Re: AdjOPS versus W/G scatter plot for 2009

Tentative quantum calculation for a tubular sphere contacting a
qualitative sphere and resulting trajectory postulates. (How to hit a
home run)

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The bizarre effects of Special Relativity, introduced by Albert
Einstein in 1905, are manifest as time dilation, length contraction,
and varying mass. Thus, as an object moves faster, time (t) passes
more slowly for it, its length in the direction of motion (l) shrinks,
and its mass (m) increases. At the velocity of light, time would stand
still, length in the direction of motion would shrink to zero, and
mass would become infinite. These distortions occur so that the
velocity of light will always appear to be a constant (c), regardless
of relative motions and one's own inertial frame of reference (i.e.
coordinates at a constant velocity). In Einstein's view, this simply
preserves the absolute universality of the laws of nature, since the
velocity of light turns out to be an artifact of Maxwell's Equations
for electromagnetic interactions. Maxwell's Equations, and so the
velocity of light, are equally valid for every inertial frame of
reference (the original "Galilean" form of Relativity), which means
however it is that one is moving, as long as one is moving at a
constant velocity (which means the same speed and direction), the
velocity of light (in a vacuum) will be a constant.

The change in mass itself explains why ordinary objects cannot attain
the velocity of light: They would have an infinite mass there and so
would need an infinite force to accelerate themselves to that
velocity. This circumstance is not always appreciated, even by great
science fiction writers like Robert Heinlein, who has one character in
a story ask why we can't go faster than the velocity of light and the
answer is given that "we don't know" but "we'll see when we get
there". Another science fiction story speculates that a ship hitting
the velocity of light would be bounced back into the past.

The original formulae for the transformation of coordinates, from one
frame of reference moving past another in the x axis [as given in The
Universe and Dr. Einstein, by Lincoln Barnett, Mentor, 1948, 1950,
1957, footnote pp. 54-55, and Michael Berry, Principles of cosmology
and gravitation, Cambridge U Press, 1976, pp. 35] are at left. These
are the "Lorentz Transformations," proposed prior to Einstein by the
Dutch physicst H.A. Lorentz to account for the anomaly of the
Michelson-Morley experiment in 1881, where the velocity of light had
not varied regardless of the direction in which it was measured. The
form of the equations is for the x coordinate, and time (t). The y and
z coordinates are unaffected. The equation is also given for the
addition of velocities (v). Lorentz did not know, however, why this
effect had occured, so these were just ad hoc mathematical
descriptions. Einstein provided the reason. Simpler equations for the
length (l) contraction for the object, the dilation for a unit of time
(t), and for the increase in the mass (m) of the moving object are all
given at right [versions given in Physics, The Foundation of Modern
Science, Jerry B. Marion, John Wiley & Sons, 1973, pp. 197-205].

The Lorentz Transformations are not mathematically very difficult, but
they do not transparently relate space and time to each other, and
they do not relate to any intuitive sense of why this all would
happen. Another equation that does provide a better sense of things,
called the "Separation Formula," is given at right, where s is the
"separation" or "proper time," which is the elapsed time for a moving
object, while t, x, y, and z are the changes in the coordinates in
time and space as an object moves [cf. Michael Berry, Principles of
cosmology and gravitation, Cambridge U Press, 1976, pp. 48-49, and
Roger Penrose, The Emperor's New Mind, Oxford U Press, 1989, pp.
195-196]. At first this may not seem like an improvement over the
previous equations. But, as with many equations, it can be simplified.
First of all, the Greek delta, which indicates the change in the
coordinates, can be left out, as in the first equation at left, giving
us variables for the movement in each of four dimensions. Then we
should take into account that the Separation Formula is really an
extension of the Pythagorean Theorem. The basic Pythagorean Theorem is
in two dimensions (x2+y2), but it can be generalized into three
dimensions (x2+y2+z2). In four dimensions we get an anomaly, since all
the dimensions of space are added together, while they are all
subtracted from the dimension of time (t2-((x2+y2+z2)/c2)). This all
by itself is revealing, since it answers the question whether time is
treated exactly like a dimension of space in Relativity. It isn't.
Since what concerns us is the relation between time and space, the
separation formula can be simplified by replacing the x2+y2+z2 term
with its simple equivalent, r2 (r = "radius"). This can then be
further simplified by picking the right units. The velocity of light
can be set equal to one with the choice of light years (LY) and years
(y). The velocity of light is, indeed, one light year per year (that
is the definition of a light year). With all those simplifications,
the Separation Formula ends up as a very simple equation indeed:
s2=t2-r2. Although the velocity of light term has been elimated, it
should be remembered that its units of velocity are there and that the
"separation" comes out in units of time.


