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Deep Unity of Classic and Quantum Physics at the Space Thermostat Presence with Technical Applications

Authors:

Abstract

The paper demonstrates deep unity of classic and quantum physics at the space thermostat (ST) presence, which fulfilled all space by the temperature T 0 = 2.73 K. The ST presents itself the Cosmic Microwave Background (CMB). From the main quantum position we consider the ST/CMB as the wave function carrier (" quantum background "). The paper is devoted to ST/CMB medium the classic conservation laws of mass, momentum and energy. We show the soliton like solutions of our classic model correspond to Schrodinger's quantum solutions, demonstrate the atom hydrogen specter and other quantum peculiarities. The paper contains typical technical examples classic/ quantum simulation at the ST presence.
Journal of Applied Mathematics and Physics, 2017, 5, 801-812
http://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI: 10.4236/jamp.2017.54069 April 12, 2017
Deep Unity of Classic and Quantum Physics at
the Space Thermostat Presence with Technical
Applications
M. Ja. Ivanov1, V. K. Mamaev1, Guanghua Zheng2
1Gas Turbine Department, Central Institute of Aviation Motors, Moscow, Russia
2Northwestern Polytechnic University, Xian, China
Abstract
The paper demonstrates deep unity of classic and quantum physics at the
space thermostat (ST) presence, which fulfilled all space by the temperature
T
0
= 2.73
K.
The ST presents itself the Cosmic Microwave Background (CMB).
From the main quantum position we consider the ST/CMB as the wave fun
c-
tion carrier (quantum background). The paper is devoted to ST/CMB m
e-
dium the classic conservation laws of mass, momentum and energy. We show
the soliton like solutions of our classic model correspond to Schrodinger
s
quantum solutions, demonstrate the atom hydrogen specter and other qua
n-
tum peculiarities. The paper contains typical technical examples classic/
quantum simulation at the ST presence.
Keywords
Space Thermostat, Conservation Laws, Schrodingers Solutions, Technical
Applications
1. Introduction
Some achievements in experimental physics and astrophysics during of a few last
decades show that it would be useful additionally to study important back-
ground theoretical aspects for classic and quantum physics. First of all among
such experimental achievements we name the registration of the Cosmic Micro-
wave Background (CMB) with the finite temperature
T
0 = 2.73
K
[1]-[5]. The
second significant achievement is the discovery of Dark Matter/Energy (DME)
[6] [7], which is also called physical vacuum. Now we know that 96% whole
substance in our Universe consists of DME. The baryonic substance accounts to
only near 4%.
How to cite this paper:
Ivanov, M.J., Ma-
maev, V.K.
and Zheng, G.H. (2017)
Deep
Unity of Classic and Quantum Physics at
the Space Thermostat Presence with
Tech-
nical Applications
.
Journal of Applied
Mathe
matics and Physics
,
5
, 801-812.
https://doi.org/10.4236/jamp.2017.54069
Received:
October 25, 2016
Accepted:
April 9, 2017
Published:
April 12, 2017
M. J. Ivanov et al.
802
The paper is devoted to CMB as the Space Thermostat (ST) medium with the
known (no zero) temperature and finite mass particles and presents the classic
conservation laws of mass, momentum and energy [8]. We show the soliton like
solutions of our classic model correspond to Schrodingers quantum solutions
[9], demonstrate the atom hydrogen specter and other quantum peculiarities.
From the main theoretical position we consider the ST/CMB as the carrier of
quantum wave function and quantum phase trajectories (quantum back-
ground). At present time it seems to be indeed that quantum mechanics cannot
be formulated by simply considering a statistical approximation from a classic-
al-like deterministic theory.
Alternatively, it can be also interpreted as the evolution of a quantum flow in a
hydrodynamic form of quantum mechanics [10]. In 1952 Bohm proposed a
physical hidden-variable model [11]-[13] which reproduced the predictions of
the standard quantum theory without violating any of its postulates. The main
goal of this work consists of helping to develop a common united appropriate
classic and quantum physics similar Bohmian mechanics at the ST/CMB pres-
ence.
2. The Space Thermostat: Properties and Parameters
In physics a thermostat is called a greater thermodynamic system, the number of
particles which far exceeds the number of particles in a studied system with her
in thermal contact [14]-[16]. In our case as such thermostat (the large thermo-
dynamic system) we take the real Cosmic Microwave Background (CMB) of our
Universe and dark matter/energy (DME) medium with certain massive particles
(see below), which are holders and carriers of thermal radiation (and, in partic-
ular, CMB). Any smaller size natural or technical system we believe in the ther-
mal contact with the specified Space Thermostat (ST). The considered ST essen-
tially differs from ordinary Gibbsthermostat [14], in which studied systems are
in thermal contact by
T
2.73
K
and have the same temperature.
Our ST is radiate compressible medium and bearer of quantum wave function
and quantum phase trajectories (quantum background”). Following the [8] we
believe that the thermal radiation (and, in particular, CMB) behaves like an ideal
gas with adiabatic factor
γ
= 4/3 and is synonymous in this sense with photon
gas. Coming back from some fundamental ideas (first of all, from recommen-
dations by M. Planck, A. Einstein and L. de Broglie) allows us to indicate a cor-
relation linking energy
E
with mass value
m
, frequency
ν
and temperature
T.
E = mc
2
= hν = kT,
(1)
where
с
: the light velocity;
h
and
k
: the Planck and the Boltzmann constants. The
last equality in (1) is the law of evenly distributed energy on freedom degrees.
