** **

** **

**COMPLETED QUANTUM
MECHANICAL THEORY (CQMT)**

Nimit Theeraleekul B. Eng. (Communication)

(April 17, 2008, 1st revision Oct. 1, 2011)

**Abstract:**
Quantum mechanics is one of the most important theories
in modern physics, despite of their** **predictability** **and** **accuracy; it is facing with crucial
philosophic problems. Indeed the problems could be solved, and the theory could be improved
by using **“Vacuum Mechanics” **i.e. the mechanism of vacuum medium concept!

** ****Content:**** **1) Background.
2) Introduction.
3) Philosophy behind natural wave.
3.1) Crisis of wave in modern physics. 3.2) Wave-particle duality. 4) **Mechanism of particle
wave.**
5) Particle wave vs. light wave. 6)** **Mathematical formula for particle wave. 6.1) Original
derivation of de Broglie wave**. **6.2) Improved de Broglie wave**. **7)** **Better view of
Schrödinger equation. 8) Quantum vacuum mechanics. 8.1) **Wave collapse
interpretation. **8.2)
Mechanism of the uncertainty principle. 9) Particle wave or matter wave?** **10) Classical
mechanics vs.
Quantum mechanics. 11)** **Physical
meaning of Plank constant. 12)** **Is energy continuous or discrete?** **13) Is quantum
entanglement right or wrong? 13.1)** **Einstein-Podolsky-Rosen paradox. 13.2) Bell
inequality is a mathematical tool.** **13.3) Einstein hidden variable vs. quantum
mechanics. 14) Problem with hydrogen atom model. 15) Conclusion & Discussion.** **16) References.** **

** 1)
Background.** This article is the fourth expanding part of the original article**
“Vacuum mechanics a New Approach to the Theory of Everything” **(VMTE) [1].
There are four most important theories involved in VMTE; Maxwell
electromagnetic field theory (EMT), Einstein special theory of relativity
(STR), Einstein general theory of relativity (GTR) and Quantum mechanical
theory (QMT). Because what has been talked in VMTE is rather a philosophical
one, so the first three articles have been expanded into a more scientific
detail with the titles **“Completed Maxwell electromagnetic field theory” **(CEMT),
**“Completed Einstein special theory of relativity” **(CSTR) and “**Completed
Einstein general theory of relativity”** (CGTR) respectively [1].

**In VMTE**
we have started with the fundamental hypothesis that “**Vacuum medium is the
fabric structure of vacuum space****”**, and following
is with the consequence hypothesis “**Electrons and proton are **(something like)** tiny
black holes **of vacuum medium”. We have proved the existence
of vacuum medium space instead of the conventional empty vacuum space, also
showing how “condensed”
vacuum medium were formed to be electron and proton, which are the** origin of
all the material matters existed in the universe!**

Also we had
shown that **“vacuum mechanics”** (i.e.** **the mechanism of vacuum
medium) which
is the action of vacuum medium or of condensed vacuum medium and the
interaction between them, is responsible for** all natural phenomena.** Then
we used it for explaining the unsolved philosophic problems in EMT, STR and GTR
and here** **we will show how to use it to solve the **unsolved philosophic
problems in QMT either!**

** **

** 2)
Introduction.** Let
us start with Nobel Prize Laureate (in quantum electrodynamics) Richard. P.
Feynman word **“**I think I can safely say** nobody understands quantum
mechanics”** [2], but **why?** Is it because it was prohibited by God or
just due to our ignorance? Here we will find out together!

Another
point of view comes from David Bohm, a famous physicist in quantum theory, who
said that **“**Quantum mechanics should be called as quantum** non**mechanics”
[3]. This is because even through the theory was called “quantum mechanics”
which mean that it should be something as **“mechanism” **of
quantum system, but it seems that there is no such thing which could regard as
system of separate parts working together according to **causal
laws!**

Could this means that **there is
no need any mechanism **for quantum mechanics? Of course, it should
not be that, otherwise it is a** magic, **not
science!

Then it
is not surprise that Einstein said that “Quantum mechanics** is not a
complete theory **even though it is correct”** **[4], and
even “What Einstein had in mind by a ‘completion’ of quantum mechanics is not
entirely clear, …” [4’]! Anyway we know that a complete theory, by definition,
should include with a philosophical idea which explain **how the
theory work! **

And we
also know that “**Understanding** is the progression of science” then it will
give us a more confidence that we will not **misinterpret** the
theory and lead to **something crazy!** Quantum mechanics **provide us
only** mathematic formulas which enable us to calculate a correct result
without any deeper understanding, so this is the reason why it is so mysterious
that no one could understand it.

Now to**
improve QMT, **we will first point out detail of the main problems that
involved. Then we will use the new propose concept of** vacuum mechanics **(i.e.**
**the mechanism of vacuum medium) to solve the problems.** **Finally
what we will get is **“Completed quantum mechanical theory” **(CQTM)**! **

** 3) ****Philosophy behind
natural wave.**** **Let’s start with the most familiar natural wave that is **water
wave**. When a stone was dropped in a pond, it will disturb the water and
creating the circular surface waves and propagate away. Here it is easy to see
the **mechanism of water wave**. First the energy of the falling
stone was used to disturb the clam (undisturbed) state of the water. Next, disturbance
of the water was oscillated back and forth due to elasticity of water and then creating
waves. Finally, the energy was transported away from disturbance point to the
rim of the pond by the created waves! So in this sense, we could define that **“wave is
nature’s mechanism for transporting energy without degradation and without
transporting matter” **[5]. (Note this definition is given by admitted that energy
degradation effects are secondary and that waves are fundamentally conservative
phenomena.)

** **

** 3.1) Crisis of waves
in modern physics theories. **In classical mechanics, all kind of natural waves which we
can see in our daily life are so familiar that there is no need to ask **“what is wavy” **or **“wave of what”?** Oscillation of
water, vibration of string, vibration of earth quake etc. all is the **“wavy of the medium” **which is the carrier
of the wave involved.

** For sound wave,** at first sight there
is a little doubt because we can not see the medium “air”. Fortunately we still
could feel it, and later we had proved that **sound wave is the wavy of “air” medium!
**Actually** **any** **physical** **involved medium can
be used; it could be water, string or metal rod. The most crucial point is that
**there
must have something to be the carrier of wave**, whether or not we could touch it!

When Maxwell created the electromagnetic field theory, he used **elastic
solid ****(aether****)**** to be the carrier of
electromagnetic waves**. Soon after Einstein had created special theory of relativity then
the aether was ignored.** **And physicists have explained that** **electromagnetic waves could be
propagated by
mutual creation between the changing of electric and magnetic field! But the
problem is that **both electric and magnetic field are created and changing at
the same time** from one common time varying current source. So how could
they create each other? **It violates causality! **(Please see detail explanation
in CEMT.)

