but two small peaks at the reflective angles of 100 and -100 are due to the diffraction of the He atom. H2

molecule showed also analogous results.

Fig.1. The diffraction of electronsin nickel, (a) is experimental device, (b) is the result[9, 20, 21].

At the same time, the diffraction of the neutrons has been observed also. Because the fact that the mass of the neutron

is greater (about1.66x10-24g), then the neutrons with energy of 0.01 eV can also lead to a stronger diffraction relative

that of the X-ray. The diffraction effect of neutrons (Laue diffraction) on the NaCl crystal was observed [6,9].

The result of distribution of reflected strength of the He atom beam on LiF crystal at T=2950C was observed, where

the result at the reflective angle 00 corresponds to the normal reflectance of the He atom. Its strength is maximal, but

two small peaks at the reflective angles of 100 and -100 occur also due to the diffraction of the He atom. The

analogous results for H2 molecule were also attributed to them.

Fig. 2.Diffraction of electrons in multi-crystal [9,20,22-24].



Fig.3. The image of diffraction of electron in multi-crystal [6,20].

On the other hand, the diffraction of the neutrons has been observed also. Owing to the fact that the mass of the

neutron is m0 = 1.66 x 10-24g, then the neutrons with energy of 0.01 eV can generate a stronger diffraction resembling

that of the X-ray mentioned above. The diffraction effect of neutrons (Laue diffraction) was observed experimentally

[6,9].

It is known that Young’s slit interference experimental results of proton, electron and photon and their double-center

diffraction images can indicate the wave feature of these microscopic particles. In fact, Tuan and Gerjuoy [25]

discussed and observed an atomic version of Young’s double slit experiment for matter waves in ion-atom collisions.

They also debated the capture processes in collisions of protons with H2 and predicted that the diffraction of the

protons from the two atomic centers of the molecule can lead to interference effects, in which the captured cross-

sections are related to the molecular orientation, which may be due to the interference effects [26-29]. Schmidtet al.,

[30] observed the interference effect caused by the pronounced matter wave in multiple differential momentum

spectra of recoil ions produced in dissociative capture in collisions. However, the interference effects in

ionization processes are quite difficult to observe [31] because the final state of the collision involves three unbound

particles, opposed to only two in a capture process, where the phase angle between the amplitudes associated to the

two molecular centers is accessible through both the momentum information of the collision and the molecular

orientation. Data for ionization, in which the phase angle was determined, are only available for electron impact [32].

For ion impact, interference in the projectile diffraction was observed for fixed projectile energy losses in the scatter

in angle dependence [33, 34]. The interference structures were also discussed in double differential energy

distributions of the ejected electrons in fixed emission angles [35–41]. Two-center interference should also occur in

the ion impact induced excitation of a molecule. Zhang et al., [42-43] studied the diffraction process, using an indirect

approach, by investigating simultaneous excitation of projectile ions, resulting in dissociation, and target

2H He+ +

2H +



ionization, in collision with He, i.e. , in a kinetically complete experiment providing

the full phase information. They observed the interference structures due to scattering of helium matter waves from

the two atomic centers in the ions.

The above results shown in Figures 1-3 indicated clearly that the microscope particles including the electron, photon

and neutron all have a wave feature, no matter what the nature of these particles. This means that wave feature is a

basic nature of microscopic particles. However, the microscopic particles have also a corpuscle features. This

statement can be confirmed by the following experimental results.

(1). The microscopic particles all have certain sizes and masses, for example, the masses of electron, proton, and

neutrons are 9.10953 x 10-31 kg, 1.67265 x 10-27 kg, and 1.67495 x10-27 kg, respectively, the electron’s classical radius

is 2.817938 Fm.

(2). The radiation phenomenon of black-body recognized and supported the corpuscle feature of microscopic

particles [1-9]. This phenomenon was researched by Planck, and he explained perfectly it by the radiation theory

based on the assumption. So-called is that the body can absorb completely all incident lights without any reflection. In

this case Planck proposed the concept of light quantum (or photon quantum) and thought that the black-body is

composed of a large number of resonant-oscillator with certain charges and quantized energies of .

Thus he obtained the formula of distribution of photon quanta radiated by the black-body, which is

(14)

where is a minimal energy, which is completely consistent with the experimental results. This verified that

the photon quanta in the radiation of the black-body has a corpuscle feature, its energy is quantized, i.e., , n

is a integer.

In accordance with Planck’s theory, the specific heat of solid was also found, where the atoms in the solid are some

resonant oscillators (or phonons). Then the average energy of the atom can be denoted by

. Thus, the specific heat can be expressed by

. Hence, he obtained that the specific heat of the solid is k at high

temperate and zero at low temperate , respectively. This means that the oscillators or

phonons in the solids also have a corpuscle feature.

(3). Einstien explained the photon-electric effect occurred in metal using Planck’s concept of photon quantum [1-9].

In this case, Einstein thought that the electrons in the metal absorbed the energy of the incident photons to escape

from the metal through overcoming the attraction force of the surface of metal on it, in which the residual energy is

used to move in the space. The relation of photon-electronic effect is represented by , where

2H He H H He e+ ++ → + + +

2H +

0 0 0 0, 2 ,3 ...... ...n

2 2 1(8 / ){exp[( / ) 1]}d h c h KT d −= −

0 h =

E nh=

1{exp[( / ) 1]}E h h KT −= −

2 1/ ( ) [exp[( / )]{exp[( / ) 1]}C dE dT h h KT h KT −= = −

( / 1)h KT ( / 1)h KT

2

0(1/ 2)mv h W= − m



and are the mass and moved the velocity of the electron, respectively, is the escaped function of the electron,

is the energy of the phonon. Therefore, the correctness of the relation affirmed and confirmed experimentally

the corpuscle features of phonon and electron.

At the same time, from the relationship of the energy-momentum in the relativity theory, , we can

obtain the relationship of the energy-momentum of the photon with light velocity c is . Thus, we can gain the

Broglie relationship:

(15)

This indicated clearly that the photons possess both corpuscle features, which is embodied by E and , and the wave

feature, which is embodied by and . This means that all microscopic particles involving the particles of

relativity or higher velocity subject all to the Broglie relationship. This implies the Broglie relationship exhibited and

embodied really essential and interactive features and nature for wave-corpuscle duality of microscopic particles.

(4). The Compton scattering effect of the photon and electron, which was observed and confirmed by Compton’s and

Wu You-xun’s experiments [1-9], respectively. Thus the corpuscle feature of microscopic particles are again verified.

This experiment indicated that the X-ray with high frequency is scattered by the electrons in matter, where the

wavelength of the X-ray increases with increasing the scattering angle. The phenomenon is, in essence, due to the

collisions of the photon with the electron, which is shown in Fig.4. Based on this case the conservation relations of the

energy and momentum in the collision process can be discovered, they are expressed by

(16)

2

/ ' / '[ ],1

mvc con c con

= +

−

(17)

20 'sin / sin '[ ],

1

mvc

= −

−

(18)

respectively, where /v c == , Eq.(16) indicated the energy conservation, Eqs.(17) and (18) manifested the

momentum conservations along the OA direction and the vertical direction of OA in Fig.4, respectively, where and

' are the angle between OB and OA as well as OC and OA, respectively. From Eqs.(16)-(18) we can obtain the

following relationships of change of frequency or wavelength of the photon in this scattering process.

or (19)

v 0W

nh

2 2 2 2 2E m c c P= +

E cP=

,h h

E h P n n kc

= = = = =

P

k

2

2

1' [ 1],

1mc

= + −

−

2

2

2' ' sin ( '/ 2)

mc − = 24

' sin ( '/ 2)mc

= − =



Fig.4. The of Compton scattering phenomenon of the photon on the electron [18].

