Quantum
Mechanics states unambiguously
That
Particles
never exist in two places at once
Kevin Aylward B.Sc.
Contents
Overview
I felt
somewhat compelled to write this little ditty due to my somewhat frustration of
the prevalence of so much misunderstanding and nonsense associated with Quantum
Mechanics.
It is
quite clear that, although there are innumerable higher/better qualified
Quantum Mechanics than yours truly, many of them simply fail to understand that
their waffles, which are actually classical descriptions, are trivially
contradictory to Quantum Mechanics.
I am by
no means the only one to notice this somewhat stunning state of affairs. For
example, Lubo
š
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Motl presents extremely clear
technical explanations as to why/how Quantum Mechanics is systematically
misunderstood. Although one might argue that that his particular presentation style
is problematic.
The
fundamental issue is the failure to understand that a quantum state vector such
as
|X〉
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does not refer to a measurement of position
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@3681@
, but only to the probability of
measuring a position,
P(X)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacIcacaWGybGaaiykaaaa@38AF@
.
Explanations
invariably confuse the two, and make arguments about the state vector as if
it was identically equivalent to discussing the actual position of an object,
instead of what it actually is, to wit, only a description of the probability of obtaining a position of an object.
The
Problem
A BBC2
episode of Horizon “What Universe Are We In” 2015, illustrates the problem.
Professor Seth Lloyd PhD., in explaining the double slit experiment produces
the usual standard twaddle “the electron went through both slits at once”. The
notable Professor Lawrence Krauss PhD also advocates this view in more than one
YouTube posted debate.
Unfortunately,
such a statement is indisputable false, according to standard Quantum
Mechanics. The fundamental issue is that there is a general refusal to accept
that the universe is subject to Quantum Mechanics, not Classical Mechanics. It
is a “new” result or axiom that stands on its own and can not, in principle, be
“explained” by any type of classical reasoning. For example, …well if the
particle did this, to account for that… it must do this…
If this classical
type of argument is applied to the double slit experiment, the result is a
result that contradicts the axioms of Quantum Mechanics. The basic mathematics
of Quantum Mechanics is settled, irrespective of any ad-hoc metaphysical
interpretations to the mathematics. Quantum Mechanics states that the result of
any measurement, i.e. a true physical fact, is that a measurement can only
result in a single eigenvalue. Applying the rules of Quantum Mechanics,
unequivocally states that the probability of a particle being measured, i.e. a
true physical fact, to have, say, both spin up and spin down simultaneously is zero
(see appendix). This means, absolutely, that if Quantum Mechanics is correct, a
particle cannot be in two locations at the same time. Period.
This is
so simple and obvious, that I would guess that most would be going, wow…if it
is so trivial under the standard rules of Quantum Mechanics, that two at once
is a no no, why do so many “experts” tout all this metaphysical claptrap? One
can only conjecture…
So,
rather than the typical waffle of “…this
physical explanation of the two slit experiment proves that a particle can be
in two places at once” the argument, essentially, is an argument that proves
that classical mechanics is false.
Quantum Mechanics gives the correct answers, Classical Mechanics does not. It’s
that simple.
I would
refer the reader to Lubo
š
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaG+aaaWdbiaa=fwaaaa@38BC@
Motl for the detailed mathematics, a
summary of which is in the appendix of this paper.
The Roots
To Confusion
One of
the most common sources of misunderstandings is the widespread inability to
understand what the symbols of Quantum Mechanics equations actually mean, and
how to manipulate those symbols. For example, consider Boolean algebra.
The
Boolean equation:
y=A+B
where A=1 and B=1 results in
y=1
not y=2
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Because
the “+” operator in the context of Boolean algebra has a different
definition and meaning, than that in ordinary arithmetic. Similarly, the
“+” sign in Quantum Mechanics is also not an arithmetic operator; it is
a probabilistic Boolean OR operator.
