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
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.
C
opyright © 2012 SciRes. JQIS

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
B
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
2.
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.
REFERENCES
[1] A. Einstein, B. Podolsk and N. Rosen, “Can Quantum-
Mechanical Description of Physical Reality Be Consid-
ered Complete?” Physical Review, Vol. 47, No. 10, 1935,
pp. 777-780. doi:10.1103/PhysRev.47.777
[2] E. Schrödinger, “Die Gegenwärtige Situation in der
Quantenmechanik,” Naturwissenschaften, Vol. 23, 1935,
pp. 807-812.
[3] A. Aspect, J. Dalibard & G. Roger, “Experimental Test of
Bell’s Inequalities Using Time-Varying Analyzers,” Phy-
sical Review Letters, Vol. 49, No. 25, 1982, pp. 1804-
1807. doi:10.1103/PhysRevLett.49.1804
[4] J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa
and Y. Hasegawa, “Experimental Demonstration of a
Universally Valid Error—Disturbance Uncertainty Rela-
tion in Spin Measurements,” Nature Physics, Vol. 8, 2012,
pp. 185-189.
[5] Heisenberg, “Uncertainty Principle.”
http://en.wikipedia.org/wiki/Uncertainty_principle
[6] J. Conway and S.Kochen, “The Free Will Theorem,”

H. YAMADA 118
Foundations of Physics, Vol. 36, No. 10, 2006, pp. 1441-
1473. doi:10.1007/s10701-006-9068-6
[7] M. F. Crommie, C. P. Lutz and D. M. Eigler, “Confine-
ment of Electrons to Quantum Corrals on a Metal Sur-
face,” Science, Vol. 262, No. 5131, 1993, pp. 218-220.
[8] Tonomura, “Electron Waves Unveil the Microcosmos,”
Friday Evening Discourse at Royal Institution, 1994.
[9] Pauli Principle.
http://en.wikipedia.org/wiki/Pauli_exclusion_principle
[10] H. Yamada, “Quantum Theory Named Dialog Principle,”
Soryusiron, Vol. 94, No. 2, 1996, pp. 53-68.
[11] H. Yamada, “Quantum Mechanical Approach to the
Meaning of Existence, Will and Life,” AIP Conference
Proceedings, Vol. 716, 2004, pp. 109-115.
[12] H. Yamada, “Ryousi Rikigakuga Akirakanisuru Sonzai,
Ishi, Seimei Noimi,” Koushiken Publish, Kusatsu, 2011,
pp. 1,172.
[13] C. Jönsson, “Electron Diffraction at Multiple Slits,”
American Journal of Physics, Vol. 4, No. 1, 1974, pp. 4-
11. doi:10.1119/1.1987592