Also Equation (1) follows from Plancks distribution in vicinities of maximum
radiation density of an absolutely black body and presents itself Wiens dis-
placement law. The relation (1) allows us to define also the vacuum particle
mass, when
T
≠ 0. The presence of these nonzero mass particles in physical va-
M. J. Ivanov et al.
803
cuum was specified in [8] and it was identified with massive particles of DM
named Hidden Mass Boson (HMB). The same way we propose simulation for
the Dark Energy (DE) [8]. To be short, we change the virtual Planck resonators
in his derivation of the famous formula for absolutely black body radiation den-
sity by real (massive) particles with
m = kT/c
2 (following from the relations (1)).
Also the possibility of radiation (including of electromagnetic waves, similar the
virtual Planck resonators) allows us to consider these real HMB particles as a
classic Hertzs dipoles.
Considering the ST particle concentration
n
and multiply (1) on
n
we can
write
n∙mc
2
= n∙kT
and go to the typical ideal gas state equation
p ≈ ρc
2
= nkT.
(2)
Here
ρ = n∙m
– density,
p
pressure in ST medium. The relation (2) is one of
the mathematical forms of Avogadros law. Now we show that the recommen-
dations by M. Planck, A. Einstein and L. de Broglie (1) may be considered as
another form of Avogadros law and the classic state equation for perfect gaseous
medium (2). The relations (1) and (2) may be used for answers on intrigue ques-
tion so to what comprises about 96% of content of the Universe (
i.e
., what and
why over 70% of the mass-energy content of the Universe is in form of the un-
known vacuum DE, over 20% of the mass is in the form of the mysterious DM).
Get through (1) and (2) refinement more accurate the value for the ST particle
mass at
T
0 = 2.73 K and the perturbation velocity
c
= 2.998 × 108 m/s. We have
obtained
2
2
00 0 0
2 40 4
0
33 3 3 3
22 2 2 2 2
5.6 10 3 10 .
av U U
AA
mv R R mm
E kT T m T RT c
N mN
m kT c kg eV
γ
γγ
γ
−−
= = = = = =
= =×=×
(3)
We calculate the gas constant
R
and the specific heat capacity
cv
и
cp
by
γ
=
4/3
17 17 17
JJ J
0.25 10 ; 0.75 10 ; 1.0 10 .
kg K kg K kg K
v pv
k
R c c Rc
m
= = × = × =+= ×
It should be stressed that the thermal radiation has the classic state equation
( )
, 1, .
v
p RT p e е с T
ρ γρ
= =−=
(4)
The ST particles (3) are the sub-atomic (non-baryonic) material particles
moving almostfree in all directions at different velocities. One half of particles
have positive charge and other half has negative identical in its value electrical
charge [8]. Besides, pairs of the oppositely charged particles from the classic
Hertz dipoles, which have with translational, rotary and oscillatory degree of
freedom. We obtain a liner size of the dipole
l
= 7 × 1020
m
and its charge
q
=
1028
C
. The value of the electric dipole moment
p
=
q·l
= 7 × 1048
C·m
. In spite
of its miniature size we consider that all known properties of electric dipoles are
M. J. Ivanov et al.
804
retained. Thus the medium as a whole is quasi-neutral; however there are so-
called collectiveprocesses possible, such as a local concentration of positive
and negative electrical charges.
3. Noethers Theorem, Lagrangian and Hamiltonian
Mechanics
Looking at some consequences of the thermostat existence the question of the
applicability of the Noether theorem in modern physics should be analyzed. The
Noether theorem declaims that every differentiable symmetry of the action of a
physical system has a corresponding conservation law [17]. By Noethers theo-
rem the symmetries of translations in space and time get to the conservation
laws of momentum and energy within this system, respectively. In the same time
Noether’s theorem takes place only for dissipative less isolate systems. In our
case with ST and energy dissipation on its level we have no possibility to use
Noether’s theorem. The same situation takes place with the Lagrangian and the
Hamiltonian mechanics.
There is used as the standard language of the particle physics to express in
terms of Lagrangians [18]. To give the flavor of the general theorem, a version of
the Noether theorem for continuous fields in four dimensional space time has
been given. The requirement of invariance of the Lagrangian for local gauge
transformations is the original principle of all modern physics theories of a mi-
crocosm. The gauge transformation is rotation at arbitrary angles around the
same axis in the Minkovsky 4-space. Here we have local time’s arrow decline and
rotation. It is one of the great differences between perspective new horizons and
modern physics. We should include into account some principal limitations of
the Lagrangian formalism and quantum field theories, which connect with
times arrow (real times arrow has no possibility to decline at any side and
twisting). The real physics has only the one-way direction arrow of time
(without any rotation).
Further in our paper we analyze deep unity of classic and quantum physics at
the presence of the external ST with energy changing (first of all, the energy dis-
sipation on the level of ST). A few going forward we would like to emphasize
that the growth of our systems entropy characterizes the amount of scattered
(absorbed by ST) energy. Now we consider some items of Bohmian quantum
mechanics.
4. Some Bohmian Mechanics Background
The wave function
ψ
supplies the quantum system with dynamical information
on each point of the associated configuration space at each time moment. With-
in quantum presentation of Bohmian mechanics [11]-[13] this information is
encoded in its phase, as can be seen through the transformation relation
( ) ( )
( )
,
12
, ,e
iS r t
rt rt
ψρ
=
, (5)
where
ρ
and
S
are the probability density and phase of
ψ
, respectively, both being
M. J. Ivanov et al.
805
real valued quantities. This relation allows us to pass from the Schrödinger equa-
tion
(6)
to the system of coupled equations [11]
0,
S
tm
ρρ
∂∇