Time has pass, physicists
seem to forget**
what the basic criterion for wave is.** For example, according to Einstein
general theory of relativity, gravity wave was explained to work by the** vibration of
space-time **itself**.** Unfortunately we could not visualize how empty
space-time (without anything) could be act as the mechanism of wave, it seem to
be** a magic wave **rather than a scientific wave**!**

** **

** For Quantum
mechanics**
which is the **description
of wavelike **behavior
of atomic particles**
**such
as electron, neutron, alpha particle etc., thing get worse! Not only because **no one knows what the
particle wave is, **but
also** no
one knows what its mechanism is!** Indeed** Max Born** whom who
gave the correct interpretation of wave function (which was used until now)
himself used to call this mysterious wave as **ghost field** [6]**!**

** **

** **Here, armed with
vacuum mechanics,** **we will show that what we called **particle**** wave**, actually
it is **the waves of vacuum medium!** It is the different aspect of the
same thing as
electromagnetic waves (explained in CEMT) and gravity wave (explained in CGTR.) Then **ghost wave** in QMT
will turn to **God wave** (nature wave), so let’s explore it together.

** **

** 3.2) Wave–particle
duality.****
**Normally
when dealing with waves, we usually refer to **continuous wave**, but there is also a
short portion of wave train which was called **“wave- packet”.** For mechanical wave
such as water wave, wave packet could be created by applying **a short duration of some
disturbance source**
such as dropping a stone in water. But if we dip a continuous vibrating fork in
water, the created waves will be** a continuous wave train.**

**For electromagnetic
waves**
which include radio waves, light wave etc., it seem at first that all are continuous
waves. But if we observe carefully, we would found that **there are both **continuous waves and
wave packets. Normally **radio waves** which are man-made waves are **continuous waves**, while **light waves** which are natural
waves are**
wave packets**
and were called as **“photons”.** And in one sense we
may say that a photon is something like a** particle! **(Please see detail in CEMT.)** **

** **

** **Now let’s look at a
particle such as **electron** which we know that
it is obviously a point like particle. Anyway, in some circumstance such as in
case of double-slit interference, **it reveals itself as if it is wave.** So we say that **it has a wave-particle
duality.**
This wave-particle duality also occurs in other subatomic particles such as
proton, neutron and up to an atom! And these waves which involved with atomic
particles were called as** “particle waves”.**

** **

** What really the particle
wave is?**
Nowadays, physicists have used the concept of particle wave successfully,
but they still do not know what it is! For example, is an electron composes of
particle and wave? Or is it just a particle but also has wave property? In the
former, we could not imagine how it is. In the latter, **it may be because
electron could generate wave, but how? **

** **

** **One interesting point** **about electron wave
is that **it
will reveal its wave nature only when it is moving! **This gives us some
hint because it reminds us about the water wave generated by a moving boat. Anyway,
water wave can be created **only on the condition** that **there is water to be
disturbed by a moving object!**

** **

** **Armed with vacuum medium
space**,
**it
is easy to understand **how
electron could generate wave**. This is because the **moving electron would
disturb the surrounding vacuum medium** in the same way as a moving boat disturbs
the water! Anyway, electron wave is not as simple as water wave, and we will
discuss next about its mechanism.

** 4) Mechanism of
particle wave.**** **It is easy to
understand particle wave such as electron wave by using method of analogy. Let
us start with **“surface
water waves”**
created by a **moving
water beetle**
as show in the diagram of fig. 1(a). Normally, we are familiar with ship waves
which occurred behind the moving ship. But for a small moving object such as a
water beetle, **the
generated waves occur around **(and in front of)** the moving beetle **[7]**!**

**Figure 1 Beetle wave vs.
electron wave**

In more technical
detail of surface water waves, the preceding wave in front of the moving object
is **capillary
wave**
(wave due to surface tension), while the wave behind the object is **gravity wave** (wave due to
gravitational force). At the first sight it seems that the mentioned water wave
phenomena are a simple one, but actually it is very difficult and was study in
the **advance
course**
of wave!

In summary, beetle
wave was classified as linear dispersion wave, so it involves with group
velocity of wave. **The
preceding waves in front of the beetle are the group velocity which contained
waves’ energy**
that move forward and exceed phase velocity of the capillary waves! Normally
water beetles typically length is less than 1 cm and they move as fat as 50 –
100 cm/s, then the generated capillary wavelength is 1.7 cm [7’].

Now we return to** particle wave **such as an** electron wave**. When an electron
moving through vacuum medium (remember that electron is immerged in vacuum medium),
it will rotate**
**while
disturbing the medium around.** The disturbed vacuum medium will adjust
itself by creating standing waves in front and around **as shown in fig. 1(b).
(Note that the diagram shown is a three dimensions which was cut horizontally
in x – z plane to show only the lower half part.)

This is the same way
as beetle create wave, but why doesn’t electron just only create turbulences
around? The answer is that **“nature” has her own elegant and effective way to transport
energy away via wave mechanism!** And of course the wave mechanism is the same
thing for any type of waves; it is due to **the oscillation from the interchanging
action between inertial force and elastic force of the medium.**

** **

** In conclusion** both beetle wave and
electron wave are **standing
wave**
which is**
co- existence and moving with the matter **(beetle and electron). Anyway there
are two different points between them; first, the former is a 2-dimension wave,
while the latter is a 3-dimension wave. Second, beetle wave is** **a vertical transverse
of water surface wave**,
**while
electron wave is a rotational transverse wave in vacuum medium.

** **

** 5) Particle wave vs.
light wave.** Up to now, we have talk a lot about particle wave; here we will
get a better understanding of it by comparing with **photon**, which
is a wave packet of light wave. And first we will summarize about what a photon
is! (Please
see detail explanation in CEMT.)

** **

**
**Let us
start**
**with
the** Mechanism
of light wave****.
**We
have learned in school that light is emitted from an exited atom, but under the
concept that **“vacuum space is empty”** it is impossible to explain how it
could be done. Instead if we accept the concept **“existence of vacuum
medium”** then it is easy to understand how light is emitted!

**
**In
diagram fig. 2(a), when **an electron** in an atom was exited by energy** E**
(say, from incoming electromagnetic wave) and move from its

** **

**Figure 2 Light wave
vs. ****particle
wave**

** **In section 4 we
already talked about the** Mechanism of particle wave** (electron wave) which
was shown now in diagram fig.2 (b).** **Here we are going to compare particle
wave with photon as follow.

a)
Both photon and particle wave are wave packet of the **same one
common transverse electromagnetic wave.** In both diagrams, electric field ** E** is a radial vector
(side view), while magnetic field

b) While photon is a **free wave packet** which propagates
with the velocity of light, particle wave is a **standing
wave packet** moving together with the particle.