(5).The quantum feature of the light spectrum of hydrogen atom. This feature was obtained by Bohr [12-14]. In this

phenomenon, Bohr assumed the electrons with quantum feature move along the orbit’s circumference around the

nucleon. In this case the quantization condition of the circle orbits is denoted by nhPdq = , or ,

where n is the quantum number, and p and q or and are the generalized momentum and coordinate,

respectively. In this case the energy of the electron in motion of circumference of the orbits can be represented by

Therefore, the energy of hydrogen atom is quantized. Bohr thought further that the light spectrum of the hydrogen

atom arises from the transitions of the electron between the energy levels, and then the frequencies of photons

radiated in the hydrogen atom can be expressed by

(20)

Utilizing Eq.(20) the light spectrum of the hydrogen atom can be perfectly explained, However, the phenomenon

cannot be explained by wave features of the electron in the atom obtained from Eq.(8). Thus we confirmed again the

corpuscle feature of the electron and approximation of quantum mechanics in Eq.(1).

(6). As it is known the different accelerators, such as the electron accelerator and proton accelerator, were widely used

in industry, medicine, and science research. In these cases we cannot thought absolutely that the electrons and protons

have only a wave feature, and can think that they have corpuscle feature, or wave-corpuscle duality because these

accelerators can accelerate these particles, but cannot s can accelerate wave. This verified and affirmed also that these

microscopic particles have the corpuscle feature or wave-corpuscle feature..

The above results studied show clearly that all microscopic particles, including electron, proton, neutron,

photon, hydrogen atom and He molecule, possess both corpuscle and wave features, therefore they have wave-

2

0p d nh

=

p

2 2 2 4 2 2/ 2 / / 2 / 2 ,( 1,2,3,....)E T U e a e a e a me n n= + = − = − = − =

)11

(2

223

42

nmh

em−=



corpuscle duality [1-9]. Just so, the natures of microscopic particles were represented in the well-known De Broglie

relationship of nhPdq = , or [15-9]. Thus we have to represent the states and properties of

microscopic particles by a wave function, or , where denoted the position of the particle at time t.

This means that embodies both its wave feature and corpuscle feature. Therefore, we can affirm and conform

that the microscopic particles should have a wave-corpuscle duality. These results are contradicted completely with

those obtained by quantum mechanics mentioned above. They exhibited and showed systematically and completely

the layer upon layer difficulties and contradictions of quantum mechanics, or speaking, quantum mechanics is only an

approximate or wave theory, and cannot describe the really wave-corpuscle duality of microscopic particles at all,

which were conformed and verified by a large experiments. This implies that these difficulties of quantum mechanics

must be overcome and eliminated, it should be also improved and developed forward to describe completely and

correctly the wave-corpuscle duality of all microscopic particles. The methods of elimination of the difficulties of

quantum mechanics and its directions of improvement and development are described as follows.

There are always the contradictions of started assumptions with final results in the investigations of quantum

mechanic problems

If the processes finding the solutions in Schrödinger equation in Eq.(1) with different potential functions are noted and

checked carefully we can find that they existed always some contradictions between the started assumptions and final

results, which are described as follows.

For example, the interactional potential between the electron and nuclei in the hydrogen atom is represented by the

Coulomb potential, V= -e2/ in dynamic equation of the electron in Eq.(10), where is distance between them. As

it is known, the Coulomb potential represented the electric interaction between the two macroscopic charged particles.

This signifies we first assume that the electron and nuclei in the hydrogen atom are all the real particles. However, we

know clearly from the solution in Eq.(11) of Eq.(10) that the electron is a wave having the damping feature as

mentioned above, it has not corpuscle feature at all. This exhibited and exposed clearly that the started assumption is

contradicted with final result. For example again, the potential rFrV .)( = , or xeExV ')( −= , which is related

Eq.(6), or Eq.(7) indicated that the particle having the charge e has the potential energy in is in an one- dimensional

uniform strength E’. This denoted also that the microscopic particle possesses the properties of macroscopic particles.

However, from the solution in Eq.(6) we knew that the particle cannot be localized at all and has also not the feature

of corpuscle in this case. This indicated obviously the contradiction between the begun assumption and final

conclusion.

The instance is again the potential of 2)( axxV = , which is related to Eq.(8), this is, in essence, a resonant potential

of macroscopic resonant-oscillator, it is now used to describe the vibration of microscopic particle. However, the

solution in Eq.(8) obtained from its dynamic equation in Eq.(1) has only a wave feature with the attenuation feature.

2

0p d nh

=

( ),r t ( ),r t r

( ),r t

r r



This means that the microscopic particle has not also the corpuscle feature. This exhibited again that the contradiction

of the begun assumption with the final conclusion.

We can give a lot of examples to verify the correctness of the contradiction between the begun assumption and final

conclusion in quantum mechanics [44-54]. As a matter of fact, the contradiction exists always in the applications of

quantum mechanics because we must assume first of all that the microscopic particles are a corpuscle and gave again

its potential according to its properties of interaction with other particle or the background field and movement

features. Therefore, the layers upon layer contradictions are always existent in quantum mechanics. This is its intrinsic

weak point. And contradictions of quantum mechanics must be overcome and eliminated, it should be improved and

developed forward to describe completely and correctly the wave-corpuscle duality of all microscopic particles. The

methods of elimination of the difficulties and contradictions of quantum mechanics and its directions of improvement

and development are described as follows.

The direction and methods eliminating the difficulties and contradictions of quantum mechanics

The investigations showed that the direction of solution of the difficulties and contradictions of quantum mechanics is

to consider the nonlinear interactions among the microscopic particles or the nonlinear interaction of the particle with

background field, which can often be represented by the form of 2

b , where b is a nonlinear interaction

coefficient, and to add it into the linear Schrödinger equation in Eq.(1) [44-59]. Thus in this case the dynamic

equation of microscopic particles is varied as the following nonlinear Schrödinger equation

222 ,

2

hih b V r t

t m

→ = − +

, (21)

where ( , )r t→

is still the wave function of microscopic particles. We can suppose and predict that the natures and

features of microscopic particles will generate considerable variations, in this case the particle could be localized and

have a corpuscle feature. Then the difficulties and contradiction of quantum mechanics can be solved or eliminated

immediately. Thus this scheme can become a perfect method to solve and eliminate the difficulties and contradiction

of quantum mechanics. Why? Following researched results can confirm the correctness of this conclusion.

In order to demonstrate and exhibit the conclusion we here find first the solutions and their features of Eq.(12) at

,V r t→

= 0.

In such a case the equation (12) at V(x,t)= 0 in one-dimensional case is represented by

02

''' =++ bi xxt (22)



where ' 2 / , ' /x x m h t t h= = .The distinction of Eq.(22) with Eq.(1) at ,V r t→

= 0 is only the 2

b to be added

in Eq.(22). If we assume the solution of Eq.(22) is of the form[55-59]

' '( ', ') ( ) i t iQxx t e += (23)

where ' 'ex v t = − . Inserting Eq. (23) into Eq.(22) we can obtain

2 2( ) (2 ) ( ) ( ) 0,( 0)ei Q v Q b b + − − + − = (24)

If the imaginary coefficient of vanishes, then / 2eQ v= .Then from2A Q= + we get that

2 / 4eA v = − .