+∇⋅ =


(7)
( )
0,
2
S
SVQ
tm
+ ++=
(8)
2
2 2 12 2 2
12
1
2 42
Qmm
ρ ρρ
ρρ
ρ


∇ ∇∇

=−= −





(9)
is the so-called quantum potential. Equation (7) is the continuity equation,
which describes the ensemble dynamics,
i.e
., the motion of a swarm of trajecto-
ries initially distributed according to some
ρ
0. Equations (8) and (9) govern the
motion of individual particles, in particular, the quantum Hamilton-Jacobi Equ-
ation (8) accounts for the phase field evolution ruling the quantum particle dy-
namics through the equation of motion
u Sm
= ∇
.
(10)
This relation indicates that one can define a local velocity field on each point
of the system configuration space and, by integrating it in time, to obtain the
corresponding trajectory.
In our study we get some next steps in Bohmian mechanics direction, in
which the ST/CMB presents as the wave function carrier (quantum back-
ground). We consider
ρ
as real ST medium density and
u
as real ST medium
velocity. By that we can write conventional equations of discontinuity and mo-
mentum (see the next section). Further the relations (1) and (2) allow introduc-
ing also pressure
p
and temperature
T
in our consideration of quantum Boh-
mian mechanics at the ST presence. Following classic mechanics we have possi-
bility now to use additionally the equation of energy conservation.
5. Some Classic Mechanics Background
We present the common conservation laws for the case of the ST presence as the
two components model of a gaseous and radiation (quantum) medium [8].
From the main quantum position we consider the ST/CMB as the quantum wave
function carrier (quantum background). There are used the index
g
for con-
ventional gas and the index
f
for radiation components of medium (for example,
for densities
ρg
and
ρf
). For the one velocity model the values of velocity com-
ponents
u
,
v
,
w
at the axis
x
,
y
,
z
are the same for each medium components. The
integral conservation laws are presented as [19] [20] for the volume
( )
t
ω
with
the boundary
( )
t
γ
M. J. Ivanov et al.
806
( ) ( )
( ) ( )( )
( ) ( ) ( )( )
2
d d,
d d d,
1dd d
2
,
d.
kk
tt
k kk
t tt
k k k kk k
t tt t
dq
dt
dp
dt
dq p K gradT L
d
k gf
t
ωω
ω γω
ω γγ ω
ρω ω
ρω γ ω
ρ εω γ γ ω
=
=−+