** **

c) Both
photon and particle wave were **created by a moving electron!** If the
moving electron is **accelerated**, then** **photon was
generated, while a **uniform** moving electron creates particle wave.

d) From
the reason of c) we often called photon as **“acceleration **or **far
field” **which will radiate after it was created. Also particle wave was
usually called **“velocity **or **near field”** which
will not radiate, but co-moving with the moving electron!

** **

** 6) Mathematical
formula for particle wave. **In** **section 4, we have explained the
mechanism of particle’s wave philosophically, now** **we will see
how it works mathematically, i.e.** de Broglie wave relation** (1) below. Historically
[8],** **de Broglie wave
relation,**
is not just a hypothesis **as we know today. Louis de Broglie had** derived it** in his Ph.D. thesis
(which Einstein had recommended the award) and later was awarded the Nobel Prize!

_{}….. (1), [where λ =
particle wave’s length, *h* = Plank constant, *p* =momentum].

** **Nowadays, almost all
text books do not talk about the **derivation’s process **because physicists
think that**
**it
is wrong, so they accept only the output! But now we will see that the derivation
is still correct (and it is **corresponds to **the concept of **vacuum medium wave** which we will talk
later) and below is the original derivation.** **

** 6.1) Original derivation
of de Broglie wave formula.** **The** **crucial point** in** **De Broglie derivation is that he assuming that
**photon
has a certain rest mass _{}!**

** **

_{} …… (2),
_{}…. (3).

Now consider the
same vibration as observed in laboratory frame in which the **photon moving with
speed v, **and
using Lorentz transformation, we have

_{}…… (4) (where_{} ), then the
disturbance described by

_{}…… (5), or _{}…… (6).

Equation (6) is **a
wave with frequency f and wave speed (phase velocity) w,**

where_{}…..(7),
and _{}…..(8).

Note that the **total energy of the
photon**
measured in laboratory frame _{} then

_{}….. (9).

The wave disturbance
described by (6) has a characteristic **wavelength λ** defined by the

ratio of the wave speed *w* to the
frequency *f*; _{}……. (10).

Making use of
equation (2), the relation _{}and equation (10), **finally we get de
Broglie wave relation**
which applicable **for
particles with rest mass** as (11) below.

_{}…….. (11).

Note
that the above derivation is **“Nonrelativistic treatment of ****de Broglie waves****”** De Broglie applied
his relativistic treatment of wavelength for photons to **particles
of nonzero rest mass.** The wave nature of particles of nonzero rest mass can also be
described consistently with a **nonrelativistic theory,** a fact
of vital importance in allowing simple solutions of many quantum-mechanical
problems. **(**This means that we use *m* in (11) as**
nonrelativistic mass of the particle considered!)**

**
6.2) Improved derivation of de Broglie wave formula.** Armed with a **physical**** **concept** **of** **wave- packet of
vacuum medium, now we will improve the derivation of the conventional **abstract **concept (of
particle wave) by using the original de Broglie’s derivation as a guideline.
Then what we got is a more** rational derivation** with its
**physical
meaning of de Broglie waves**** **as below.

a) When **a particle** with rest mass ** _{}**(not photon rest mass
as in the previous section) moving with a uniform velocity

b) Because the total
energy of the moving particle is_{}, so its **kinetic
energy** is equal to_{}. Then this
amount of kinetic energy **appeared as the energy of the created** **wave
packet!**

c) But
remember that the created (co-moving) **wave packet **could not
be existed alone without the original bare (rest) particle, so when we consider
its energy we must include the energy of the bare particle which is co-moving
with wave packet. That is the **overall energy of the particle wave
packet** is equal to_{}, which is the same thing as we
view the total energy of the moving particle **without bothering
about its composition!**

d) Now
consider** **the energy of the **particle wave packet** which is
as (12) below

_{}…… (12),
and it could be write as _{}or as

_{}…… (13),
which has the **same form** as (2) but with **different
meaning!**

While
_{}in (2) is
the rest mass of the photon, _{} in
(13) is the rest mass of the particle! Accidentally, what we got is similar to
the concept of **“Compton effect”**, which states that “In general
the energy of a photon whose wavelength is equal to the Compton wavelength of a
particle is just the **rest energy of that particle**”.

e) **Finally,** we could derive
de
Broglie derivation process similar to the procedure** **above
(form equation (2) to (11)), together with some **additional
changes in the interpretation** (i. e. we start with the
co-moving particle** **wave packet
of vacuum medium, instead of the photon** **concept). Lastly, what we got is a **better de Broglie
wave relation **as
in equation (11) as desired!

** 7) Better view of Schrödinger
equation. **In
**classical
mechanics**,
if we want to find the solution for a moving particle we will use Newton’s law
of motion. In**
electromagnetism**,
we obtain solution for **electromagnetic wave** by using Maxwell equations. But for **quantum mechanics,**** **Schrödinger** wave **equations (14) were
used to find the solution for subatomic **particle **such as **electron **etc**!**

_{} ** **…. (14),**
**

Anyway**,** this** derivation **of Schrödinger
equation**
**could
not be done from **fundamental
law**,
Schrödinger derived it by “making use of some deep formal analogies between
optics and classical particle mechanics” [8’]. And because it is a **wave equation** which was used as
the **“dynamics
equation for particle’s motion”**, so it requires additional interpretation
related to “particle’s aspect”** **i.e. **“wave function Ψ ”**, which was defined
to be the

** **

** **Now we know that** **actually particle
wave is vacuum medium wave.** **And we have learnt from CEMT that electromagnetic wave is
also** **vacuum medium wave,** but** **different aspects of
the same one**
**vacuum
medium wave!**
So it is easy to derive Schrödinger wave equation in the same way as was done
with electromagnetic wave! Here, we will use David Bohm’s derivation [3’’] as
the**
guide line** (the only difference is that his derivation was from an**
abstracted wave’s** concept, while for the author’s one is from a **realistic
wave** with philosophical idea behind!)