The
equation:
|φ〉=a|
X
1
〉+b|
X
2
〉
allows the calculation of the probability that the vector |φ〉 will be either the vector|
X
1
〉 OR the vector|
X
2
〉
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It does
not mean that there is a physical sum, where both vectors are argued to exist
simultaneously in the physical world. In fact, a “sum” wouldn’t even cut it, it
would need to be a logical, probabilistic AND function!
Furthermore,
it is stressed that the vector
|
X
1
〉
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4HaaeqabaGaamiwamaaBaaaleaacaaIXaaabeaaaOGaay5bSlaawQYiaaaa@3A08@
is not
X
1
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaacaaIXaaabeaaaaa@3768@
.
This last
point, whilst pretty much trivially obvious in the context of the rules of
Quantum Mechanics, seems to have completely escaped many. The quantum state
represented by
|
X
1
〉
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is not the same classical state
represented by
X
1
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. Confusion as to what a state
means in Quantum Mechanics is usually at the heart of most waffle descriptions
as to what QM means.
In classical mechanics, equations
refer to, for example, the actual physical positions given by
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
. That is, the classical state
is represented by
X
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.
In Quantum Mechanics, state equations
only represent the probability P, for example, of obtaining
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
. That is:
|X〉→P(X) not |X〉→X
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That is, a
quantum state is represented by
|X〉
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and its associated probability,
P(X)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacIcacaWGybGaaiykaaaa@38AF@
of
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
is not
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
itself.
The vector
|X〉
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itself has no direct or “instantaneous”
meaning in an equation. It only has a meaning with respect to, what values of
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
are allowed, and what is the probability that
such
X
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBaaaleaaaeqaaaaa@36AD@
’s are obtained in a final calculation.
Understanding
these points is key to understanding why a statement such as “the state vector
is in a superposition of states” does not mean that, for example, a particle is
actually in two X positions at once. It simply means that there is a probability
that the particle may be at one X position or another X position.
Relative State
A key
assumption in classical mechanics is that it is possible to assign unique values
of properties to objects independent of the measuring system.
Quantum
mechanical experiments prove this assumption to be false. It is thus surprizing
that many “two places at once” arguments ignore this key point. For example:
https://en.wikipedia.org/wiki/Stern
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaG+aaaWdbiaa=nbiaaa@388D@
Gerlach_experiment
The
Stern-Gerlarch experiment proves that the assignment of unique numbers, say
spin, to objects is false. It is simply impossible to label objects with characteristics
independent of its measuring system. For example, the assumption of Newton that
one can simple declare an object to have a unique mass, is therefore now known to
be false. The Stern-Gerlach experiment shows that spin is a function of its interaction
with other objects. The value of an electron’s spin is not simply a function of
the electron itself. Its spin is relative to what it interacts with. However,
many misinterpret that this means that the object itself doesn’t exist!
The point
here is that the numbers assigned to objects are man-made. Nature has no such
numbers. It only deals with the object. There is no demand of Gods that the
numbers man assigns mean anything at all. Much erroneous debate has been made
of this misunderstanding, e.g. the EPR Paradox on “elements of reality”. Stern-Gerlarch
proves that there is no general element of reality in the sense of assigning unique
numbers to objects, but this does not mean that the object itself is not real.
Even in classical
mechanics, an object may have many simultaneous values of measured velocity, despite
actually only having one state of motion. Walking on the Earth at 5 mph, also means that
you are moving at 67 kmph around the Sun.
It’s
Quantum Mechanics, Not Classical Mechanics, Dah…!
The reader
should refer to the standard accounts of the two slit experiment for a general
overview, for example, https://en.wikipedia.org/wiki/Double-slit_experiment.
A key
feature of the experiment is that single particles are fired through the slits
and the appearance of a diffraction pattern appears over time as many electrons
build up a pattern. In an effort to explain this result classically, it
is assumed that there are two wavicles at different positions on the
imaging screen side of the two slit barrier. The maximum and minimum of the
wave functions of these wavicles are then argued to interfere with each, other hence
accounting for the variations in density of the electrons on the screen. It is
thus argued that, as this is the only way to “explain” that there is a wave
pattern on the screen, it “proves” that a single electron gets split up into
two as it passed through the slits, and magically combines to one on hitting
the screen… Bollocks!
