Some Comments on the Real Meaning of Schrodinger’s Equation Revealed by the Fact that Electron Is Always a Particle

doi:10.4236/jqis.2012.24017

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Journal of Quantum Information Science, 2012, 2, 112-118

http://dx.doi.org/10.4236/jqis.2012.24017 Published Online December 2012 (http://www.SciRP.org/journal/jqis)

Some Comments on the Real Meaning of Schrödinger’s

Equation Revealed by the Fact that Electron

Is Always a Particle

Hironari Yamada

Faculty of Science and Engineering, Ritsumeikan University, Kusatsu, Japan

Email: [email protected]

Received October 7, 2012; revised November 8, 2012; accepted November 16, 2012

ABSTRACT

We all physicist have long been believed that an elementary particle is a wave as well as a particle, but we discuss in

this paper that an electron (probably all fermions) is always a particle. Author claim that quantum mechanics (QM) is

not such mysterious as Bohr stated that the wave turn to the particle by observation. We can understand QM by natural

human sense. The wave nature of electrons is only an appearance or a phenomena but not intrinsic or substantial. An

electron is an individual body, which interferes with other individual electrons. Interference is the key word instead of

the wave to understand the quantum mechanics. Interference produces the wave nature and the uncertainty. When we

determine that an electron is nothing but a particle, we will see the true meaning of wave function and the Schrödinger’s

equation.

Keywords: Quantum Mechanics; Interference; Uncertainty Principle; Pauli Exclusive Principle; Schrödinger’s Equation

1. Introduction

There is no doubt on the wave nature of elementary par-

ticles, since Schrödinger’s equation was so powerful to

reveal the nature in the microscopic world last 90 years.

As Nils Bohr, however, stated that “an electron immedi-

ately shrinks to a particle from a wave by observing, it

created great confusion to the meaning of existence. The

observing is an action by human. If we read this state-

ment correctly we human can change the nature of ele-

mentary particle. If the nature is changed by observing,

we lose our foundation of physics. Einstein and Schrö-

dinger wondered this Bohr’s statement and offered an-

other opportunity of thinking in EPR [1] and Schrö-

dinger’s cat [2]. If we follow the experimental results on

EPR it turned out that Einstein is wrong [3], but this

doesn’t automatically mean that Bohr is correct.

Recent empirical result by Hasegawa [4] raised the

question that the Heisenberg’s uncertainty principle [5] is

not applicable to neutron or electron propagating free

space. The momentum and the position are measured

precisely and simultaneously. We think that it is the time

to think over the foundation of quantum mechanics. We

discuss in this paper the meaning of uncertainty, the

meaning of wave, the meaning of observation and so on,

and conclude that electron has a kind of will as Conway

proposed the word free will [6].

Author claims in this paper that Fermions can be al-

ways a particle and that the wave nature appears only

when electron is captured in a potential. Wave nature

appears by the interference between elementary particles.

2. No Explicit Evidences Indicating the

Electron Wave

2.1. Experiment by Don Eigler

If we look at carefully the past experimental results we

see there are no evidences of electron being a wave. You

might think that there is a beautiful demonstration of

electron wave as show in Figure 1. This picture is ob-

tained by Don Eigler [7] which is measured by an atomic

force microscope (AFM). The 48 As atoms are circularly

deposited on a Cu coated wafer like a coral reef. The

wave structure appears inside the coral reef which is

similar to water’s standing wave in a circular pan. Author,

however, thinks that this is not the evidence of one elec-

tron wave. AFM measures the density distribution of

electrons. This structure is not made of one electron, but

is made of many electrons. Yes wave nature appears as a

statistics. This wave nature is very similar to that of wa-

ter pun. We know that the water’s ripple is a nature of

water surface, but nobody feels so that water or H2O

molecule is a wave.

H. YAMADA 113

Figure 1. The 48 As atoms are circularly deposited on a Cu

coated on a wafer like a coral reef, named quantum corral.

The wave structure appears inside the coral reef which is

similar to water’s standing wave in a circular pan. But au-

thor thinks that this is not the evidence of one electron

wave.

2.2. Experiment by Tonomura

Another example, Figure 2(a), is demonstrated by Tono-

mura [8]. The experimental setup of this measurement is

illustrated in Figure 2(b). Electrons are emitted one by

one from the top of thin needle shaped cathode. Electron

propagates through two separated passes. Then finally

each electron stains a spot on a two dimension digital

camera. Here each electron appears as a spot (see in the

right low) which indicates a particle. When the number

of electrons is increased interference pattern appears as a

whole nature. Here again, the interference pattern ap-

pears by the statistical nature. We don’t see again the

evidence of each electron being a wave. Tonomura states

that each time only one electron is ejected, so this elec-

tron must pass both slits simultaneously. Thus electron is

a wave, and turned to the particle when the observation is

made as Bohr stated. People think that we can’t under-

stand the quantum mechanical phenomena by our normal

human sense. Author will give a correct interpretation

soon.