+ =− ⋅+ ⋅+


=
∫∫∫ ∫∫∫
∫∫∫ ∫∫ ∫∫∫
∫∫∫ ∫∫ ∫∫ ∫∫∫
u nr
un n
(11)
Here
2
q
- the square of the velocity vector
u
and
( ) ( )
, .
g gf f g g f gf g f f
L CTT Q L CTT Q
′′
= −+ = −+
Energy conservation laws are written for heat transfer gas and radiation com-
ponents (the second terms in the right side of these equations,
g
K
and
f
K
correspondently thermo transfer coefficients for gas and radiation parts). The
last terms in the right side of initial energy equations describe an energy ex-
change between gas and radiation parts (the space thermostat). The terms
g
Q
and
f
Q
are an additional energy souses.
6. Some Examples of Quantum Hydrodynamic Analogy
The achievement of a mutual systematic understanding of quantum and classical
phenomena has been considered in [10]-[13]. In these papers, the standard
quantum mechanics is derived as the deterministic limit of the stochastic hy-
drodynamic analogy. Now there is considered deep coordination of classic and
quantum solutions on the example of a hydrogen atom specter and screening
spaces of elementary particles.
At first we analyze a hydrogen atom specter (in particular, the Ballmer series,
Figure 1) on the base of the system (11). The main steps of classic solution rea-
lization for a hydrogen atom specter are presented in [8] [21]. We also give
Figure 1. Decomposition task of initial contraction for sequence of solitons for the Ball-
mer series.
M. J. Ivanov et al.
807
calculated solutions of the decomposition task of initial contraction for sequence
of soliton solutions (Figure 1).
Detail differential ST simulation allows proposing also internal structures of
molecules and atoms in chemical physics, based on electron, proton and neutron
particle thermodynamics. By that we follow to
L
. de Broglies and D. Bohms
methodologies [11]-[13]. This and next sections consider in detail important ST
items from elementary particles up to nuclear simulation.
We can derive equations, which describe the distribution of electrical poten-
tial and concentration of particles in polarized spaces of electrons, positrons,
protons and neutrons [8]. Equation for electrical potential
φ
in polarized ST can
be written in form
22,D sh
ϕϕ
∆=
where
φ
is related to its characteristic value
0
Te
ϕ
=
;
22
0
4
D T ne
π
=
: the
Debye radius;
e
: the value of ultra-elementary electrical charge;
0
n
: characteris-
tic concentration of DM particles (HMBs).
In case of spherical symmetry the following equation can be written
2
2
2
2.
Dd d
r sh
dr dr
r
ϕϕ