Thus from Eq.(24),we obtain[55-59]

3 0b A + − = (25)

This equation can be integrated, which results in

2 2 4( ) D A = + − (26)

where D is an integral constant. The solution ( ) of Eq.(26) is obtained by inverting an elliptic integral:

0 2 4

d

D A b

=

+ − (27)

Let2 2 4 2

1 2( ) ( ' )( ' ) ' (2 / ) 'P A b D = − − = − + + , where1/4' ( / 2)b = , then we can get from Eq.(27)

[ ( ) ( ', )]K k F k − = , where K (k) and ( ', )F k are the first associated elliptic integral and incomplete elliptic

integral, respectively, and

1/2 1/2 2 1/2

1 2 1 1,2[( ) / ] , / (2 ) [ ( / 2 )]k A b D A b −= − = +

Using these and ,2

1,2 = , then we have

1/4 ' '2 '2 2 1/4 1/2

1 1 2'[( / 2) ] {1 [(1 / ) ( ( / 2) , )]}b sn b k = − − (28)

When' ' ' ' 1/4

1 0 0 00, , 1, ' sec [ ( / 2) ]D k h b → → → → , where' 2 1/4

0 (2 / )A b = , thus, the soliton solution

of Eq.(22) can be finally represented [45-59] by

2 1/22( ', ') sec [ ( ' ')]exp[ ( '/ 2 '/ 4 (2 / ) ')]e e e

Ax t h A x v t i v x v t A b t

b = − − + (29)



Then the solution of Eq. (22) eventually can also write as the following form in coordinate of (x,t)

( ) ( )( )'

0[ ]'00 0, sec ei mv x x Et

e

A bmx t A h x x v t e

− − = − −

(30)

where'2 '2 ' '

0

( 2 )e e e cmv E m v v vA

b b

− −= = ,

'

ev and '

cv is the group and phase velocity of motion of the electron,

respectively,' '2 e cE mv v= = in the coordinate system . The solution of Eq. (30) can be also found by the inverse

scattering method[60-61]. This solution is completely different from Eq.(2)in quantum mechanics, and consists of a

envelop and carrier waves, the former is '0

0 0( , ) sec { [( ) ]}e

A bmx t A h x x v t

h = − − and a bell-type non-topological soliton

with an amplitude A0, the latter is the '

0exp{ [ ( ) ] / }ei mv x x Et h− − [62-67].This solution in Eq.(30) can be

represented in Fig.5.

Fig.5. The representation of solution in Eq. (30)

Therefore, the particle described by nonlinear Schrödinger equation (8) is a soliton according to the soliton theory[22-

23] .The envelop φ(x, t) is a slow varying function and the mass centre of the particle, the position of the mass centre is

just at x0, A0 is its amplitude, and its width is given by 0' 2 /W h mbA= .Thus, the size of the particle is

0 ' 2 /AW h mb= and a constant. This shows that the particle has exactly a mass centre and determinant size, thus

is localized at x0. According to the soliton theory [62-67], the bell-type soliton in Eq.(30) can move freely over

macroscopic distances in a uniform velocity v in space-time retaining its form, energy, momentum and other quasi-

particle properties. Just so, the vector r→

or x has definitively physical significance, and denotes exactly the positions

of the particle at time t. Then, the wave-function ( , )r t→

or φ(x, t) can display exactly the states of the particle at the

position r→

or x and time t. These features are consistent with the concept of particles. Thus the feature of corpuscle of

the microscopic particle is displayed clearly and outright.



At the same time, we simulated numerically also the collision feature of two soliton solutions of Eq. (22) using the

fourth-order Runge-Kutta method, its result is shown in Fig. 6. From this figure we see clearly that the two particle s

can go through each other while retaining their form after the collision, which is the same with that of the classical

particles. Therefore, the particle depicted by the nonlinear Schrödinger equation (22) is a soliton and has an obvious

corpuscle feature according to soliton theory [60-61].

Therefore, the particle described by nonlinear Schrödinger equation (8) is a soliton according to the soliton theory[22-

23] .The envelop φ(x, t) is a slow varying function and the mass centre of the particle, the position of the mass centre is

just at x0, A0 is its amplitude, and its width is given by 0' 2 /W h mbA= .Thus, the size of the particle is

0 ' 2 /AW h mb= and a constant. This shows that the particle has exactly a mass centre and determinant size, thus

is localized at x0. According to the soliton theory [62-67], the bell-type soliton in Eq.(30) can move freely over

macroscopic distances in a uniform velocity v in space-time retaining its form, energy, momentum and other quasi-

particle properties. Just so, the vector r→

or x has definitively physical significance, and denotes exactly the positions

of the particle at time t. Then, the wave-function ( , )r t→

or φ(x, t) can display exactly the states of the particle at the

position r→

or x and time t. These features are consistent with the concept of particles. Thus the feature of corpuscle of

the microscopic particle is displayed clearly and outright.

At the same time, we simulated numerically also the collision feature of two soliton solutions of Eq. (22) using the

fourth-order Runge-Kutta method, its result is shown in Fig. 6. From this figure we see clearly that the two particle s

can go through each other while retaining their form after the collision, which is the same with that of the classical

particles. Therefore, the particle depicted by the nonlinear Schrödinger equation (22) is a soliton and has an obvious

corpuscle feature according to soliton theory[60-61].

we can re-write the solution Eq. (30) as the following form[60-61]：

( ) ( ) ( )'0[ ' ']' 2

0

2, 2 sec 2 ' '

e civ x x v t

ex t k h k x x v t eb

− −

= − −

(31)

where 23/2k/b1/2= A0,

b

vvvA ece

2

22

0

−= , ve is the group velocity of the particle, vc is the phase speed of the carrier

wave. For a certain system, the values of '

ev and '

cv are determinant and do not change with time. From the above

results we can affirm clearly that the particle is a soliton.



Fig.6. The feature of collision of two soliton solutions represented by Eq. (22).

According to the soliton theory[60-67], the soliton has determinant mass, momentum and energy, which can be

represented in the coordinate of (x,t) by [60-67]

2

0' 2 /sN dx A bm

−= =

( )* * '/ 2 x x e sp i dx mv N const

−= − − = =

22 4 '2

0

1 1

2 2 2x sol eE b dx E M v

m

−

= − = +

(32)

in such a case, where 02 /sol sM N A bm= = is just effective mass of the microscopic particles, which is a

constant, ve and vc are defined in the coordinate of (x,t). Obviously, the energy, mass and momentum of the particle

cannot be dispersed in its motion. This manifests again that the particle represented by ( , )r t→

or φ (x,t) is localized

and has a corpuscle feature. This means that the nonlinear interaction, 2

b , related to the wave function of the

particle balances and suppresses really the dispersion effect of the kinetic term in Eq.(22) to make the microscopic

particle localize eventually [60-67], thus the corpuscle feature of the particle is exposed.