With this simple equation, we can inspect some Relativistic effects.
The graph at left, in vertical units of years and horizontal units of
light years, shows two trips in space-time. The red path is an object
(a spaceship) moving at 60% of the velocity of light--it travels 3
light years in 5 years. The blue path is the movement of light
itself--5 light years in five years. The green line is a stationary
object in this inertial frame of reference--we could think of it as
the Earth, from which the spaceship travels 3 light years away. We
discover from the Separation Formula, that while 5 years have elapsed
on Earth, the ship has arrived at its destination in only 4 years,
according to its clocks. Meanwhile, the ray of light experiences no
elapsed time--it's separation is zero (25-25=0). That is called "light-
like" separation, in comparison to the "time-like" separation of the
other object. It can be seen from this that the longest path in space-
time is the shortest separation. On the other hand, what if an object
had gone faster than light? If so, the r term would be larger than the
t term, resulting in a negative number, and the separation would be
the square root of that number. The separation would therefore be an
imaginary number (-1 = i). That is called "space-like" separation, and
it signals us that such motion is impossible. You can't get there from
here in space-time, though this does not tell us why--we need to know
about the change in mass for that. If the light ray is rotated around
the vertical axis, this would generate a cone, a "light cone," which
defines the area in space-time accessible from the point at the
origin.

The previous graph gives rise to a paradox. If the spaceship is moving
at a constant velocity away from the Earth, then it has its own
inertial frame of reference, in relation to which it is the Earth that
is moving away from the ship. Thus, time should be passing more slowly
on Earth than in the ship. The graph at right illustrates this
situation. The red line represents the ship, at "rest" in its own
inertial reference frame, experiencing the passage of 4 years. At a
distance of 3 light years, the Earth (green line) would only have
experienced the passage of 2.65 years (s=(16-9)=7=2.65). This is
peculiar, since back on Earth, everyone has experienced the passage of
5 years (purple line). What this paradox illustrates is that in
Special Relativity simultaneity is relative. In the simultaneous space
for the spaceship, 2.65 years have elapsed on Earth and 5 years there
would be in the future, while in the simultaneous space for Earth,
defined by the thin diagonal purple line to the spaceship, 5 years
have passed. A graph of simultaneous space for Earth, placed on the
inertial frame of reference for the spaceship, would be systematically
distorted, a "Poincaré motion," so that simultaneous events in one
frame of reference will be in the future or the past for another [cf.
Roger Penrose, The Emperor's New Mind, pp. 199-200].

The paradox of simultaneity can be solved by reuniting the objects in
a single frame of reference. For this to be done, of course, at least
one object must experience an acceleration. This provides the simple
rule for Special Relativity: The "absolute" time we end up with is
determined by the unaccelerated frame of reference. Thus, in the graph
at right, we are back to the Earth's frame of reference, since the
spaceship has turned around (which is a change in velocity and so an
acceleration) and returned to the Earth. It returns at 60% of the
velocity of light again and so experiences a separation of 4 years.
Overall, 10 years have elapsed on Earth while only 8 years have
elapsed for the spaceship. Simultaneity is reconciled. Longer trips at
higher velocities produce more dramatic differences. But if the
spaceship did not return to Earth, there would be nothing special
about the Earth's inertial frame of reference and simultaneity between
the ship and Earth would not be reconciled.

The only truly absolute frame of reference, velocity, and time in all
of this is that of light. The only absolute time is none--"light-like"
separation. This sounds like nothing less than Platonism, where that
which truly exists is eternal and unchanging. At the velocity of
light, no time will pass at all for the entire history of the
universe, which sounds like Socrates imagining that, if death is like
dreamless sleep, then "all of eternity will be no more than a single
night."