** New derivation of
Schrödinger equation.**
As mentioned in section 4 that electron wave is **standing wave** which is** co-moving with the electron
**so
the suitable mathematical tool is Fourier analysis. This is because **Fourier analysis** enables us to
represent an arbitrary function as a sum of standing plane waves of all
possible wavelengths and amplitudes of **electron’s wave packet!**

** **

**
**_{} …. (15)** **

The Fourier integral
(16) is for **free
electron particle**
written in complex function at *t =0*, and the propagation vector *k *for** **electron’s wave
packet must oscillate with angular frequency, _{}. Hence the value of *Ψ*
for all times is given by multiplying each j (*k*) by exp - _{},

_{} …. (16).** **

** **

**
**This
tells us what happens to an arbitrary **wave function** as time passes** **for the case of free
particle. Now to get the **partial difference equation satisfied by Ψ** , first
differentiate equation (17) with respect to time, then we get

_{} …. (17).** **

Let us now evaluate

_{} , then we get _{}…. (18).** **

Combining (18) and (19) we obtain,
_{}…. (19).

This is a special
case of**
Schrödinger equation **(same
as (14))**
**which
is the standing wave packet of vacuum medium that was created and propagate
with the moving particle!

** 8) Quantum vacuum mechanics.
**Up
to now we have seen how to improve the mysterious **basic concept** of the conventional
quantum mechanics with a better philosophical idea. We have explained
what the mechanism of **particle
wave **is,** **and shown how** **the **de Broglie wave
formula **was
derived**.
**Then
it permits us to derive an understandable** Schrödinger equation.** All these could be
done via **vacuum
mechanics **(the
mechanism of vacuum medium), so it should be called as** **“quantum vacuum mechanics”!

Now, we have a** completed quantum
mechanical theory **which
we are looking for. Next in order to show some confidence in our new concept, we will
use it to
solve some unsolved basic problems in conventional QMT, i.e. waves collapse interpretation
and the concept of uncertainty principle as follow.

** **

** **

**Figure 3 Wave
collapse interpretations**

** **

**8.1) Waves collapse
interpretation. **Conventionally,
wave collapse interpretation in QTM is something which could explain as show in
diagram fig.3 (a). But the problem is** there is no explanation** “how each electron
converts itself into waves, passing through both slits and then the two part of
waves **collapse** back into a single
particle on screen”. Instead, armed with the concept that electron particle
wave is the co-moving wave of vacuum medium, it is easy to visualize and
understand**
how electrons do the magic **as below**.**

** **

** **In diagram fig.3 (b),
all the electron particles **moving together with their own created waves** (i.e. electron
particle is the wave source). Some of electrons pass through slit 1, while
others pass through slit 2 randomly. While each electron passes through slit 1** or** 2, its co-moving
waves will pass through both slit 1**and** 2 at the same time. After passing through
both slits, both the split waves will interfere each other while moving toward
screen and forming to be interference pattern.

Finally, because** **all electron
particles **must
move together **with
their own created waves** **(they are co-existence),** **then it is not surprise that all electron
particles are located in the constructive interference zone. **This is the reason** why all electron
particles are forced to arrange and forming to be interference pattern!

Note that, even
though that the interference pattern which occurred
on screen comes from stream of electrons, but it is not the interference
between two electrons; instead it comes from** one
single electron which interferes with itself! **Of course,
one electron cannot create interference pattern, only a large amount of them
could do. And **this principle is also applied to photon,** and
which means that each photon will interfere with itself forming interference
pattern!

** 8.2) Mechanism of
the uncertainty principle!**** **We are
familiar with Heisenberg’s uncertainty principle which state that **“It is
not possible to determine both the position and the momentum of a particle with
unlimited precision”. **Instead, it could be done only according to
the relation (20) below,

** **

_{} …..** **(20), where *h*
is Planck constant = _{}

It was
said that **“**the uncertainty principle** protects **quantum
mechanics”!** **Heisenberg recognized that **if it were possible** to
measure the momentum and the position simultaneously with a greater accuracy, **the
quantum mechanics would collapse **[9]. The only question is why it
is so**,** or what is its mechanism**?**

** To
answer** the question, we will start** **with**
classical mechanics** by using the guide line from “Physics” [10] with some minor
change. When we deal with **“wave packet” **which is
a portion of a continuous wave train as shown in the diagram of fig. 4 (a),
then what we got is the relation between length *x* and wave number *k *is
as (21)
below.

** **

_{}…..** **(21), _{} …..** **(22), _{} …..** **(23)

If we
reduce ∆*x* then ∆*k* is increase, and in the reverse when we reduce
∆*k* then ∆*x* is increase. This means that we cannot determine both
the position and the wave number of a wave packet with unlimited precision!** **So this
is** **the** ****uncertainty**** ****principle**** in
classical mechanics **which is occurred** due to the property of wave
packet!**

** **

**Figure 4 Mechanism of**** the uncertainty
principle**

** **

Note
that the
uncertainty relation of wave packet (21) is **valid for any kind of waves,** so we will apply it to** photon** which is the wave
packet of light wave by using **de Broglie wave** (22). (Remember
that for** **De Broglie derivation
in sec 6.1**
**was
derived from photon.) Finally what we got is Heisenberg’s uncertainty principle
as (23) or (20) above. Here we can conclude that “It is not possible to determine both
the position and the momentum of a **photon **with
unlimited precision” **because it is a**** wave packet of light wave****”!**

** **

Now we
will apply the
uncertainty relation of wave packet (21) to **electron wave** via **“single-slit
diffraction”**
experiment in the diagram of fig. 4 (b). (Please remind that electron wave is a
standing wave packet.) Consider a beam of electrons of speed _{}pass
through the slit and create diffraction pattern as shown. We take somewhat
arbitrary value of _{}as a
rough measure of the **uncertainty of our knowledge** of _{}and call
it_{}. In
the diagram we could see that _{} and
_{}, so the
combination of the two relations is_{}. After
apply**
**de
Broglie wavelength_{}, what we get is _{}. This
means that** we cannot determine both the position and the momentum of
electron with unlimited precision!**

** In
conclusion,** we could see that the **uncertainty measurement of
electron** is due to the wave packet (of vacuum medium) created by the
moving electrons. And the energy of the wave packet will guide the moving
electrons to form the interference pattern. (This forming of interference
pattern is the same way as what happened with double- slit). **This is
the mechanism of Heisenberg’s** **uncertainty principle****!**

** It is
interesting** to note that if there were **no vacuum medium**, then
the moving electrons cannot create electron wave. So the moving electrons will
go straight pass though the slit direct to screen (without guiding from wave)
and pile up as a smooth curve (due to statistical randomness, not interference)
shown at the top of the diagram. So in principle, **we can
make a measurement of electron with unlimited precision!**

**9) Particle wave or
matter wave?**
Up to
now, we are quite confidence in our new “quantum vacuum mechanics”. Here we
will use it for reconsidering what which conventional called as **matter wave**. So far, we have
found that all particles in atomic level, starting from electron, proton, and
neutron up to atom, have their own wave property. Anyway, the concept was extended
to cover **any size of particles** (**of matter**), such
as bullet golf ball up to a moving car or even a moving planet like earth, **also has
wave property! **

** **

** **Now the problem
is** **the **matter
wave** concept right or wrong?** **To answer
this, we will** **start to consider by using Richard P. Feynman explained about
“interference experiment with bullets” [9’] as shown below.