However, the envelope of the solution in Eq.(30),or (31) is a solitary wave. It has a certain wave vector and frequency

as shown in Fig. 7, and can propagate in space-time, which is accompanied with the carrier wave. The feature of

propagation depends only on the concrete nature of the particle. Figure 7 shows the width of the frequency spectrum

of the envelope φ (x,t), but the frequency spectrum has also a localized structure around the carrier frequency ω0. This

shows that the microscopic particle also has a wave feature.



Fig.7. The feature of wave vector and frequency of the soliton denoted in Eq. (30).

Thus, the particle has exactly a wave-corpuscle duality [60-67], which is first obtained. The Equation (30) or (31) and

Figure 5(a) are just a most beautiful and perfect representation of wave-corpuscle duality of the microscopic particle.

This consists also of de Broglie relation, and , of wave-corpuscle duality of wave-corpuscle

duality and Davisson and Germer’s experimental result of electron diffraction on double seam in 1927 as well as the

traditional concept of particles in physics mentioned above [50-59]. This means that the wave-corpuscle duality of the

microscopic particle is obtained and affirmed further also from this improvement and variations.

From the above investigations we know that if the nonlinear interactions 2

b is considered and added into the

linear Schrödinger equation in Eq.(1), then the new dynamic equation in Eq.(21) formed can eliminate completely

these difficulties and contradictions of quantum mechanics and can result in the localization of the microscopic

particles and their wave-corpuscle duality, which are completely different from those of the quantum mechanics, thus

the problems and disputations existed in the quantum mechanics were solved also in virtue of the above theory. In this

case we can affirm and confirm that the nonlinear interaction, 2

b , and corresponding nonlinear Schrödinger or

dynamic equation in Eq.(21), which are used to describe the dynamic properties of microscopic particles, are correct,

suitable and successful. Then, the nonlinear Schrödinger equation in Eq. (21), instead of the linear Schrödinger

equation (1), can be served as the correct and basic dynamic equation of microscopic particles in microscopic

systems.

The effects of nonlinear interaction on the difficulties and contradictions are not changed with the different

potentials

In the above investigations we discussed only the eliminated effects of nonlinear interaction on the difficulties and

contradiction of quantum mechanics at ,V r t→

= 0. In this case it is very necessary to know whether the eliminated

effects of nonlinear interaction on the difficulties will be changed, if the externally applies potentials is varied, or

speaking, at 0),( trV [47, 51-59,62-67]. We now researched this problem.

E h = = p k=



If cxxV += ')'( +c in Eq. (21), where and c are some constants [51-59]. The wave function, ( )tr ,

, can be

written as

( ) ( ) ( )trietrtr ,,, = (33)

where both the amplitude ( )tr ,

and phase ( )tr ,

of the wave function are functions of space and time.

Substituting Eq. (33) into Eq.(21) in one-dimensional case we can obtain

)35...(..............................02

)34.().........0....('

'''''

22

''''

=++

+=−−−

txxxx

xtxx bcxb

Now, let

(36)

and describes the accelerated motion of .The boundary condition requires to approach zero rapidly,

then the equation (35) can be written as:

(37)

where '/.

dtduu = . If 0/2.

− u , then equation (37) may be denoted by

)2//(/[)'(.

2 utg −=

or

(38)

The integration of Eq. (38) yields

(39)

where )'(th is an undetermined constant of integration. From Eq. (39) we derived

(40)

2 '( ', ') ( ), ' ( '), ( ') (t') t dx t x u t u t v = = − = − + +

( '. ')x t ( )

2

2u 2 0

− + + =

'

' 2

g(t ) u

x 2

= +

20

''( , ) ( ) ( )

2

x dx ux t g t x h t

= + +

02 2 20

''( ) ( )

2

x

x

dx gu gu ug t x h t

t

=

= − + + +



From Eq.(34) we can obtain

(41)

where is a real parameter and is defined by

−= )()(0 VV (42)

with

(43)

Clearly, in the case discussed, 0)(0 =V . Obviously, it is the solution of Eq. (41) when and g are constant. For

large , we may assume that+

1

, when is a small constant. To ensure that22 / dd and approach

zero when → ,only the solution corresponding to 00 =g in Eq. (41) remains stable. Hence we choose g0=0 and

obtain the following from Eq. (38)

(44)

Thus, from Eq. (43) we obtain

, . (45)

Substituting Eq. (45) into Eqs.(40) and (43), we obtain

(46)

Finally, substituting the above into Eq. (41), we see that:

(47)

When > 0, Pang gives the solution of Eq. (47), which takes the form

23 32

02/b g

= − +

2

' 02

u u gu( '+c)+ ' h(t')+ ( )

2 4xx x V

=+ + =

'

u

2x

=

2u ux'+c ' h(t')

2 4x = − + − − 2 2 3 2( ') ( / 4 ) ' ' / 3 ' / 2]}h t v c t t v t = − − − +

2 2 3 2( ' / 2) ' ( / 4 ) ' ' / 3 ' / 2]t v x v c t t v t = − + + − − − +

23

20b

− + =

2 / sec ( )b h =



Pang [1-13] finally obtained the complete solution in this condition, which is represented as

(48)

This is a soliton solution.

Thus, if V(x’)=c, the solution can be represented as

(49)

At '2)'( xxV = and b = 2 we can also derive a corresponding soliton solution from the above process.

However, in this case Chen and Liu [68-69] adopted the following transformation:

(50)

to make Eq.(21) become

. (51)

Thus Chen and Liu [68-69] represented the solution of Eq. (21) at ')'( xxV = and b = 2 by

(52)

At the same time, utilizing the above method Pang [47,51-59] also found the soliton solution of Eq. (21) at

)(')(')'( 2 tBxtAkxxV ++= . Pang obtained finally its solution in accordance with the above method , which is

represented by

(53)

where

, )()'2cos(2)'( 0 tutktu ++= (54)

( ) ( )

( )

' 2

0

' 2 2 3 2

0

2', ' sec ' ( ' ' )

exp{ [( ' / 2) ' ( / 4 ) ' ' / 3 ' / 2]}

x t h x x t vt db

i t v x x v c t t v t

= − + − −

− + − + − − − +

( ) ( ) ( )' ' ' 2

0 0 0

2', ' sec ' ( ' ) exp{ [ ' / 2 ( / 4 ) ']}x t k h x x v t t i v x x v c t

b

= − − − − − − −

( ) 2 3 2', ' '( ', ')exp[ 2 ' ' 8 ' / 3], ' ' 2 ' , ' 'x t x t i x t i t x x t t t = − + = − =

22 0x xt

i ' + + =

( ) ( ) ( )' 2 '

0 0

2 2 2 3 2

0

', ' 2 sec 2 ' (2 ' 4 ' ') exp{ [2( ' ') '

4( ' ) ' 4 ' / 3 4 ' ' ] }

x t h x x t t i t x x

t t t

= − + − − − − +

− + − +

( , )( ( )) i x tx u t e = −

( ) ( ) '

0

2', ' sec ' ( ')x t h x x u t

b

= − −



(55)

where L is a constant related in A(t’).

If ,0)()( == tBtA then the above solution is still Eq. (53), but

,

(56)

For the case of 22 ')'( xxV = and b=2, where is constant, Chen and Liu [68-69] assumed , thus

they represent the soliton solution in this condition by

(57)

where =2 is the amplitude of microscopic particles, '4 is related to its group velocity in Eqs. (52) and (57).