3. Pauli Exclusion Principle Is an Evidence

of Fermion Being an Individual Particle

Pauli exclusion principle [9] is a strict evidence of Fer-

mions’ particle nature but not wave nature. All elemen-

tary particles are classified into either Fermion or Boson

depending on the symmetry of the wave function and

spin. Fermion has a half integer spin, thus the wave func-

tion is asymmetry. Boson has an integer spin, and the

wave function is symmetry. It is well known that Fer-

mion follows the Pauli exclusion principle but not for

Boson. More than two Fermions can’t stay in the exact

(a)

2D detector

electron

biplism

electron gun

monitor

(b)

Figure 2. Tonomura’s experiment. (a) Each electron makes

a stain one by one on a 2D detector. When the number is

increased interference pattern appears; (b) These stain is

observed when electron pass through the double slit or by-

plism.

same phase space and level, but Boson can. Pauli prince-

ple is here interpreted in the framework of wave function,

but other interpretation is possible.

Fermion can be a particle in the sense of an individual

[10,11]. Since it is an individual thus is exclusive. Every

individual’s existence is exclusive. We understand natu-

rally that more than two particles can’t stay in the exact

same phase space, because they are individuals. We can

say that every individual for instance human, cell, DNA,

molecule, atom etc. follows the Pauli exclusion principle.

Let’s consider the scattering problem. This is experi-

mental evidence that two electrons scatter each other in

the head on collision and the scattering cross section is

finite. Because electron is Fermion and two particles

H. YAMADA

114

can’t stay in the same phase space. On the other hand

there is no report on the finite cross section of two pho-

tons head on collision, because it is Boson and can stay

in the same phase space according to the Pauli’s exclu-

sion principle. Photon is truly a wave as we know as a

radio wave. We should know that the meaning of the

wave nature is different for Fermion and Boson.

We know that we can’t form laser by electron, because

it is Fermion, but we know that the ensemble of Fermion

can form a Boson and leads to a laser. We shouldn’t,

however, call this ensemble the wave. We are not sure

whether every such ensemble stays in the exact same

phase space or not. We understand that it is a train of

ensembles aligned in an equal spacing. Hereby we un-

derstood the real meaning of Pauli exclusion principle

that Fermion is always an individual particle and Boson

is always a wave. We know that the particle nature of the

photon can be interpreted as a wave packet in the wave

dynamics. Thus it is not necessary to think that Boson is

a particle. We can understand the particle and wave in a

classical concept, since it is always a particle or wave.

4. Uncertainty

4.1. Heisenberg’s Thought Experiment

Demonstrates that Electron Is a Particle

Uncertainty principle is based on the wave nature of par-

ticle, and if uncertainty is broken, this must be another

evidence of author’s claim that Fermion is always a par-

ticle.

The Heisenberg’s thought experiment that is a meas-

urement of electron momentum and position by optical

microscope is actually the case of the famous Compton

scattering. An electron and a photon can be treated as a

particle, and we calculate the momentum and energy of

both scattered photon and electron by kinematics. Hei-

senberg treats the photon as a classical wave, and intro-

duces the diffraction limit. When the microscope is used,

the optical aperture has to be certainly wide to avoid the

occurrence of diffraction, which introduces uncertainty

of momentum. If we, however, use an ideally small scin-

tillation detector instead of the microscope, no interfere-

ence should appear. If we use a cloud chamber under

magnetic field, we can measure exactly the electron posi-

tion and momentum simultaneously. In the high energy

physics we always measure accurately the momentum

and energy of elementally particles. Do you think this is

because the energy is high, and the wavelength is short?

No we can always measure the momentum and position

accurately as Hasegawa group measured in the low en-

ergy neutron case. In the Heisenberg thought experiment

the uncertainty never appears. This indicates that electron

is a particle. If so question arises. When and how the

uncertainty principle appears? This must be an important

question to understand the foundation of quantum me-

chanics.

4.2. Uncertainty Appears when Elementary

Particles Are Captured

Author thinks that uncertainty appears only when Fer-

mions are captured in a potential [12]. The captured elec-

trons shows particular distributions in a closed system

like atom, which is solved by the Schrödinger’s equation.