** **

**Figure 5
Interference experiment with bullets**

** **

** **In the
diagram fig. 5 (a), is the curve shows the probability of bullets predicted by **quantum
mechanics**, and the rapid wiggles are supposed to **represent
the interference pattern** one get for wave of very short wavelength.
But in the actual experiment we can not see the interference pattern and what
we saw is only a kind of average; a **smooth curve** as show
in fig. 5 (b). The reason is because the **bullets wavelengths were so tiny** that the
interference pattern became very fine. So fine,** **that with any
detector of finite size one **could not distinguish the separate maxima and minima.**

** **

** **Now let us see, **how long the** **de Broglie wavelength
of the bullet is!**
For bullet of mass 41 g. travels at 960 m/s. its wavelength is _{} And when we compare
it to the diameter of electron (radius_{}), we would see that it
is an **incredible
tiny wavelength!**
Then a new problem arises, **is this realistic case or just because we think it is? **To answer, we have to
consider it in more detail as below.

** **

According to the
conventional concept of de Broglie wave, it is hopeless to answer the above
problem because **we
do not know what actually the particle wave is.** But instead, armed
with the new concept that de Broglie wave is a physical wave packet of vacuum
medium, so we could understand and visualize it** **and **not so difficult to
solve the problem!**

** **

Technically, to set up the
experiment above, both the two slits’ width should **comparable or smaller** to the wavelength.
Anyway, at the same time the slits’ width must **wider than the
bullets size**
that is in order of millimeter_{}, which is_{}times bigger than the
wavelength! **This is crazy, how could we make such slits and how could we got
the interference pattern **as shown in (c)?** **

Now return to** water beetle wave **discussed in section
4. If we study in more detail we would found that **it is not any size** or **at any speed** of the moving beetle
that could create wave! **Only within a certain range **of size and speed
(which correspond to some mechanical properties of water’s surface tension)
that **enable
the beetle to create the** **capillary wave** in front. For example, the beetles have
typical length less than 1 cm. and it comparable to the capillary wavelength
which less than 1.7 cm. This is the reason why **we never seen
capillary wave in front of a moving ship! **Also the moving speed
of beetle is about 50 – 10 cm./s. which is faster than the minimum phase speed
of wave (about 23 cm./s). If the beetles move slower than this speed, what we
could found is only a highly localized disturbance of water surface; **no wave pattern could
be created!**

Analogous to beetle
wave, a moving particle should disturbed and create particle wave in vacuum
medium for a **certain
suitable condition!**
This will correspond to what was called as **“the domain of wave mechanics”** which is a rough
criterion for the applicability of ordinary particle mechanics (either Newton or relativistic) [8’’]. And it states that **“the de Broglie wavelength must less than
some characteristic linear dimension l of the system.** For

** **

In conclusion, we
may say that **a
particle wave can be created **if its wavelength is** **comparable** **or** **longer than the
diameter of the moving particle!** **So it is likely that de Broglie wave could be
found only in the realm** **of** microscopic **(atomic) world, but not in** macroscopic **one. Then we should
limit **de
Broglie wave **to
**“particle
wave” **rather
than** “matter
wave”, **shouldn’t
we?**
**

** 10) Classical mechanics
vs. quantum mechanics. **Up to now, we have only talked about quantum mechanics;
here we will see how the unfamiliar quantum mechanics relate to the familiar classical mechanics which we use in our
daily life as follow;

First, let us start
with a **classical mechanics** concept of Newton
law of motion used for the trajectory of gun’s bullets as show in the top diagram fig.6.
In the
experiment, stream of bullets pass through both slits, then we got the **probability** of them (P12) as
shown in (b). If one of the two slits was closed, then we will get the
probability (P1) or (P2) as shown in (a).

Second,
we deal with
a classical
mechanics
concept of **wave motion** as show in middle diagram fig.6. In this experiment, if we close one of the two slits, then what we got
is the **intensity
**of
wave, I1 or I2 as shown in (a). And if we add I1 with
I2, then we get I1+2 as show in (c). But if we open both slits, then we will
get **interference** pattern of intensity I12 as shown in (b).

[Note,
actually the light wave of just talked above is not continuous, it is **photon **(wave
packet of light) as shown in fig. 2(a). And the experiment result is the same as in the
second case! The difference is that the interference
pattern of **intensity** (of light wave) could equally call as **probability** of light
particles (photons). Anyway, for simplicity here we will limit our
consideration to the continuous wave.]

Third,
now we will deal with **electron wave-particle** duality as show in the lower diagram fig.
6. And what we got is the interference pattern of** probability** (P12) of
electrons (b) like the photon case. Also if we close one of the two slits, then what we got is
the** probability** (P1 or P2) of electrons wave as shown in (a),
while the sum of them is (P1+2) as show in (c).

Fourth,
now we will compare in more detail about the **detection result** between
classical particle mechanics (first case), classical wave mechanics (second
case) and quantum mechanics (third case) as below.

**Figure 6 ****Classical mechanics vs.
quantum mechanics**

Classical **particle**
mechanics - Total
probability of bullets;

If one of the two slits close, _{}… (p).

Classical **wave**
mechanics - Total
intensity of light wave;

If
one of the two slits close, _{}… (w).

If
both the two slits open,
_{} … (w’).

Quantum **wave-particle**
mechanics - Total
probability of electrons;

If one of the two slits
close, _{}… (q).

If
both the two slits open, _{}… (q’).

[Where _{}are wave
amplitudes via slit 1 and 2, and _{}are
probability amplitudes

via slit 1 and 2 respectively]

Fifth, from equation (w), (w’) and (q), (q’), we can rearrange and what we got are

_{} …..(s),
_{} …..(s’).

[Where Ө is
the phase difference between _{} and
_{}]

The last
term in (s) is the** **interference term which arisen from wave property of light wave,
while the last term in (s’) is the **interference** term
which arisen from **wave property of electron**!