From the results in Eqs.(49)-(50), (52) and (56), which are completely different from those of Eqs. (2),(6)-(8) in

quantum mechanics, we know that the solutions of the differently nonlinear Scrodiner equation having different

potentials, V(x)=c,

cxxVxxV +== ')'(,')'( ,

222 ')'(,')'( xxVkxxV == and )()()( 2 tBxtAkxxV ++= are still a soliton,

which are also analogous with Eq. (29, or Eq.(30), i.e., they are all a bell-type solitons with a certain amplitude A0,

group velocity ve and phase speed vc,, and they are all composed of a envelope and a carrier waves; They have also a

certain mass center and determinant amplitude, width, size, mass, momentum and energy, which can be found out if

these solutions in Eqs. (49), (50), (52) , (53) (56) and (57) are substituted into Eq.(32). The differences among them

are only the outlines and the sizes of value of their parameters. This means that These microscopic particles are still

localized and have still the wave-corpuscle duality, although their amplitude, size, frequency, phase, group and phase

velocities have some differences. Thus we obtained and confirmed that the externally applied potentials cannot vary

the eliminated effects of nonlinear interaction on the difficulties and contradictions, namely, the nonlinear

Schrödinger equation (21) can eliminate thoroughly the difficulties and contradictions of quantum mechanics, it is

correct dynamic equation of microscopic particles.

( ) ( )'

' 2 2

0 0 00

0 0

', ' [ 2 sin(2 ' ) ( ') / 2)] ' {[ (2cos(2 '' ) ( '') / 2)]

( '') [ 2 sin(2 '' ) ( '') / 2]} '' '

t

x t kt u t x x kt u t

B t kt u t dt Lt

= − + + − + − + + −

+ − + + + +

0( ') 2cos(2 ') ( ')u t kt u t= +

( ) ( )'

' 2

0 0 00

0 0

', ' [ 2 sin(2 ') ( ') / 2)] ' {[ (2cos(2 '') ( '') / 2)]

[ 2 sin(2 '') ( '') / 2]} ''

t

x t k kt u t x x k kt u t

k kt u t dt

= − + − + − + +

− + +

( ') (2 / )sin(2 ')u t t =

( ) ( ) ( )' ' '

0 0 0

2 ' 2 '

0 0 0

', ' 2 sec 2 ' (2 '/ )sin(2 ') exp{ [2 ' ' cos 2 ( ' )

4 ( ' ) ( ' / )sin[4 ( ' ) ]}

x t h x x t i x x t t

t t t t

= − − − − −

+ − − − +



The nonlinear interactions are real and extensively existed in all physucal systems

The essential effects of the nonlinear interaction

The above investigations tell us that if the nonlinear interaction,2

b , is added into the linear Schrödinger equation

in Eq.(1), then the microscopic particles are localized and have the wave-corpuscle feature, thus the difficulties and

contradictions of quantum mechanics are eliminated completely. In this case we want know what are the essential

effects of the nonlinear interaction? In order to response this question we observe the nonlinear motion of water wave

arising from the macroscopic movement of sea water in approach the coast , which is shown in Fig.6. We see clearly

that the appearing of distortion of sine water wave with increasing the time under affection of nonlinear interaction.

This means that the nonlinear interaction deforms the outlines of the wave. In this process, when the water-wave

approaches the beach, its shape is varied gradually from a sinusoidal cross section to triangular, and eventually a crest

which moves faster than the rest, as shown in Fig. 8, which gives change as time goes on. As the wave approaches the

beach, it will be broken up due to the fact that the nonlinear interaction is enhanced. Then, we can affirm that the

nonlinear interaction destructs mainly the wave shape

Fig.8. The changes of wave with increasing nonlinear interactions

and its propagation. In this case the speed of wave propagation depends on the height of the wave. If the phase

velocity of the wave is denoted by cv , since it depends weakly on the height of the wave, h, then we can express the

relation by

1 ,c cov v h

k

= = + (58)

Where 01

ch h

v

h =

=

, 0h is the average height of the wave surface, cov is the linear part of the phase velocity of the

wave, 1 is a coefficient denoting the nonlinear effect. This expresses that the nonlinear interaction results in changes

in both form and velocity of waves. This is the same for the dispersion effect, but their mechanism and rules are

different [62,63,70]..

In dispersion medium the relation between the wavelength and frequency of the light (wave) is expressed by

( )k = or ( , ) 0G k = , where 2 2 2/ 0k in one-dimensional case. It specifies how the velocity or frequency

of the wave (light) depends on its wavelength or wavevector. The equation depicts wave propagation in a dispersive



median and is called as dispersion equation. The linear Schrödinger equation (1) in quantum mechanics is just a

dispersion equation, in which there is ,E p k= = . The quantity cv k= is called the phase velocity of the

microscopic particle (wave), but the wave vector k is a vector designating the direction of the wave propagation. Thus

the phase velocity can be denoted by2( )cv k k= . This is a standard dispersion relation. Therefore, the systems

described by quantum mechanics constitute a dispersed medium, then the solutions Eq. (1)-(9) of the linear

Schrödinger equation (1) is affirmatively a dispersive wave. Thus the microscopic particles represented by Eq.(1)

possess only a wave feature and are instable. When the dispersive effect is weak, the velocity of a wave can be

denoted by

' 2

2/ ,c cov k v k = = + (59)

where '

cov is the dispersion less phase velocity, 0

2 2

2 ( / )c k kv k == is the coefficient of the dispersion feature of

the wave. Generally speaking, the lowest order dispersion occurring in the phase velocity is proportional to k2, and the

term proportional to k gives rise to the dissipation effect [62,63,70].

Therefore the dispersed effect and nonlinear interaction can all deform the form of the wave, but their mechanisms

and effects are different. If they occur simultaneously in a dynamic equation, then the natures of microscopic particles

described by the dynamic equation will be changed. If the balance between two deformation effect of wave each

other, namely, dispersion effect of the wave is suppressed by the nonlinear interaction, then the localization of

microscopic particles appears possibly, which can be verified in following example.

Because the above nonlinear interaction 2

b was added into the dynamic equation in Eq.(21), the corresponding

Hamiltonian operator of the system is changed as

22

),( btrVTH ++=

(60)

In this case the nonlinear interaction energy 22

b can deform and restrain the dispersed effect of kinetic energy of

particles through its distorted function, thus the features of microscopic particle are changed relative to those in

quantum mechanical case. If they can be balanced and counteracted each other , thus the particle is localized and have

the wave-corpuscle duality[47, 51-56]. This means that the nonlinear interaction plays important and key role in the

localization of the particle, there is not the wave-corpuscle feature without the nonlinear interaction. This is a clear

explanation and elucidation for the effect of nonlinear interaction.



The real and extensive existences of the nonlinear interaction in the microscopic systems

However, the key of use of the nonlinear Schrödinger equation (21) or (22) to describe the features of microscopic

particles is whether the nonlinear interaction, 2

b , exist really and extensively in the physical systems ? Because

the nonlinear method and theory has not any physical meaning and values if the nonlinear interaction is not really

existent in practically physical systems although it is very perfect and beautiful. Therefore, the investigation of real

existence of the nonlinear interaction 2

b is very important. In practice, the nonlinear interaction is really and

extensively existent in different physical systems, which are produced by virtue of four mechanisms, which are

described simply as follows.

1) The self-interaction mechanism of produce of nonlinear interaction and its features.