The
mathematical rules of Quantum Mechanics, independent of any physical
interpretation, are absolute in that the probability of an observable having
two simultaneous eigenvalues of position, is zero. Period. So, if Quantum
Mechanics is true, any argument that leads to declaring that “a particle is in
two places” must be false. In this case, it means that the assumption
that a particle splits up is false. There is no realistic way of escaping this
conclusion.
The
correct explanation for this dilemma, is that physical reality cannot be
explained by classical reasoning. End of story. The universe is Quantum, not
Classical. Dah… There is simply no explanation from Classical axioms that
explains why there is a diffraction pattern.
So, it is
astounding that many experts, and sometimes major TV personalities, spout of
the laws of Quantum Mechanics in the same sentence as claiming to the masses
that particles are at many places at once. They are completely confused on the
point that Quantum Mechanics and their waffle Classical descriptions are
contradictory. They simply have not learnt to think Quantumly and are
hopelessly trying to cling to their Classical baggage.
Quantum
Reality
The
reality of Quantum Mechanics, is that pretty much 99% of all professional
practitioners of Quantum Mechanics simply don’t concern themselves with waffle
interpretations as to what is alleged to be actually happening. For the most
part, it is irrelevant to the 90% of Physicists engaged in real world work
designing new semiconductor processes, and the remainder engaged in designing
the new legal high by simulations of the Shrödinger equation.
Many
Worlds - Ho…humm…
Might as
well put in my 1 cents worth on this topic…
Quantum
Mechanics states that particles do not split up by the notion that all
measurements must result in single eigenvectors and eigenvalues. Sure, it is
possible to postulate that a particle split ups and vanishes into another
universe, such that it can not be measured in our universe, in order to avoid
the inherent contradiction of the postulates of Quantum Mechanics and charge
and energy conservation laws. However, if it can not be measured, ever, it is
simply irrelevant to our universe. We can never know, as any interaction that
could “prove” alternate universes would constitute a measurement of charge or
mass-energy violation in our universe, which would destroy the Quantum Mechanical
argument that led to many worlds theory in the first place.
The root
cause of the pathological desire for interpretations such as The Many Worlds,
Bohmian Interpretation and so forth, is the complete failure to understand the
fundamental point proven by Kurt Godel. That is the notion that any complete
system description, must contain statements that are true, but not provable
(derivable/explainable) within the system.
All
interpretations, apart from the Copenhagen and Ensemble interpretations,
essentially and futilely, attempt to give a classical, touchy-feely description
to the experimental facts of Quantum Mechanics. The proponents of such
interpretations are quite unable to remove the straightjacket clothed on them
since childhood. They produce never ending classical pictures that try to
“explain” how rational, common sense ideas can account for the strange results
of QM. Listen up mate. Quantum Mechanics is simply one of the additional Godel axioms
that need added to the cooking pot. Live with it.
Appendix
Derivation
of the probability for the occurrence of two simultaneous observables.
For
simplicity, consider the case of an electron having spin up or down.
Quantum
Mechanics states that:
Prob=〈φ|P|φ〉
Where P is the projection operator
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajqgaG9VaaeiuaiaabkhacaqGVbGaaeOyaiabg2da9OWaa4raaKazaa2=baGaeqOXdOgabeGaayzkJiaawEa7aiaadcfakmaaEiaajqgaG9FabeaacqaHgpGAaiaawEa7caGLQmcaaOqaaKazaa4=caqGxbGaaeiAaiaabwgacaqGYbGaaeyzaiaabccacaWGqbGaaeiiaiaabMgacaqGZbGaaeiiaiaabshacaqGObGaaeyzaiaabccacaqGWbGaaeOCaiaab+gacaqGQbGaaeyzaiaabogacaqG0bGaaeyAaiaab+gacaqGUbGaaeiiaiaab+gacaqGWbGaaeyzaiaabkhacaqGHbGaaeiDaiaab+gacaqGYbaaaaa@6990@
The projection operator for the probability of two simultaneous observables A and B is given by
P=
P
A
.