=


(12)
We shall bring typical solutions of the Equation (12) for polarized electron
space (Figure 2). A principal important particularity of distribution presented is
potential pit and barrier on external border of polarized space with distribution
φ
(
r
) break [8]. Induced by electron core electrical charge is concentered at the
external boundary. Due to charge symmetry the provided results are also correct
for the structure of polarized positron space (in this case the sign of
φ
should be
changed).
Figure 2. Potential distribution in electron screening space for two
D
.
M. J. Ivanov et al.
808
The scheme of polarized electron space is shown on Figure 3(a). There are
shown the electron core with the radius
r
~ 10 17 m, part of the polarized
space (the screening sphere filling HMBs) with the radius
r
~ 5 × 1011 m.
It is possible to model protons and antiprotons in such way. It is supposed
that whole positive charge of proton is situated in its center (Figure 3(b)). The
dimension of this center is less than 10−17 m. It is surrounded by sphere of pola-
rized liquidDM. Dimension of this area is about 0.8 × 10−15 m. It is defined by
characteristic size of proton. There is a sphere of gaseousDM around the liq-
uidone. The modeling of proton bi-layer is possible via integration of Equation
(14) for electrical potential with the help of diverse equations of state (for liquid
and gaseous phases of DM). The obtained solutions are similar to the electron
space solutions [8]. It should be noted that there are two potential holes and two
barriers with their radiuses about
r
0.8 × 10−15 m and
r
≈ 5 × 1011 m. The ob-
tained proton structure is stable. Antiproton structure is similar with proton due
to charge symmetry (except for
φ
sign).
Following our methodology we show now shortly the scheme of neutron as a
positive nuclear of proton with Debye screening (up to
r
0 = 0.8 × 10−15 m) by liq-
uid layer of HMB dipoles and stationary electron, presenting on boundary of
Debye region (Figure 3(c)). Decay of neutron gives naturally proton, electron
and antineutrino, which presents itself isolated soliton in DM gaseous medium.
Exact soliton solutions for neutrino and antineutrino were obtained in [21].
The model of hydrogen atom consists of positive charge in the center (
r
< 0.4
× 10−16 m) and two layers of liquid (0.4 × 10−16 <
r
< 0.8 × 10−15) and gaseous (0.8
× 10−15 <
r
< 0.5 × 10−10) polarized DM space (Figure 4). There is also stationary
electron at the external boundary of polarized space (
r
= 0.5 × 10−10 m). This
motionless state of electron is achieved by potential distribution in electron and
proton models and equilibrium between
grad φ
and
grad p
.
The existence of polarized space around atoms with their spherical shape (van
der Waals spheres) has been proved by chemistry of crystals and scanning probe
microscopy. It is possible to manipulate the separate spheres with needle of tun-
nel microscope. We can move these spheres or build some figures of them.
7. Some Technical Applications
Some basic complexities of designing high temperature air-breathing engines
(a) (b) (c)
Figure 3. Scheme of screening spaces of electron (a),
proton (b) and neutron (c).
M. J. Ivanov et al.
809
Figure 4. Scheme of hydrogen atom.
relate to origin of quantum effects (so-called “unexpected” heat) of working
process. This additional heat essentially complicates coordination of engine basic
components (for a turbojet - compressor, combustor and turbine, for a high-
speed air-breathing engine - air inlet, isolator, combustor, and nozzle). In our
simulations we include into account unexpected heat as radiate heat and heat
transfer to ST using system (11).
Propulsion combustion chambers with temperatures above 3000 K, scramjet
chambers with temperatures near 2500
K
and other jet engine combustors with
temperatures near 2000
K
are strongly influenced by radiate heat transfer. Ra-
diate heat loads depend on temperatures fourth power. Radiate heat transfer
simulation usually requires the solution of radiate transfer equation, which de-
pends on special, directional and spectral variables. Examples such types model-
ing were realized for rocket combustion chambers and have good enough
progress. At the same time the practice realization of scramjet propulsion has
great difficulties. One of these reasons is a detrimental operation mode called
unstart.
Using our theoretical model we would like to explain the similar unstart
mode, which dont allow us to realize supersonic burning in scramjet with posi-
tive thrust. This operation is the unstart mode, when radiation and shocks
(pseudo-shocks system) are destroying supersonic flow in combustor channel.
Figure 5 presents typical results this mode simulation and shows Mach num-
ber and pressure counter lines inside channel without burning (a) and with
burning (b). In the last mode heat addition destroys supersonic burning and
shock wave system is located in the inlet zone. Our simulation based on system
(11) is relevant to high-class models based on real 3D geometry of flow path.
The typical examples of investigation of steady and unsteady working modes
of the bypass gas turbine engines are shown in Figure 6. We present simulation