This mechanism is occurred often in atom, molecules and the system of many particles. This mechanism was

proposed first by Fermi [71]. In order to describe the dynamical property of radiation due to the charge distribution in

an atom, Fermi [71] proposed his first nonlinear Schrödinger equation, in which the nonlinear interaction is due to the

moved electro in an atom, which accepted an interaction by variation of electromagnetic radiation- of the electron

itself. The nonlinear interaction is denoted by a reaction principle combined with the Schrödinger interpretation of the

wave function, which is denoted by )*(RRV , which is same with the above nonlinear interaction of 2

b , where

rdceVRR .)3/2( 3−= is a radiation-reaction potential of a dipole generated by the Schrödinger 's charge distribution

)()(*)( rrer = .

Davydov and Pestryakov [72] studied the nonlinear localization phenomenon of a complex scalar field of spin less

quasi-particles with inertia less self-interaction. The Hamiltonian density of the scalar field with self-interaction in

one-dimensional infinite space is given by

(61)

where H’ =H is the Hamiltonian density, its nonlinear interaction is still 2

b , b is a nonlinear parameter

independent of the velocity, the “-” and “+” signs correspond to attraction and repulsion self-interactions,

respectively. Such a self-interaction can be realized by local interaction between the microscopic particles, which

correspond to the field, or by local interaction of these quasi particles with the field of inertia less displacements of the

other particles, which are not considered explicitly here. If the self-interaction in the systems is absent, i.e., b=0 , the

),( tx field describes the states of non-interacting quasi particles with mass m. Their motion is described by a plane

wave with wave vector k and energy mk 2/22 . If 0b , the corresponding equation of motion is

224

'2

H bm x

=



(62)

This is a standard nonlinear Schrödinger equation and the same with Eq.(21), then it has a soliton solution as given by

(30). Therefore, electrons in these organic conductors are self-localized due to the self-interaction. In other words,

equation (62) provides stationary solutions in the form of modulated plane waves with an inhomogeneous spatial

distribution of the density of the microscopic particle with a small constant velocity.

(2).he self-focusing mechanism of nonlinear interaction.

This mechanism is existent in some fiber systems, such as optic fiber, and some polymers. We now elucidate the self-

focusing mechanism of nonlinear interaction. As it is known as mentioned above, the nonlinear interaction in this

mechanism is due to the anomalous dispersion effect or a nonlinear effect of the systems or materials. We now

establish a corresponding dynamic equation of microscopic particles in the system using this mechanism. It is well

known that a linear wave equation in linear quantum mechanics is often denoted by a linear operator

L which

consists of t and , i.e.,

(63)

Its solution can be approximately denoted by a monochromatic wave }].(exp[ txki −= , corresponding

dispersion relation of frequency and wave vecto k can be represented as

or (64)

If comparing Eq.(62) with Eq.(63) we find that and k correspond to 'x

i− and 'ti− , respectively [47,51,54].

Thus, Eq.(63) becomes

(65)

where 'x

is the gradient with respect to 'x and )('x

i− is the pseudo-differential operator obtained by replacing

k with 'x

i− in )(k .

In a weak nonlinear medium, which responds adiabatically (or instantaneously) to a wave of finite amplitude, the

nonlinearity is expected to affect the dispersion relation of the carrier wave (in addition to the generation harmonics of

smaller amplitudes), thus, the frequency of the microscopic particle relates to its intensity.

2 22

22 0

2i b

t m x

+ =

'( , ) 0.tL =

( , ) 0,L i ik− = ( ).k =

( ' ')

' '( ) 0i k x t

t xi i e − − − =



Sulem et al.[73,51]replaced the frequency )(k by a function ),(22 k with )()0,( kk = due to the

nonlinear feature of the medium. This effect is just called a self-focusing effect. In this case a complex wave

amplitude is no longer a constant but is modulated in space and time, and thus becomes dependent on the slow

variables and 'tT = . In Eq. (65), the derivatives 't and 'x

should also be thus replaced with Tt + '

and + 'x

, where now denotes the gradient with respect to the slow spatial variable X. Then the equation (64)

becomes

.

It is natural to obtain the weakly nonlinear dispersion relation

where is small. If we expanded various quantities in this equation to powers of and keep the terms up to the

second order, the following equation is obtained [73, 51]

(66)

where kg = is the group velocity, the coupling coefficient b is related to the expansion coefficient of wave

function of microscopic particles (or intensity of wave) and is given by )(/2

evaluated at 02= and at the

carrier wave vector k . According to Sulem et al.’s method [73], equation (66) may serve as an initial value problem in

time. This equation is conveniently written in a reference frame moving at the group velocity by defining

gTX −= . Rescaling the time in the form , the following nonlinear Schrödinger equation can be obtained

(67)

where the spatial derivatives are now taken with respect to , its nonlinear interaction is still 2

b . This equation

is the same as Eq. (21) and the dynamic equation of photon in light. This affirmed that the photon soliton in the light

fiber is formed by the self-focusing mechanics by means of the Kerr effect, a nonlinear effect.

The above discussion illustrates that the motion of a microscopic particle is always described by the nonlinear

Schrödinger equation under this self-focusing mechanism, which results in the nonlinear interactions in the systems.

'X x=

22 ( ' ')

' '( , ) 0i k x t

t T xi i i i e − + − − − =

22( , ) 0.Ti k i + − − =

2( ) 0.T gi v b + + + =

T =

22 0i bd

+ + =



(3).The self-trapping mechanism of generation of nonlinear Interaction and its properties.

This mechanism occurs widely in condensed states, molecular crystal and bio macromolecules, such as protein

moles and is extensively studied by plenty of researchers, such as Landau, Pekar [74], Frohlich [75,76], and Holstein

[77] discussed Landan’s suggestion [78-79] in detail. Other examples of the self-trapping include electromagnetic

energy in a plasma and hydrodynamic energy in a water tank. Subsequently, Davydov and co-workers [80-84]. The

orm of the mechanism is described as follows.

For example, molecular crystal-acetanilide (ACN) and protein molecules are composed of the peptide groups (N-C=0)

or amino acid molecules in proteins to form a chain structure of …H-N-C=0…H-N-C=0…H-N-C=0…,.. Davydov

[64,80-84]used this mechanism to research the properties of nonlinear excitation (soliton), which is formed by virtue

of the self-trapping interaction of the excitons having 0.205eV with the displacement of peptides (phonons), and

transport of bio-energy of 0.43 eV released by hydrolysis reaction of adenosinc triphosphate (ATP) molecules along

the molecular chains. In this case Davydov [80-84] gave the Hamiltonian of the excitons with energy eV2005.00 =

by the following form:

(68)

where )( nn BB+is the exciton’s creation (annihilation) operator. However, he abandoned traditional average –field

approximate method used often in quantum mechanics , and considered and gave further the low-frequency

vibrational Hamiltonian of the peptide groups and the interaction Hamiltonian between the exciton and the vibration

of the peptide group by

, (69)

(70)

respectively, where M is the mass of the peptide group, w is the spring constant of the molecular chain, nP is the

conjugate moment of nn uJu = /, ,is the coupling constant. In such a case, Davydov rewrote the Hamiltonian of

the system, which is

(71)

and gave the wave function of collective excitatio states of exciton and phonon in the system by

( )0 1 1( )ex n n n n n n

n

H D B B J B B B B + + +

+ + = − − +

2

1

1( )

2 2

nph n n

n

PH w u u

M−

= + −

int 1 1( )n n n n

n

H u u B B +

+ −= −

int .ex phH H H H= + +



(72)

where ph

0 is the ground state of the phonon. Using the functional )()( tHt and the variational approach, in

the continuity approximation, Davydov et al.[80-84] got the dynamic equations of the exciton and phonon and their

associations, which are denoted by

(73)

and

(74)

Or

(75)

and

(76)

respectively, where, MwrvWJ /,2 000 =+−= , is the sound speed of the molecular chain.