There is actually another paradox in the first diagram above
(reproduced at left): If, on the spaceship, it seems that 4 years
have passed, then the spaceship, to travel 3 light years in 4 years,
had to have been going 75% of the velocity of light (3LY/4y), not 60%
(3LY/5y). The 60%c velocity is only the way the velocity would appear
in the Earth's inertial frame of reference. However, this answer will
not do; for if the spaceship went 4 light years in 5 years, or 80% of
the velocity of light (4LY/5y), as in the diagram at right, only 3
years would have passed for the ship ((52 - 42) = 3), which would mean
that the ship went 4 light years in 3 years, at a velocity then of
167% of the velocity of light (4LY/3y). But the whole point of Special
Relativity is that it can't go faster than light, however we measure
it. What should handle this, however, is that as the spaceship is
moving, the distance (4 LY), as measured in the Earth's frame of
reference, contracts (using the equation l = lo(1-v2/c2)): 4 (1-(0.8)
2) = 4(1-0.64) = 40.36 = 4(0.6) = 2.4. The ship sees the distance it
traverses as only 2.4 light years--at high velocities the universe
literally becomes smaller, at least in the direction of motion. 2.4
light years in 3 years is, indeed, 80% of the velocity of light (2.4LY/
3y = 0.8LY/y). Similarly, the ship that goes 3 light years in 5 years
(60%c), experiences that distance as 2.4 light years also, which also
confirms the velocity of 60%c (2.4LY/4y = 0.6LY/y).




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Philosophy of Science

Home Page


Copyright (c) 1998 Kelley L. Ross, Ph.D. All Rights Reserved

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Historic Equations
in Physics and Astronomy

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Being more of a spectator on mathematics than a participant, my
favorite historic equations are those that the ignorant layman (like
me) can play with just by plugging in the right values and seeing what
happens. As it does happen, this doesn't get one very far with the
more recent and more important equations, which often feature complex
operators whose use requires a mathematical education in itself and
whose solutions are often still matters of contemporary exploration
and controversy. Here I give both kinds, and the mathematically
literate are welcome to sneer at my rudimentary understanding of the
more sophisticated material. What I can now do, however, is refer the
reader to a recent examination of historic equations by the
specialists themselves. Just published is It Must be Beautiful, Great
Equations of Modern Science, edited by Graham Farmelo [Granta Books,
London, New York, 2002, 2003]. Here we have the likes of Roger Penrose
and Steven Weinberg doing what I would really like to be doing here.
Farmelo, however, has not chosen an essay about Kepler's Laws, which I
think are more my speed and that I think I explain rather well here.

Constants for these equations are given in the table at "Physical
Constants."

Equations for Kepler's Laws: Kepler's First Law is that the orbits of
the planets are ellipses, with the Sun at one focus. At right is a
general equation for a conic section (circle, ellipse, parabola, or
hyperbola) in polar coordinates, together with commonly used physical
dimensions for the orbits of planets, asteroids, comets, etc. The
"major axis" (2a) is the longest line that can be drawn in an ellipse;
the "minor axis" (2b) is the shortest line that can be drawn through
the center. The shortest distance from the focus to the curve (q) is
"perihelion" for a planet, "perigee" for an object in orbit around the
earth, or, in general, "periapsis," "closest approach," for any kind
of orbit. The longest distance from a focus to the curve (Q) is
"aphelion" for a planet, "apogee" for an object in orbit around the
earth, or, in general, "aphapsis," "furthest approach," for any kind
of orbit. The distance across the curve, through the focus, at right
angles to the major axis (2d), is the "latus rectum". The periapsis
(q) and semi-latus rectum (d) are physical dimensions of any conic
section, not just ellipses-- though a hyperbola might be thought of as
an ellipse with a negative major axis and an imaginary minor axis. The
"eccentricity" (e) defines the shape of the curve: e=0 is a circle;
0<e<1 is an ellipse; e=1 is a parabola; and 1<e is a hyperbola.
Finally, the angle made by the radius (from focus to orbiting object)
with the point of periapsis is the "true anomaly" ().

Kepler's Second Law is that the radius from the focus to a planet
sweeps out equal areas in equal times. The mathematics of this is
still not easy to deal with. Given the mean angular motion (n), which
would be 360o divided by the period of the orbit (in radians: 2/p),
and the time elapsed since the "epoch of perihelion" (T, a benchmark
time when the planet was at perihelion), the "Mean Anomaly" (M) can be
calculated, which would give the angle with perihelion if the planet
had been moving with a uniform angular speed. With the mean anomaly in
hand, it is not the true anomaly that is calculated first, at least
for elliptical and hyperbolic orbits, but the "eccentric anomaly" (E),
which is the point on a superscribed circle that corresponds to the
point where the planet is on its curve. This is illustrated in the
diagram at left. Calculating the eccentric anomaly from the mean
anomaly is difficult because the equation M = E - e*sin E (for
ellipses) cannot just be solved for E. Instead, we can write the
equation as M + e*sin Em = E and, starting with Em = M, solve the
equation over and over, substituting each new approximation of E for
Em. This permits calculating E as accurately as desired. With the
eccentric anomaly determined, the true anomaly () can be calculated
directly. Equations are given at left for circular, elliptical,
parabolic, and hyperbolic orbits. For circular and parabolic orbits,
the true anomaly can be calculated directly either from the mean
anomaly or just from the time since perihelion (t - T). The hyperbolic
equations work much like the elliptical ones, usually just with some
opposite signs and hyperbolic functions.