Lastly,
if electron particle has no wave property, then (s’) will reduce to the same
form of (p), which is the probability of bullets, and Newton classical
mechanics could be applied. So we could say that in classical mechanics we deal
with particle and wave **separately**, while in quantum mechanics **particle **and** wave **(wave-particle
duality) being considered** together! **

In
conclusion, Newton **classical mechanics** (with empty space concept) was used to handle **normal**** particle** such as
a moving
gun’s bullet, in which the existence of vacuum medium has negligible effect. In contrary to **quantum
particle** such as a moving electron which creates a co-moving wave of
vacuum medium, so
**quantum
mechanics**
was needed to explain its behavior. Anyway, in an abstract empty space
(or vacuum
medium
effect was neglected), then the
complicate quantum mechanics could be replaced by the simple Newton
mechanics!

**
11) Physical meaning of Planck constant. **Up to now**, **we would notice that Planck constant ** h**,
is always involved in QTM. Conventionally, it is a

** **

**
**_{}…… (24),
_{}…… (25),
_{} …… (26).

Anyway,
we do not have any idea about physical meaning of this Planck constant ** h**, but armed
with the concept that

** **
Start from section 6, we have a **photon momentum**_{}, which could be
written as in (26).Then
we could see that (25) is the same as (26)!** **

** **Now,
armed with the concept that photon is the wave packet of vacuum medium,** **which
corresponds to**
de Broglie**
**concept
****for a photon** with **rest mass _{}**(in section
6.1). Then we could rewrite (24) and (25) and (26) as (27),
(28) and (29) below.

_{} …… (27),
_{}…… (28),
_{} …… (29).

After we combine (27) and (28)
together and using_{}, then **what we
got is**

** **

** _{}**… (30),

Finally, we can
interpret the “physical meaning of Plank constant ** h**”

** **

a) The
first two components
in the right hand side of (30) formed the **momentum of vacuum medium’s wave
packet **(momentum of a photon)**.** While** **the third component _{}is the **distance**
of one wavelength of the wave packet, so *h***is one unit of the**
**“action of wave packet of vacuum medium” **and has a dimension of
Joule-second (linear momentum-distance).

b) Because the entire
three components in the right hand side of (30) are the **basic properties of the
uniform vacuum medium **and they are all constant, so* h ***is constant!** (Note that ** c** is not only the velocity
of light wave in conventional sense, but actually it is the velocity of wave
packet which propagates in the

c) For equation (31), the entire three
components in the right hand side of (31) is constant because
when ** m** is increasing then

d)
Finally, we return to the mechanism of light wave (photon) concept in section
7, which we sew that actually it is the mechanism of **wave packet
of vacuum medium!** And when we combine it with what we have talked about the Planck constant ** h
**in a), b) and c), then we may conclude that the

Note that for **particle wave**, the Planck constant ** h
**in

** 12) Is energy
continuous or ****discrete? **In modern physics, **it was
believed that energy quantity is a discrete, not a continuous value as was told
by classical theories.** The reason comes from Planck’s radiation law
– the quantization of energy which he assumed that atoms behave like tiny
oscillators with a characteristic frequency oscillation. And the assumption
behind the law is that **the energy of an atomic oscillator is
quantized;** the oscillators emit or absorb energy** E** in a
discrete set, defined by

** E
= nhf**
…… (32)

** h **is Planck
constant and

Beside quantum
mechanics, **it
was assumed**
that **the**** concept
of quantized****
energy is also true for classical mechanics **such as the** **mechanical oscillation** **system of clock spring
and pendulum. To support the quantized energy concept, it was explained that
we can not
detect the graininess (of the discrete energy) of the system because it is too
small to be observed. Anyway, **this is not the evidence which decide whether
the assumption is right or wrong! **

**Now we will look in
the opposite side**
i.e. to show that it is likely that **the**** concept of quantized**** energy was limited
within quantum ream and not for the classical one! **And indeed we will
found that it is easy to understand and to visualize how it behaves in this way,
by using the concept of **quantum vacuum mechanics**. Here are the details.

First it is easy to
visualize the between **a short portion of wave train **(wave packet or
discrete amount) and **a continuous wave train** which occurred in water
wave. By applying **a short duration **of the disturbance source such as
dropping a stone in water then wave packet of water could be created, while a
continuous wave train could be generated by dipping a **continuous vibrating**
fork in water!

Analogously for electromagnetic
waves, wave packet of light (photon) is a **short portion of wave train** of electromagnetic
wave, while **radio
wave**
which is man-made electromagnetic wave is a **continuous wave. **The reason is because** **radio wave was
created from a continuous varying current source (please see detail in CEMT).

Now we come back to
the **mechanism
of light wave packet**
mentioned in section 5, which says that when an electron (in an atom) was in an
exited state, it will jump back to the normal state. At the same instant
of time, it will transfer its **impulse force** disturbing and vibrating the
surrounding vacuum medium, then radiate as the emitting photon (light wave packet).
This corresponds to Planck’s radiation law which says that the
energy of an atomic oscillator is quantized. And here we can see that the** atomic
oscillator is not a continuous oscillator, **instead** it acts
as****
an impulse oscillator.** This is the reason why the energy from an atomic
oscillator is **quantized****! **

So the crucial point** **is in the **interpretation** of Planck idea, in
which **we
must not think that the atomic oscillator is a “normal” continuous oscillator
which generates the discrete energy** as was misunderstood. Instead, it is the
oscillator which oscillates in discontinuous (discrete) manner i.e.** it is an impulse
oscillator!**
So this atomic oscillator is not the same thing as a classical mechanics
oscillating system such as clock spring and pendulum.

** **

By the way, **normally** when talking about
an oscillator, we usually mean a** continuous oscillator **both a classical
mechanics oscillating system or electromagnetic oscillating system. And in our
daily experience **we
could see both types of oscillator** as mentioned in the early paragraph, thing
is different **in
atomic ream**
in which **only
the impulse oscillator is possible!**

** **

In conclusion,
because of our **ignorance** about mechanism of Planck’s
radiation
energy, so we think that** the**** concept of quantized**** energy is also true
for classical mechanics**. Now based on vacuum mechanics concept i.e. the
mechanism of vacuum medium, then we found that actually** **quantized energy** was limited within
quantum ream **and
not for the classical one!

** ** **13) Is quantum
entanglement right or wrong?** Nowadays, main part of modern
physicists seems to believe in quantum entanglement (correlation), such as one
electron can **affect** another electron simultaneously,
no matter how far they are separated! This was claimed to be the nature of the
conventional QTM, and a mathematical formula called Bell inequality was used
for proving the idea.

Anyway, the problem
is that they do not know how electron can do the** **magic! For me (the
author), this is crazy and we will show that the true problem is because of our
**ignorance
**in the
**mechanism** of quantum mechanics,
which is then give us with **misinterpretation** in the theory. Here we will break the quantum** magic** and turn it back to the
real science by using our new concept of quantum **vacuum** mechanics, as
follow.