Now let )]1(/[4, 22

0 swGvtxx −=−−= , JGJrmvvs 4/,2/,/ 2

0

2

0 === ,

.

Equation (75) is a nonlinear Schrödinger equation, which is the same Eq.(21). It has a soliton solution as given by [

[62-64,80-88]

(77)

( ) ( ) 0 exp ( ) ( ) 0n n n n n nex phn n

it t B t P t u +

= − −

2 2

2

( , )2 ( , ) 0,

2

x ti x t

t m x x

−+ − =

2 222 0

02 2

2( , ) ( , ) 0

rv x t x t

t x M x

− − =

2 22

20

2i G

t m x

− + + =

2

0

2

2 ( , )( , )

(1 )

r x tx t

x w s

= −

−

2 2

1

( ) ( ) ( ) , ( ) ( ) ,

1( ) ( ) .

2

nn n n n

nn n

n

t t u t t P t Mt

W M wt

−

= = =

= + −

0 02

0 0

( , ) sec ( ) exp ( )2 2

v Etx t h x x vt x x

r Jr

= − − − −



Thus, from Eqs.(76)-(77) we can give the solution of Eq. (74) as follows

(78)

Equations (74)-(77) show clearly that the exciton and phonon are localized to become the solitons due to the

nonlinear interaction 2

G , which is same with 2

b , and )1(/2 22

0 sMr − ,respectively. The soliton, which

is formed from the exciton ,has the properties of the energy of 2

02

1vmEE sol+= , binding energy of

JwEB

24 3/−= and the rest energy of JEJwJE ex 23/2 00

22

00 −=−−= and the effective mass of

.

This is just the description of self-trapping mechanism of nonlinear interaction of 2

b , which is extensively existed

and applied in condensed state physics, molecular physics polymers and biophysics. In this mechanism the motions of

microscopic particles and background fields or crystal lattice as well as their interactions are considered

simultaneously, this is a very perfect and complete model, which eliminated thoroughly the short and weak points of

quantum mechanics, in which the displacements or movements of the background fields, or crystal lattice are

neglected in general, the interaction of microscopic particles with he background fields, or crystal lattice are

approximated and replaced by an average field. Therefore the nonlinear model is a progress, development and

precision to researched problems.

(4). the self-localized mechanism of the nonlinear interaction and its dynamic equations.

This mechanism occurs often in systems having anisotropic structure, hydrogen-bonded systems, biological systems

and polymer, such as ice, solid alcohol, water, carbon hydrates, DNA and proteins as well as some many condensed

matters [89-93].

We here elucidated this mechanism of nonlinear interaction by the hydrogen-bonded system.

The system is composed of hydrogen ions and heavy ions or a heavy ion group through covalent bonds and hydrogen

bonds, such as in ice crystals they can be denoted in the chain of OH__H…. OH__H…. OH__H….OH__H .

Experiments show that there is considerable electrical conductivity, such as ice crystal exhibit considerable electric

conductivity; along the chains it is about 103-104 times larger than that in perpendicular direction [90-92]. The

invesitions show that this phenomenon is formed and carried out by proton (or hydrogen ion) transfer along the chain

systems [90-92]. This is a very interested problem and worth to study deeply.

2

002

0

( , ) tanh ( )(1 )

rx t x x vt

w s r

= − − −

−

2 2 4

2 2 2 3

0

4 (1 3 2 2).

3 (1 )sol

s sm m m

w Jv s

+ −= +

−



Obviously, the two kinds of arrangements of the type OH__H…. OH__H…. OH__H…. OH__H and the type

H__OH….H__OH….H__ OH….H__OH have the same energy in the normal state in this system. This means that the

potential energy of the proton should have the feature of nonlinear double-well with two minima corresponding to its

two equilibrium positions as shown in Fig.9. In the double-well potential, its barrier has the order of magnitude of the

binding energy of the covalent bond H__O, which is approximately 20 times larger than that in a hydrogen bond,

OH…H. Thus we may infer that when the system is perturbed by an externally applied field, the states and positions

of the protons in the double-wells are moved from one well to another by means of a translation and a jump or

migration in the inter bonds and intra bonds in an ionic defect, hydroxonium, H3O+, and another ionic defect,

hydroxyl, OH-. Their movements can be denoted. In practice, the proton transfer along the hydrogen-bonded chain

can be carried out by means of the migration of hydroxonium and hydroxyl ionic defects in the intra bonds. However,

when a proton approaches a molecule occupying a boundary of the chain, the transfer is not possible in the same

direction, but can be achieved with a re-orientation of OH groups by the Bjerrum (or bonded) defect, a pair of D and L

defects, which is induced by the rotations of OH groups. Thus the proton (H+) can be transferred along the hydrogen-

bonded chains of OH__H…. OH__H…. OH__H…. OH__H …by virtue of movement of the two kind of defects.

Therefore the proton transfer is carried out by a combination motion of the ionic and bonded defects in the hydrogen-

bonded system, in which the nonlinear double-well potential of the proton in 2

00 )]/(1[)( rRURU nn −= as shown in

Fig.7 plays an important role.

Fig.9.One-dimensional lattice model of double-well potential in the hydrogen-bonded systems.

Plenty of researchers [93-108] investigated the properties and rules of proton transfer in hydrogen-bonded systems

using different models. In Pang’s model the features of structure of the molecule and nonlinear property of the proton

transfer are sufficiently considered, a new model of proton transfer with two components, shown in Fig.9, and the

corresponding Hamiltonian of the systems are proposed. In the one-dimensional hydrogen-bonded chains, if

considering again the elastic interaction, which is caused by the covalent interaction and the coupled interaction

between protons and heavy ions, and the resonant or dipole-dipole interaction between neighboring protons and the

changes of relative positions of neighboring heavy ions, resulting from this interaction, as well as the harmonic model



with acoustic vibrations of low frequency for the heavy ionic sub lattice, then the Hamiltonian of the systems is

expressed by [101-108]

(79)

where the proton displacements and momentum are nR and nn RmP•

= , respectively, the first being the displacement

of the hydrogen atom from the middle of the bond between the nth and the (n+1)th heavy ions (oxygen atoms in ice),

0r is the distance between the central maximum and one of the minima of the double-well, 0U is the height of the

barrier of the double-well potential. Similarly, nu and nn uMp

•

= are the displacement of the heavy ion from its

equilibrium position and its conjugate momentum, respectively. nu= /2

01 and nu= /2

12 are coupling

constants between the protonic and heavy ionic sub lattice, which represent the changes of the energy of vibration of

the protons and of the dipole-dipole energy between the neighboring protons due to a unit extension of the heavy ionic

sub lattice, respectively. 2/1

2

1 +nn RRm shows the correlation interaction between neighboring protons caused by

the dipole-dipole interactions, and 0 and 1 are diagonal and non-diagonal elements of the dynamic matrix of the

proton, respectively. W is the elastic constant of the heavy ionic sub lattice. m and M are the masses of the proton and

heavy ion, respectively. 2/1

00 )/( MWuC = is the sound velocity in the heavy ionic sub lattice, and uo is the lattice

constant. The 0r , Rn and rn and 0U are shown in Fig.12. The part HP of H is the Hamiltonian of the protonic sub

lattice with an on-site double-well potential )( nRU , Hion being the Hamiltonian of the heavy ionic sub lattice with

low-frequency harmonic vibration, and Hint is the interaction Hamiltonian between the protonic and heavy ionic sub

lattices. This model is different from the ADZ model [113], and the Pnevmatikos etal. models [95-100].