Neutron or proton behaviors in a nucleus are also the

problem of the closed system. Electrons or neutrons stay

in a particular state and level in the closed system. This is

the observed phenomena. These levels have a certain

band width, which determines the uncertainty. The life-

time of the level Δτ and the band width Δε follows the

uncertainty principle, 2πh







  as well known.

Uncertainty remains in nucleus and atoms. These cases

are indeed the intrinsic quantum mechanical uncertainty.

Nobody can predict when radio isotope decays, or when

photon will be emitted from an atom. This is the real

meaning of uncertainty.

Then we answer to the double slit problem [13].

Tonomura believes that electron passed two slits simul-

taneously, thus it must be a wave. It is clear for the elec-

tro-magnetic wave in the double slits problem. We can

say that wave passed the both slits simultaneously. If we

express this phenomenon by the word photon, we can say

that photons are localized at the detector after passed two

slits in a particular density distribution. This we call the

wave nature. Not only in the double slits but also in a

cavity localization of photon appears. Photon shows the

uncertainty and diffraction leading to the mode with par-

ticular distribution statistically.

How about the case for electron? We think that the

double slits problem is also the case of captured particle

regardless of electron, neutron or photon. See Figure 3.

There, an electron gun, or photon or neutron sources,

double slits or mirrors, and an imaging device, compose

the closed system or cavity. When one of the slit is

closed the interference pattern on the imaging device

disappears. We understand that when one of the slit is

closed the system turns to the open cavity, and then the

interference pattern disappears. As far as electron or neu-

tron stays in the region A or B of cavity with both slit

open, we will see the interference pattern, but if one of

the slit is closed the interference pattern disappears, even

if it is closed when particle is in the region B. Seems

timing has no important role. The important thing is that

particles are captured or not. This will be the key in this

problem. When particles are captured in a cavity as well

as in an atom or a nucleus, mode appears and localizes

by the interference.

If electron is an individual particle as described in the

H. YAMADA

115

or photon source

Slit or mirror 2

A Imaging device

Slit or mirror 1

Electron gun

Figure 3. Double slit problem is equivalent to the closed system or cavity. Electron gun, slit or mirror 1 and 2, and imaging

device form a cavity. When particles are captured in the closed system, interference appears. No matter when one of the slit is

closed, it makes the open cavity and the interference disappear. This is same for macroscopic as well as microscopic prob-

lems.

expect the others position to be there. Nobody knows

what the next motion of human will be exactly. It is the

same for electrons. Electrons as well as humans behave

as they have a sort of “will” [6,10-12]. Electron has a

freedom to decide his position under constrains such as

the energy conservation low, the momentum conserva-

tion low, and the principle of least action. This is the

same for human body. Human has to follow the same

regulation which electron follows. In addition human has

to follow regulations of societies.

Section 3, people think that electron cannot split into two,

but author say that electrons are the individual body

which interfere with other electrons. Many other elec-

trons are there in the slits, mirrors, imaging device and

electron gun composing the double slit problem. Let’s

think that one electron is emitted from the gun and trav-

els in the “field” created by many other electrons in this

cavity. This “field” is changed by closing or opening the

slits. This field might not be an electro-magnetic field,

but might be a “Schrödinger’s field” or “quantum field”

which has a structure of Hilbert space. The traveling

electron feels the change in the “Schrödinger field” and

decides where to go on the imaging device. This is the

possible interpretation when electron is the individual

body and not wave. Wave nature is represented by the

“Schrödinger’s field”. Schrödinger’s equation descries

the Schrödinger’s field, but not the electron wave. Indeed

the wave function actually defines the state or configure-

tion of electrons. We should call the state function in-

stead of the wave function. We discuss more about the

meaning of the Schrödinger’s equation in Section 5.

The above described is a common nature in both mi-

croscopic as well as macroscopic world. Humans and

electrons follow the similar regulation, and both are the

individual which interfere each other. It is for sure that

we don’t call that human is a wave, and electron is as

well. Why every individual interferes? This is the impor-

tant question to be answered, instead of the discussion

which slit has electron passed.

Interference appears when electron or neutron inter-

feres with other electrons or neutrons composing the cav-

ity. No matter how the cavity is small like atom or large

like the double slit problem. Interference is a nonlocal

phenomenon. We know the wave function how it spread.

If only one electron exist in the universe, this wave func-

tion spread all over the universe.

Hereby we understand the existence of one electron is

defined by all other electrons surrounding and forming

the circumference of this electron. This is the real mean-

ing of existence. This electron behaves differently in the

different circumference changed by opening or closing

the slits. The path of this electron is influenced by all

other electrons, and finally decides the location in the

imaging device to form the well known interference pat-

tern.