The diagram at right gives the conventions for the physical
characteristics of a planetary orbit. The celestial equator is the
Earth's equator projected onto the sky. The apparent path of the Sun
in the sky, however, the ecliptic, is at an inclination () to the
equator. The point where the Sun crosses the celestial equator and
enters the northern hemisphere is the Vernal Equinox, providing the
benchmark to celestial longitude, which is then measured along the
ecliptic to the East, the direction in which the Sun and planets move
against the background of the stars. The first physical feature of an
orbit is the Ascending Node, the point where the object (planet,
asteroid, comet, etc.) crosses to the north of the ecliptic. The
"longitude of the ascending node" () is thus the celestial longitude
of that point, measured East from the Vernal Equinox. At the ascending
node there is also the angle of inclination of the oribt (i) to the
ecliptic. Angles of inclination are small for the planets but can be
very large, up to 90o, for asteriods and comets. If the motion of the
object is retrograde, i.e. from East to West, the angle of inclination
will be larger than 90o. Once on the plane of the orbit, the angle of
longitude is measured to the East until the perihelion (perigee, etc.)
point is reached. That angle is the "argument of perihelion" (). The
longitude of the ascending node and the argument of perihelion can be
added together for the "longitude of perihelion" (), but the two
angles are not measured in the same plane, so the angle may differ
from the true angle of separation between the Vernal Equinox and the
perihelion. A similar caution holds for the "true longitude" of an
object, which is the sum of the longitude of perihelion and the true
anomaly (+).

Kepler's Third Law is that the square of the period of a planet's
orbit is directly proportional to the cube of the semi-major axis. By
the same token, the square of the mean angular motion is inversely
proportional to the cube of the semi-major axis. However, the equation
for the period can be considerably simplified by choosing the right
units. If (Earth) years and Astronomical Units (the semi-major axis of
the Earth's orbit) are chosen, then the rest of the equation can be
put equal to 1, which means that the Third Law can simply be written:
p2 = a3. This was useful in Kepler's day when the true physical
distances, let alone the Gravitational Constant, were unknown. It is
still useful today.


Newton's Equation for Gravity: The force exerted between any two
bodies with mass -- "G" is the gravitational constant; "m1" and "m2"
are the masses of the two bodies; and "r" is the distance between
them. Below the equation for force is the equation for the
acceleration of gravity produced by a single body. The acceleration of
gravity on the surface of the Earth is 9.8 m/s2 (aE = g).

Coulomb's Law: The force exerted between two electrical charges --
"k" is the electrostatic force constant; "q1" and "q2" are electrical
charges; and "r" is the distance between the charges. Electrical
charges can be positive (+) or negative (-). Opposite charges ("+" &
"-") result in a positive (attactive) force; like charges ("+" & "+"
or "-" & "-" ) result in a negative (repulsive) force. The second
version is the way I see this equation written now, with the
"permittivity of empty space" () used instead of the electrostatic
force constant. I have not yet seen it explained what the
"permittivity of empty space" is supposed to mean. Factoring out 4
must have some relevance to space.

Bode's Law: Stated by J. Bode in 1778 but discovered by J. Titius in
1766 -- and so now frequently called the "Titius-Bode Law." A simple
numerical sequence that produces the mean distance from the Sun in
Astronomical Units for most the planets. The construction of the
series begins with 0 and 3; 3 is then doubled as many times as desired
for subsequent terms in the series; 4 is added to each number; and
then each number is divided by 10. The results are shown in the
following table at left, up to the tenth term. The period of the orbit
can be calculated using Kepler's Third Law (p2 = a3), where the mean
distance (a) is in Astronomical Units and the period (p) is in Earth
years. The results were very close to the planets known at the end of
the 18th century, up to Uranus, with one conspicuous gap: There was
no planet corresponding to B5. However, on New Year's Eve of the year
1801 the first asteroid, Ceres, was discovered, orbiting at exactly
2.8 AU. Neptune and Pluto, discovered later (1846 & 1930) did not fit
Bode's Law, however, as can be seen from the actual physical data in
the table at right.