Because the proton is quantized, thus the Hamiltonian of the systems should be quantized by using the standard

transformation [101-108]:

(80)

where )( nn aa + is the creation (annihilation) operator of the proton. Then Eq. (79) becomes

0

2( )1 11 21

1 1 1 12 2 2 2 2 2 2[ [1 ( ) ] ]

0 1 1 0int 2 2 2 2

1 2 ( ) ] ( )

12

[

(4.156)

RR R R

r

W m u u R Rn n n nn n n

nH H H H p m m U Pp n n nion n nn nm M

u u m u u Rn nn

+ −+ ++

= + + = + − + − + +

+ − + −−

0

2( )1 11 21

1 1 1 12 2 2 2 2 2 2[ [1 ( ) ] ]

0 1 1 0int 2 2 2 2

1 2 ( ) ] ( )

12

[

(4.156)

RR R R

r

W m u u R Rn n n nn n n

nH H H H p m m U Pp n n nion n nn nm M

u u m u u Rn nn

+ −+ ++

= + + = + − + − + +

+ − + −−

1/2 1 2

0 0(2 / ) ( ), ( / 2) ( )( ) ( 1)nn n n n n nR m a a p m R m i a a i •

− + += + = = − − = −

2

0 1 1 1 1 1[ ( 1 2) ( / 4)( )n n n n n i n n n n

n

H a a a a a a a a a a + + + + +

+ + + += + − + + + −



(81)

In this system the collective excitations of the proton and vibrations of the heavy ions, arising from the nonlinear

interactions, which is caused by the localized fluctuation of the protons and the deformation of structure of the heavy

ionic sub lattice, have a coherent feature, thus the wave function describing the collective excitations can be

represented by [101-108]

(82)

wherepr

0 and ph

0 are the ground states of the proton and the vibrational excitation of the heavy ionic sub lattice,

respectively.

)(tn ,

= nn PtP )( and = nn utu )( are three unknown functions, λ′ is a normalization

constant.

From |Φ> in Eq.(82) and the Hamiltonian (81), as well as the relations:

we can get the equation of motion for the heavy ion, it is

(83)

In the continuum approximation, Eq.(83)is represented by

(84)

If using the time-dependent Schrödinger equation )()( tHtt

i =

and in the continuum approximation we

can get

(85)

2 2 2 4 2

0 0 0 0 0 0( / )( ) ( / 4 )( )( )( )( )]n n n n n n n n n n n n n n n nU mr a a a a a a a a U m r a a a a a a a a + + + + + + + +

=+ + + + + + + +

2 2

1 1 0 1 1[ ( ) ] ( / 4 ) [( )( )]2 2

n n n n n n n n n n n n n

n n

M WP u u u u a a a a a a a a + + + +

− + −+ + − + − + + + +

2 0 1 1 1 1 1 0( / 2 ) [( )( )] / 4n n n n n n n n n n

n

u u a a a a a a a a U + + + +

+ + + + +− + + + +

| | | (1/ )(1 ( ) ) | 0 exp{ 1/ [ ( ) ( ) ]}| 0q n n pr n n n n ph

n n

t a i u t P t u + = = + −

( )| | | [ , ] | ,i

n n

u ti u i u H

t t

= =

| | ( ) | [ , ] | ,n n ni P i t P H

t t

= =

22 2

1 1 1 0 1 12

* * * *

2 0 1 1 1 1

( , ) ( 2 ) / 2 ( )

/ 2 ( )

n n n n n

n n n n n n n n

M u x t W u u ut

+ − + −

− − + +

= + − + −

+ + − −

2 22 2

0 1 2 02 2( , ) ( , ) ( ) | ( , ) |M u x t Wu u x t u x t

t x x

= + +

22 2

0 2

( , )( , ) ( , ) | ( , ) | ( , )

x ti x t Ju x t G x t x t

t x

= − −



where g’ is an integral constant,

(86)

(87)

The above results showed that the proton transfer in the hydrogen bonded systems are determined by the nonlinear

Schrödinger equation (85), which is the same Eq.(21), where the nonlinear interaction ),(),(2

txtxG is same with

2

b , which is generated by the double-well potential, or nonlinear interaction G due to the non uniform structure

feature of hydrogen bonded systems. This is the self-localized mechanism for the nonlinear interaction 2

b .

Conclusion

In this paper we first revealed and exhibited systematically and completely the difficulties and contradictions of

quantum mechanics by using its dynamic equations. In order to elucidate clearly the features and essences of quantum

mechanics we investigate and give systematically the solutions of the Schrödinger equation, which is basic dynamic

equation of microscopic particles. We found through the comparisons completely between the experimental and

microscopic results for the features of different particles that the solutions of the Schrödinger equation have only a

wave feature, have not the wave –corpuscle duality. At the same time, they are all different with the properties of the

microscopic particles with different natures obtained by different experiments. From these results we can affirm and

conform that quantum mechanics and the Schrödinger equation are not a correct theory and dynamic equation, which

are used to describe the properties of microscopic particles. This means that this theory must be transformed and

varied, or speaking, they should be developed forward. However, how can be they developed? We verified that if a


b is added in the Schrödinger equation, in which the nonlinear interaction restrains and

suppress completely the dispersion effect of kinetic energy term in the Schrödinger equation, or speaking, to use the

nonlinear Schrödinger equation to substitute into the original Schrödinger equation in quantum mechanics, then these

difficulties and contradictions of original theory can be thoroughly, the microscopic particles can be automatically

localized and have completely a wave-corpuscle duality. Therefore, these investigations make us find the direction

and methods to eliminate thoroughly the difficulties and contradictions of quantum mechanics, i.e., to add the


b into the Schrödinger equation to establish the nonlinear Schrödinger equation or

nonlinear quantum mechanics. We can verified that the nonlinear interactions of 2

b are widely existed in all

physical systems.

2

0, 0 0 0 0 1 2 0 0/ 3 / 2 ( ) / 4 / ( ) /os v C G t U U mr g u = = − + − + +

,,4/),2/(32/ 000

2

1

2

0

4

0

2

0

2

0

2

1 ==+− JrmU

2 2 4 2 2 2 2 2 2 2

0 0 0 0 1 2 0 0{3 / 2 [ ( ) / ( (1 ))]},4

GG U m r u MC s

J = − + − =



Finally we can concluded from these investigations that (1). The properties of microscopic particles should be

described using the nonlinear quantum theory containing the Schrödinger equation with nonlinear interaction 2

b ,

in which the nonlinear Schrödinger equation is only correct dynamic equation of microscopic particles. (2) To use the

nonlinear theory of quantum mechan can eliminate thoroughly the difficulties of contradictions of quantum

mechanics. (3) The nonlinear quantum mechanics is a correct and only the direction and method of development of

quantum mechanics.



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