P
B
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGubGaaeiAaiaabwgacaqGGaGaaeiCaiaabkhacaqGVbGaaeOAaiaabwgacaqGJbGaaeiDaiaabMgacaqGVbGaaeOBaiaabccacaqGVbGaaeiCaiaabwgacaqGYbGaaeyyaiaabshacaqGVbGaaeOCaiaabccacaqGMbGaae4BaiaabkhacaqGGaGaaeiDaiaabIgacaqGLbGaaeiiaiaabchacaqGYbGaae4BaiaabkgacaqGHbGaaeOyaiaabMgacaqGSbGaaeyAaiaabshacaqG5bGaaeiiaiaab+gacaqGMbGaaeiiaiaabshacaqG3bGaae4BaiaabccacaqGZbGaaeyAaiaab2gacaqG1bGaaeiBaiaabshacaqGHbGaaeOBaiaabwgacaqGVbGaaeyDaiaabohacaqGGaGaae4BaiaabkgacaqGZbGaaeyzaiaabkhacaqG2bGaaeyyaiaabkgacaqGSbGaaeyzaiaabohacaqGGaGaamyqaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaamOqaiaabccacaqGPbGaae4CaiaabccacaqGNbGaaeyAaiaabAhacaqGLbGaaeOBaiaabccacaqGIbGaaeyEaaqaaiaadcfacqGH9aqpcaWGqbWaaSbaaSqaaiaadgeaaeqaaOGaaiOlaiaadcfadaWgaaWcbaGaamOqaaqabaaaaaa@8FBB@
The projection operators for spin operators A and B are given by:
P
A
=|
up
〉〈
up
| and
P
B
=|
dn
〉〈
dn
|
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGubGaaeiAaiaabwgacaqGGaGaaeiCaiaabkhacaqGVbGaaeOAaiaabwgacaqGJbGaaeiDaiaabMgacaqGVbGaaeOBaiaabccacaqGVbGaaeiCaiaabwgacaqGYbGaaeyyaiaabshacaqGVbGaaeOCaiaabohacaqGGaGaaeOzaiaab+gacaqGYbGaaeiiaiaabohacaqGWbGaaeyAaiaab6gacaqGGaGaae4BaiaabchacaqGLbGaaeOCaiaabggacaqG0bGaae4BaiaabkhacaqGZbGaaeiiaiaadgeacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaadkeacaqGGaGaaeyyaiaabkhacaqGLbGaaeiiaiaabEgacaqGPbGaaeODaiaabwgacaqGUbGaaeiiaiaabkgacaqG5bGaaeOoaaqaaiaadcfadaWgaaWcbaGaamyqaaqabaGccqGH9aqpdaGhcaqabeaacaWG1bGaamiCaaGaay5bSlaawQYiamaaEeaabaGaamyDaiaadchaaeqacaGLPmIaay5bSdGaaeiiaiaabggacaqGUbGaaeizaiaabccacaWGqbWaaSbaaSqaaiaadkeaaeqaaOGaeyypa0Zaa4HaaeqabaGaamizaiaad6gaaiaawEa7caGLQmcadaGhbaqaaiaadsgacaWGUbaabeGaayzkJiaawEa7aaaaaa@8A26@
Hence:
P=
P
A
.
P
B
=|
up
〉〈
up
|
dn
〉〈
dn
|=0
as 〈
up
|
dn
〉=0
because |
up
〉 and |
dn
〉 are necessarily orthogonal to each other as they are eigenvectors of a Hermitian operator.
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C3E4@
Hence, there is zero probability,
according to standard Quantum Mechanics, of two simultaneous observables.
These
papers may be freely copied,
provided
its source here is referenced.
© Kevin
Aylward 2014
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37A2@
2016 all rights reserved
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