** ** **13.1) Einstein-Podolsky-Rosen
paradox**
Historically, quantum entanglement was criticized by Einstein, in which he called
it as **a
spooky action at a distance**! And this give rise to a debate between Einstein and Neil
Bohr (a famous quantum physicist), via a thought-experiment on a two-particle
system, called EPR paradox.

Because the
conventional interpretation of quantum mechanics, in which measurement with one
particle can affect the far away particle simultaneously, for Einstein this was
completely unacceptable! In Einstein opinion, there should have something (attribute)
hidden, i.e.**
hidden variable**,
in which we cannot observe, that **responsible** for the mentioned “**affect**”.

Anyway, there are
many experiments which tried to prove **hidden** **variable** concept of the EPR paradox, by
comparing to the conventional quantum entanglement. And most of the results
seem to agree with quantum mechanics, not** **hidden variable. In order to see, **whether the
experiments prove is justify or not**, we will consider in detail about Bell
inequality which is a new concept that is not familiar to us.

** **

** ** **13.2) Bell inequality
is a mathematical tool. **In 1964 John Bell had invent a mathematical formula called
Bell inequality which was used to show that quantum entanglement is right,
while hidden variable theory is wrong as mentioned. Here we will summarize what
the Bell inequality is.

**Figure 7 ****Bell inequality test
for EPR**

** **

** **According to diagram fig.7,
a pair of spin-0 photons is produced in a single process from a source S. The
state of the two photons, a and b are **correlated and can be described by a
two-particle wave function** via quantum mechanics. Next, we assume that polarizers
at A and B can detect only two values of two types of photons i.e. (+1) for
parallel polarize photon and (-1) for perpendicular polarize photon, then
_{} for detector A
and B respectively.

To do the experiment
test, two orientations are allowed for each detector, that is _{}and since each
detector has two possible orientations called 1 and 2, we shall denote their
responses as _{}respectively.

Now let us define the
quantity _{} is what was
called as the **correlation
function**
of the system.

[where the symbol <> denote the mean value over many measured events.]

Then the BCHSH
inequality (or Bell inequality) reads _{} And
it was proved by using Bell’s Theorem that this inequality is correspond to the
hidden variable theory [11].

When the experiment was done by counting and recording several events for some period of time, then the coincidence circuit will register the events in which two photons are detected in cascade. In this way, what we got is the four kinds of separate counts:

_{} _{}

_{} _{}

And from these four kinds of separate counts we can calculate the correlation function mentioned above.

Finally, when
the experiment was done, at some value of Ө (the angle between the orientations of the
two detectors) such as
Ө =22.5
degree, it was found that the result violate the inequality, i.e. _{}. And the result corresponds
to the prediction of quantum mechanics, i.e. _{} then **it was concluded** **that the hidden
variable theory is not correct**!

Up to now, it seems
everything should be alright, anyway the only one crucial problem is left; **is the proves justify**, or **could Bell inequality
is a right tool?**
As we have seen, Bell inequality is purely mathematical, it is not involves any
physics [11’], or relate to any property of the particles in the experiment.
Indeed, BCHSH inequality was criticized to be just a set of positive
probability which could be derived solely on the basic of locality condition
[12]. So we have to make clear that it is a **proper tool** used for proving the
hidden-variable! Then what we have to do next is to consider the validity of
the inequality.** **

** ** **13.3) Einstein hidden variable vs. quantum
mechanics. **The
aim of a hidden-variable theory is to consider quantum mechanics as a
statistical theory in a sense that it furnishes probability of possible events,
which in reality are **fixed
by non-observable properties**. This is analogous to classical mechanics,
which incorporates random fluctuations as a property of thermodynamical system;
although a discussion of individual atoms should yield complete deterministic
behaviour. It would be advantageous t find similar substructures for quantum
mechanics to preserve locality and determinism. Of course a theory including hidden-variables
has to reproduce all the experimentally confirmed the result of quantum mechanics.

Unfortunately,
Einstein had died before giving more detail about the hidden-variable. Anyway, In
1952 David Bohm had developed a thought - experiment test between quantum
mechanics and the hidden-variable [13]. In the experiment, **a real spin vector** analogous to a
classical angular momentum vector was introduced as the hidden-variable for
system of two-particle with total spin zero.

It is interesting to note that Bohm hidden-variable parameter is not just an abstract one, as was done by Bell; instead, it involves with the property of the particles i.e. spin vector! In summary, the derivation of the experiment result was done in the same way as Bell‘s one, i.e. prediction probability of the spin component of particles was measured in the experiment.

According quantum mechanics, probability of the measured spin component of particles was defined as follow:

_{}

[where
_{} = spin angle
reference to z axis]

Next we define the average value of the spin measurement as

**Correlation coefficient** _{}

And according to the hidden-variable theory,

_{}

Then the** Correlation coefficient** _{}

When the experiment
was done, we found that the result of the Correlation coefficient is corresponding
to quantum mechanics’ prediction, so this mean that the proposed **real spin vector **as the hidden-variable
is still not a correct one!

Anyway, by compare between
the** **two correlation
coefficients, we could see that it has indicated something interesting; there
is a **linear
variation**
in the correlation of the hidden-variable case, while the correlation **varies as the cosine** of the angle in the quantum
mechanics case. Could this reflect something about the property of the spin
particle, in which we are considering?

Armed with our new
concept of quantum **vacuum** mechanics, we could
see that it will **guide
us** **to Einstein
hidden-variable**
we are looking for as follow.

First, let us reconsider
the famous Schrödinger equations (14), which is the foundation equation in
quantum mechanics. Remember that it is a **wave equation**. And because it
contains the imaginary number i, so its solution is a complex wave function [3’],
for example, wave function of an electron:-

_{}[where *c* =
constant], which then give the probability_{}, correlation coefficients
<> (mentioned above) proportional to cosine (trigonometry) function.

** **

Second, according to
conventional quantum mechanics, what we have found in the previous paragraph
was derived without understanding why it is so! But anyway, we also have seen
that based on our quantum vacuum mechanics we found that actually the **wave concept was
arisen from the rotation of the moving quantum particle** (that disturbed the
surrounding vacuum medium)!

Third, now it is not
surprise why the **real
spin vector **(as
the hidden-variable) is not correspond to quantum mechanics prediction; it is
because the real spin proposed is just a classical spin concept, which **not involved with any
wave nature**!

Finally, we could
see that all the proposed hidden-variable so far is **not relate to any
property of the quantum particles under testing**. Instead, it is just
a simple classical parameter, which is not complicate as what really occurred in
quantum realm. **This
is the reason**
why the experimental results using Bell inequality test do not conform to quantum
mechanics!