Maxwell's Equations: In The Emperor's New Mind (p. 186) Roger Penrose
shows us Maxwell's equations, though I can't say that I understand
much of the mathematics. Penrose says that the "curl" and "div"
functions are "certain combinations of partial derivative operators,
taken with respect to the space coordinate." The two equations on the
left relate the rate of change (the partial derivative with respect to
time) of the electric field (above) and the magnetic field (below) to
changes in the magnetic field and electric current (above) and to
changes in the electric field (below). At least the symmetry between
time on the left and space on the right is evident, though I can't say
that the meaning and beauty of the equations is otherwise obvious to
the non-mathematician (like me). The equations on the right Penrose
says are versions of the inverse square law (above) for the electric
field and, for the magnetic field, the fact that there are no isolated
magnetic poles (below). Paul Dirac predicted magnetic monopoles, but
none have yet turned up. Penrose doesn't explain what I think is an
important aspect of equations like this: the units. Return to
"Relativity and Separation Formula"




Planck's Law: The energy of Black Body radiation for a given
temperature and wavelength. A "black body" is called that because it
does not reflect any radiation; it only emits radiation because of its
temperature. Stars are natural black bodies, though the effect can be
duplicated by heating a box with only a small hole in it. The light
that comes out of the box from the hole is black body radiation. --
"h" is Planck's Constant; "k" is Boltzmann's Constant; "c" is the
velocity of light; "" is the wavelength in meters; "T" is the
temperature in Kelvins; and "e" is the base of natural logarithms,
Napier's Constant.

Wein's Law: The wavelength at which Black Body radiation for a given
temperature peaks. -- "c2" is the "second radiation constant"; and "T"
is the temperature in Kelvins.

Stefan-Boltzmann Law: Power emitted by a Black Body per unit area for
a given temperature. Given the temperature and size of a star (and so
its surface area), its total power could be calculated. -- "" is the
Stefan-Boltzmann Constant; and "T" is the temperature in Kelvins.


Einstein's Equation: The upper equation at right is Einstein's field
equation for gravity. Roger Penrose explains and discusses it in
detail in It Must be Beautiful, Great Equations of Modern Science (pp.
180-212), along with the Separation Formula and other things. As I
understand it, whereas a "vector" is a quantity with a direction (one
dimension), like velocity, a "tensor" is quantity expressed in two
dimensions (a "scalar" quantity is dimensionless). It is nice to read
that when Einstein got interested in tensor calculus, he "had to
enlist the help of his colleague Marcel Grossmann to teach him" (p.
199). This is the basis of General Relativity, where space-time
curvature, Black Holes, the Big Bang, and the whole business comes
from. The second equation at right is Einstein's Equation with the
addition of the "cosmological constant." The negative signs in both
equations indicate the attractive nature of gravity. The positive sign
on the cosmological constant makes it repulsive. Einstein famously
said that this was the "biggest mistake" of his life, adding a
repulsive force in order to get a static universe; but it now turns
out that there may very well be a cosmological constant, one big
enough to make the expansion of the universe accelerate.


Schrödinger's Equation: Roger Penrose discusses Schrödinger's
Equation in The Emperor's New Mind (p. 288), which is the
deterministic equation for the undisturbed wave function, , in quantum
mechanics. The application of the equation ends when the wave function
is "disturbed" by observations, or even inferences, that can determine
the location of particles. Then the square of the wave function is
interpreted, as by Heisenberg, as the probability distribution of
where the particle can be found. Schrödinger, like Einstein, didn't
like that indeterministic aspect of quantum mechanics: Indeed, he
said "I don't like it, and I wish I'd never had anything to do with
it." Here the rate of change (the partial derivative with respect to
time) applies to the "state vector," , of the wave function. The
"state vector" notation is discussed by Penrose on page 257. This is
multiplied by the imaginary (i) "reduced" Planck's Constant (). (My
non-mathematician's impression is that imaginary numbers often occur
in equations of periodic functions [note], but Penrose does not
discuss the significance of the imaginary number here.) That whole
side of the equation is all equivalent to the "Hamiltonian" of the
state vector of the wave function. Penrose explains (p. 288) that the
"classical Hamiltonian" represents the total energy of the system but
that the "quantum Hamiltonian" substitutes partial differential
operators with respect to the momentum for the simple occurrence of
momentum in the original Hamiltonian. Thus, as in Maxwell's equations
above, a great deal of the mathematics of this equation is presupposed
by the symbolism. The simplicity of the equation thus conceals a level
of mathematical sophistication that is rarely explained, in any way,
to the lay public. Penrose's own effort to intelligibly present the
details of much of this stuff, although limited and not always
successful, is therefore exemplary, and in stark contrast to Stephen
Hawking's A Brief History of Time, which only contained, on the
publisher's advice, one equation (E = mc2).