of whole flow passages for bypass gas turbine engine with afterburning trust 80
kN.
Figure 6 illustrates streamlines and Mach number counter distributions in
whole flow passage of multi regime engine with bypass ratio 0.17 and the 3 stag-
es fan, 6 stages high pressure compressor, annular main combustor, one stage
M. J. Ivanov et al.
810
high and low pressure turbines and afterburner. Additional pressure losses in
main combustors demands very accurate determination high pressure turbine
cross section and correlation of power for compressor driving. Here we have the
decreased flow capacity of the compressor and led finally to increased turbine
inlet temperatures.
At the last we present solutions of external reentry shuttle problems. Flow
structure and surface temperature distribution for two typical shuttles show on
Figure 7. On simulate regime the flight Mach number equals 5 with the angle of
attack 5˚. These calculations use the equilibrium approach based on (11).
(a)
(b)
Figure 5. Mach number lines without burning (a) and pressure counter lines with burn-
ing on unstart mode (b) inside high speed channel.
(a)
(b)
Figure 6. Streamlines (a) and Mach number (b) counter distributions in whole flow pas-
sage of multi regime bypass GTE.
(a) (b)
Figure 7. Flow structure (a) and surface temperature (b) distribution for two typical
shuttles.
M. J. Ivanov et al.
811
8. Conclusions
Deep unity of classic and quantum physics at the space thermostat (ST) pres-
ence, which fulfilled all space by the temperature
T
0 = 2.73
K
,
has been demon-
strated
.
Real physics is based on the principle of Galileos relativity (invariance)
and Newtons laws of motion. After linearization procedure we get quantum
modern physics. The modern theoretical physics bases on the Lorentz inva-
riance. The Noether theorem, the formalism by Lagrange and Hamilton cant by
right in such type cases of opened systems with the ST presence.
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Conference Paper
The full class of exact solutions of the two-fluid plasmadynamic equations is presented. A few important solutions are classical and particle-like solitons, which can propagate to very long distances without energy losses. The other solutions have cnoidal or ordinary wave structures. Presented exact solutions may be used for simulations of some space phenomena. As examples we consider the solutions for gamma-ray bursts, ultra high energy cosmic rays and cosmic jets in the frame of the Newtonian approach. Also we discuss the extended Lorentz invariance in the linear case of wave and particle dynamics and some its consequences.
Article
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.
Article
Measurements of the effective zenith noise temperature of the 20-foot horn-reflector antenna (Crawford, Hogg, and Hunt 1961) at the Crawford Hill Laboratory, Holmdel, New Jersey, at 4080 Mc/s have yielded a value of about 3.5 K higher than expected. This excess temperature is, within the limits of our observations, isotropic, unpolarized, and free from seasonal variations (July, 1964 - April, 1965). A possible explanation for the observed excess noise temperature is the one given by Dicke, Peebles, Roll, and Wilkinson (1965) in a companion letter in this issue.
Article
In this paper, we propose a physical model leading to the causal interpretation of the quantum theory. In this model, a set of fields which are equivalent in many ways to a conserved fluid, with density |psi|2, and local stream velocity, dxidt=∇Sm, act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid. By introducing the hypothesis of a very irregular and effectively random fluctuation in the motions of the fluid, we are able to prove that an arbitrary probability density ultimately decays into |psi|2. Thus, we answer an important objection to the causal interpretation, made by Pauli and others. This result is extended to the Dirac equation and to the many-particle problem.
Article
The location, amount, and nature of the dark matter (DM) in the universe are addressed. Observational results are reviewed which show that the DM matter is less centrally concentrated than is the luminous matter, that DM is clumped about galaxies, is more extended than luminous matter, and that the gravitational potential of DM is more nearly spherical than flat. It is shown how dynamical arguments about dark matter lead to a value of omega = 0.2, and the theoretical implications for the amount of DM of taking omega = 1 are considered. Possible forms for the DM and their consequences for cosmology are briefly discussed.
Article
Es wird gezeigt, da man die Schrdingersche Gleichung des Einelektronen-problems in die Form der hydrodynamischen Gleichungen transformieren kann.
Article
Die gesamte Energie, die in Form von Ultrastrahlung an der Grenze der Atmosphre auf das Quadratzentimeter in der Sekunde einfllt, ergibt sich zu 3,53 10–3 erg cm–2 sec–1. Auf die ganze Erdoberflche ergibt diese 2,4 Millionen PS. Dieser Energiestrom ist nahe gleich demjenigen, der in Form von Licht und Wrme von den Fixsternen kommt. Die Auswirkung dieser Tatsache auf die Temperatur des interstellaren Raumes wird diskutiert.