In the word interference we expect that the meaning

“recognize”, “communicate”, and “influence” are in-

cluded, and is a nonlocal phenomena. We may apply

such meanings to the word interference in quantum me-

chanics [10-12]. As a natural consequence electron must

recognize the number of slits before traveling into the slit,

even recognize the closed slit afterword in the region B,

because the “state function” or “Schrodinger field” spread

all over in the cavity. Then in the cavity particular den-

sity distribution appears, and makes the interference pat-

tern on the imaging device.

Let’s say Fermion is an individual particle which in-

terfere each other. “Interfere” means the quantum me-

chanical interference described in the wave function, and

is different from interaction. We know the word inter-

ference, which is used for the relation between humans.

Electron has a nature which interfere each other like hu-

mans. How interfere is different for electron and human,

but the concept of interference is same. Human changes

its position and momentum due to the appearance of

other individuals. It is the same for electron. Nobody can

4.3. Another Thought Experiment

Let’s think another though experiment which represent

another digit evidence of “Fermion individual particle

H. YAMADA

116

theory”.

In the double slit problem of Figure 3, if two electrons

are injected to this cavity simultaneously what will hap-

pen. Author thinks that the interference pattern will dis-

appear. If electrons are truly wave, each electron path

through independently the both slits simultaneously and

creates the interference pattern as indicated in the Figure

When we however follow the Fermion individual par-

ticle theory, the case more than two particles are injected

simultaneously, one particle to the slit 1, and another

particle to the slit 2, we shouldn’t see the interference

patter. Each particle recognizes each other, the “quantum

field” generated by many other particles composing the

cavity will be distorted by another particle. This “quan-

tum field” is shared by two or many particles like in an

atom, and the space is shared. In each hemisphere only

one slit is there and the interference pattern will be dif-

ferent from the case that the double slits are recognized.

If photons are used the result is obvious that the interfer-

ence pattern is same for both cases, two photon and one

photon cases since each photon occupy the whole cavity

quantum filed as they are a Boson and truely a wave. We

can prove in this way that Fermions are the individual

body and exclusive to others, which is the most impor-

tant features of the particle nature. This experiment

proves that Fermions propagate as a particle and detected

as a particle. We will see that the wave like nature is dif-

ferent for Fermions and Bosons.

5. What Is the Meaning of the Schrödinger’s

Equation?

What is the meaning of the Schrödinger’s equation if

electrons are particles and individual bodies, which in-

terferes each other? Author thinks that the wave function

represents the Schrödinger or quantum field where the

interference among individuals appears [12]. Wave func-

tion represents the state of particles. We can call it the

state function. Schrödinger’s equation can be named as

interference equation.

We can interpret the meaning of ΨΨ*. The complex

conjugate of Ψ, Ψ* is the field generated by other indi-

viduals which are recognized as others under considera-

tion. The Ψ* is a time reversal but not a signal from fu-

ture. It is a sum of the reflections from all other particles.

The ΨΨ* express the density distribution in the steady

state after the communication is finished between indi-

viduals engaged in this problem. By the communication

individuals changes their position and shows particular

distribution, which is observed for instance as shown in

Figures 1 and 2. We understand here that the existence

of one electron is defined by other electrons surrounding.

This is the meaning of existence in the microscopic

world. This definition of existence is also applicable to

macroscopic world and human body. My existence is

determined by you and other human. This philosophy is

also applicable to the microscopic world.

Hereby the meaning of the electron existence is de-

fined quantum mechanically. Consequently our human

existence is also defined by the quantum mechanics. The

existence must be an individual body which interfere

each other. This definition of existence is different from

either that of the philosophy of materialism or that of

idealism. Materialism and idealism are combined in the

definition of quantum mechanical existence. We may call

this new philosophy the quantum mechanical material-

ism.

We can now understand the QM by natural human

sense or common philosophy. By this philosophy we

learn more about the Schrödinger’s equation. It is appli-

cable to economy and sociology too. We treat human as

individual which interfere each other as same as elec-

trons. DNA, protein, chromosome, bacteria, cell, fungi,

insect, plant, animal for instance are also same individu-

als. We may treat planet, galaxy as the body interfere

each other by gravity, although the interference occurs in

extremely slow manner as we human can’t recognize.

This is our world. Interference is the key word. We

should introduce “interference” in the foundation of

physics, and then we can expand the application field of

Schrödinger’s equation to dynamics of protein, sociology

and economy. Every individual interfere each other and

then we can describe the relationship by the framework

of Schrödinger’s equation. We can introduce the “poten-

tial” in the field of economy and sociology which is fol-

lowed by statistics and probability of money flow or hu-

man flow. We know that the potential is made of massive

humans as well as massive electrons.