Dirac's Equation: The equation at right is Paul Dirac's equation for
the electron. Frank Wilczek explains and discusses it in detail in It
Must be Beautiful, Great Equations of Modern Science (pp.132-160, with
explanation of the terms in the equation in an appendix, pp.268-270).
Dirac's equation reconciles Schrödinger's Equation with Relativity in
the description of the electron. Wilczek mentions that the spin of
electrons (up or down, right or left) was derived by Dirac in a
natural way from the equation, which also resulted in the prediction
of anti-particles, which shortly thereafter were actually observed.
The way the equation is written, however, the variable x is simply
said to take four values, with the electron and positron each in two
spin states. This makes it look like the four values are written into
the equation, rather than derived from it. So I must have missed
something. I also understand that Dirac predicted the existence of
particles that are magnetic monopoles, i.e. have a magnetic charge
that is only North or South. These have not been observed, and Wilczek
doesn't seem to mention them.

Bell's Theorem, Bell's Inequality: The equations at right are
versions of John Bell's equations to test the Einstein-Podolsky-Rosen
(EPR) Paradox. The form at top is Bell's own, published in 1964. Below
it is a slightly rewritten form. The equations are given and
discussed, with the EPR Paradox and the attendant issues, in
Einstein's Moon, Bell's Theorem and the Curious Quest for Quantum
Reality, by F. David Peat [Contemporary Books, Chicago, 1990, pp.
111-112]. Peat thanks Bell himself (before his untimely death) for
reading the manuscript, so this may be more than the typical
scientific popularization. The actual equations here predict results
consistent with "local reality," i.e. what Einstein wanted, with the
possiblity of quantum states predetermined by "hidden variables." The
two detectors (A and B) catch the "correlated" particles that are the
issue in the EPR Paradox, i.e. they may have opposite spins, but which
is which is undetermined, in reality as well as in knowledge, until
one of them is observed. Then the spin of the other is instantaneously
fixed, violating the velocity of light limitation of Special
Relativity. In the second form of the equation, the probabilities
should range between negative 2 and postive 2. (Since probability
ranges up to 1, which is certainty, the three positive and three
negative terms of the first equation cannot add up to more than 0.)
However, the predictions of "non-local" quantum mechanics are going to
be different. Peat does not give equations because he says that they
are different for every angle. When there were experimental tests of
the equations in 1982, they were at angles where a quantum correlation
of 2.70 was predicted [pp.117-118]. The experimental result was 2.697,
much larger than the "local reality" prediction and very close to the
quantum prediction (or >2.682 and <2.712). Thus, Quantum Mechanics
violates Special Relativity, and indeterminacy is established for
particles when they cannot be observed or their states inferred from
observations, i.e. the wave function completely specifies the reality.
The Realists, like Einstein or Bell himself, were not going to like
this, but, on the other hand, it still allows a Realism for those who
take the wave function to be real, as did de Broglie or as is possible
in a Kantian Quantum Mechanics.




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Copyright (c) 1998, 2002, 2003, 2005 Kelley L. Ross, Ph.D. All Rights
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Historic Equations in Physics and Astronomy, Note

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For instance, the equations at right for sine and cosine functions,
which are periodic, contrast with the equations for hyperbolic sine
and hyperbolic cosine functions, which are not periodic. The results
of all the equations are real numbers; but for the sine and cosine we
have imaginary powers of Napier's Constant (e), and, for the sine
function, these imaginary powers are divided by the imaginary number
itself. Also, the sine equation can alternatively be written:
(because: i-1 = -i).

These relationships go back to Euler's Theorem:

Return to text



Imaginary Powers of Napier's Constant


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