In conclusion, armed
with quantum vacuum mechanics, we could see that actually Einstein (local) **hidden-variable** is the quantum mechanism
of**
vacuum medium**
- quantum vacuum mechanics. And it is this mechanism which responsible for the
mentioned quantum entanglement via the conservation law of angular momentum
between the two electrons of a total zero spin system [14]!

By the way, there is
another main problem about the incompleteness of conventional quantum
mechanics, in which Einstein had mentioned; it is the **randomness** which was told to be
the nature of quantum particle. For Einstein, who believed that **determinism** (exact knowledge of
initial condition, allow the future to be predicted exactly) is the universal
characteristic phenomena, while randomness is merely due to our ignorance!

Actually, we have
talked about the **uncertainty principle** (which stated that** **it is **not possible** to
measure the momentum and the position simultaneously with a greater accuracy) and point out that
it was due to the existence of** vacuum medium**! And if there
were no vacuum medium,
then in principle, we can make a measurement of electron precisely**. **So in
this sense, we could say that the** ****randomness **was due to the **hidden-variable** of unobservable vacuum
medium,
couldn’t we?

**14) Problem with
hydrogen atom model.**** **We are familiar with** Bohr model** of hydrogen atom as
shown in diagram fig.8 (a) below. And the circular motion of electron for the
stationary state is at Bohr radius ** _{}** (calculate by using
classical mechanics), but it was criticized that it would

** In quantum mechanics,** where what we got is
a better model of hydrogen atom as shown in diagram fig.8 (b). It has shown the
probability location of the electron around the proton nucleus for the ground
state in atom, which was calculated using** **Schrödinger equation**. **And in diagram fig.8
(c) has shown the radial probability density which is **maximum at the radius
r equal to Bohr radius.**

** Anyway, **even the model in
quantum mechanics, there is still no** **explanation** why electron can stay separate from
the proton nucleus, **and
it still depends on Bohr assumption (the postulate of stationary state)! Instead,
the problem could be explained by using the concept of **“vacuum medium space”** (instead of
conventional empty vacuum space) and **“electron and proton are **(something like)** tiny black holes of
vacuum medium”.**
Please see detail in VMTE.

** **

**Figure 8**** ****Hydrogen atom model**

** **

** Under **the concept of** “quantum vacuum
mechanics” **we
could use the model shown in diagram fig.8 (d) for explanation why** “**electron can stay
separate from the proton nucleus and having the probability density” as shown
in (a) and (b). Normally electron can **move freely** (even at absolute zero degree
temperature) in **“Bohr
shell” **for
the reason that, it is the place where** **the **“Coulomb attraction force” **(between electron and
proton) **equal
to the repelling force** of the “vacuum medium” in between**!** (Detail could be
seen in CEMT.

** **

** Finally** we also knew that **there is always a
small vibration in any atom**, except at absolute zero degree temperature! So any
electron in any atom **could
not stay only within**
**Bohr
shell**,
instead it has the probability to move to other location out of Bohr shell when
time goes on. Of course, the probability density of electron is according to
wave mechanics**
**calculated
by Schrödinger equation. So what we got now is a better explanation for
hydrogen atom model which is** more realistic **and **understandable!**

** **

** ** **15) Conclusion &
Discussion.**
Up to now, we have learnt how to **improve **conventional QMT** **by using the** new concept of
vacuum mechanics,**
i.e. the mechanism of vacuum medium. Also we have shown how a suitable and
understandable model was built, what its advantage is and how it solves the
existing problems in QMT!

** **

Actually, what we
have done is just **“putting
vacuum mechanics into quantum physics theory” **and then turns it to** **quantum vacuum
mechanics**
**(CQMT)**! **So we could eliminate
the mystery in QMT, turning its **magic** to** science** and make it understandable to anyone.
Of course, we known that QMT is the excellent tool used for exploring the micro
world, but without understanding, lead us to something crazy in the theory which
we are facing now!

** **

Nowadays, the standard
model of particle physics theory, which is the main tool used for Large Hadron
Collider**, **is based on QTM (i.e. quantum gauge field theory). Anyway,
beside its **successful in prediction **about any (non-gravitational) process
in** **the** universe**, the standard model was criticized as “an** **extraordinary**
**ad hoc and ugly** **theory that it was clearly nonsense”. Armed with a
non-mysterious CQMT would make it a better theory than it is!

** **

Lastly, we could see
that what was written in this paper is a short one, and it is not fully
mathematical rigor. What we have done is just** enough to give the new idea to
the readers with some confidences. **So what was left here should be the work
of people who are involved in the matter to complete it** **for** the
virtue of academic merit and the progression of our world’s advanced knowledge!**

** **

** **

** **

**
16)
References.**

** **

(**Precaution;** Several text books
with different authors were used as the reference and each author used
different pattern and different notation for the same formula. Here in this
paper, the author has made some change of the original pattern and notation for
the readers convenient, so please be careful!)

** **[1] By Nimit
Theeraleekul, now presenting in www.vacuum-mechanics.com.** **

** **

[2] Richard P. Feynman, “The character of physical law”, p.129.

[3] David Bohm, “Quantum theory”, p.167. For [3’], p. 85, and for [3’’] p.77-79.

[4] Jeffrey Bub, “Interpreting the Quantum World”, p. 40. For [4’] p.4.

[5] Tolstoy, Ivan, “Wave Propagation.”, p.1.

[6] Tian Yu Cao, “Conceptual Development of 20 th Century Field Theory”, p. 149.

[7] Lee A. Segel, “Mathematics Applied to Continuum Mechanics”, the first page.

For [7’] - p. 379.

[8] A. P. French and Edwin F. Taylor, “An Introduction to Quantum Physics (The M.I.T.

introductory physics series)”, p. p.55-57, for [8’] p. 65, and for [8’’] p.97.

[9] Richard P. Feynman, “The Feynman Lecture on Physics (volume III); Quantum

Mechanics”, p.1-11. For [9’] p. 1-9.

[10] D. Halliday, R. Resnick, and K. Krane, “Physics” Vol. 2 Extended Version Fourth

Edition, p. 1049-1051.

[11] Charles Ruhla, “The Physics of Chance”, p.206-208, and [11’] p.205.

[12] Tony Rothman & E.C.G. Sudarshan “Hidden Variable or Positive Probabilities?”,

arXiv:quant-ph/0004109v4 5 Feb 2001.

[13] Walter Greiner “Quantum Mechanics – An Introduction”, p. 422.

[14] John S. Townsend “A Modern Approach To Quantum Mechanics”, p. 135.

………………………..