Now the question is the relation between “will” and

“interference”. The free will theory is introduced in the

reference 6. In this article Conway and Kochen didn’t

discuss the meaning of will, didn’t assume that only an

individual particle has a free will, and discussed that all

elementary particle has a free will. But to introduce the

word will, we need the concept of individual which asso-

ciates the will.

We could approach to the meaning of “will” in this

paper. We might say that the individuals which interferes

each other, recognize others, and influence others have a

will. (Yamada proposed the word will in Japanese, 意志

(pronounce Ishi) in 1996 to explain the meaning of

quantum mechanics [10], and published in English in

2004 [11].) We can use the word “will” in a limited defi-

nition. The will is the substance, which keep the individ-

ual body integrated or united, which recognizes other

individuals, and which influence others and changes oth-

ers state. Here each individual has each will, cooperate,

H. YAMADA 117

and make the world balanced. This is the same for every

living specimens as well as electrons. Let’s introduce the

concepts individual, interference, quantum field and will

in the foundation of physics. In Japanese there is a word

Ba, which is translated to field but the meaning is more

like relationship. The Ba will be a suitable word for the

quantum field or Schrödinger’s field.

We could have a new and natural insight into the

Schrödinger’s equation. The Schrödinger’s equation de-

fines the quantum field or say Schrodinger’s field in

which particles interfere and communicate each other.

We don’t need to assume that Fermions itself are wave.

Schrödinger’s equation solves the relationship between

electrons. Author calls Schrödinger’s equation the dia-

logue or interference equations. If so we should be able

to apply the Schrödinger’s equation to not only for ele-

mentary particles but also to human society and any liv-

ing individuals interfering each other.

6. Concluding Remarks

Real meaning of Schrödinger’s equation is revealed by

the fact that electron is always a particle.

In this paper author discussed that Fermion is always a

particle and Boson is always a wave by reconsidering

past experimental evidences and Pauli Exclusion Princi-

ple. We discussed that more than two individual Fer-

mions as well as humans can’t stay in the exact same

phase space. Herein the meaning of the particle is an in-

dividual body. Individuals can’t merge each other but

photons can merge each other since it is a wave or field.

Experimental evidence in which Fermions scatter each

other but photons never scatter each other is very impor-

tant evidence that Fermions are individuals such as hu-

man bodies.

In the double slits problem, author could explain how

individual particle make interference pattern on the im-

aging device without assuming that this particle passed

two slit simultaneously. Interference appears when this

particle interferes with all other particles composing the

gun, slits, and detector. Interference appears when the

particles are captured in a cavity or potential and inter-

feres with particle composing the cavity or potential.

Particle travels the “quantum field” composed by many

particles around one. “Quantum field” is different for the

two slits open and one slit open cases then this particle

recognize the difference of the two quantum fields and

behave differently.

Interference is the key word. Interference is the word

to describe our living nature as well as the quantum me-

chanical phenomena. Interference is a word which de-

fines the relation between living individuals. We may

include the Fermion in the category of such individuals.

Thus without other particles appearance interference

never appears. The particle traveling in a free space

never shows uncertainty as concluded by Hasegawa and

Ozawa.

We can say that the uncertainty appears by the “will”

of electron. As we understood QM in the natural com-

monsense we are able to use the QM more freely. QM

defines how interference appears among individuals.

Then we learn how to apply and what sort of problems

are in a category to be solved by QM.

The meaning of Schrödinger’s equation is now clari-

fied. The wave function deals the state of particles, but

not the Fermion wave. It can be called as the state func-

tion that represents the structure of “quantum field” or

Schrödinger field. The ΨΨ* defines the existence. The Ψ*

is the message from all other particles surrounding the

particle in the problem. One’s existence is defined by all

other individuals. Consequently we learn that the mean-

ing of existence is same for electrons as well as human.

This is one of the important conclusion drown by this

paper that QM deal with the real existence or real mate-

rial which you can touch. QM is the tool to study not

only the elementary particles but also human body or

human society and living specimens.

Author believe that the problems raised by Bohr,

Schrödinger and Einstein are solved by the Fermion in-

dividual particle theory. Fermions are always particle and

no need to assume that the wave turned to the particle by

observation. No need to think that the shrink happened in

the human brain. Observation is an action taken by hu-

man body, thus human interfere the observation. Quan-

tum mechanics define the existence. No need to worry

the existence of moon without seeing. Moon existence is

defined by many other individuals including electrons as

well as human.

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