In the early days of quantum mechanics, the 1920s, the so-called "wave function collapse" or "measurement problem" arose. The problem centered around the question of at what point is the final result decided upon when a measurement of a quantum particle is made. In 1956 Hugh Everett III developed the many worlds interpretation (MWI) as his doctoral thesis at Princeton University. According to MWI, the Schrödinger wave equation doesn't ever collapse. Instead, the entire universe splits into as many parts as necessary, perhaps an infinite number, so that every possible result of a quantum measurement become realities in different universes. In the essay below, I uncover a serious mathematical problem with MWI as it is currently formulated and offer my own alternative interpretation called the "Many Alices Interpretation." I also offer a solution to the long-standing "measurement problem."

1. Many Alices Interpretation
Many Worlds According to One Amateur Scientist
by John J. Winders, Jr.
Alice image attribution: https://www.cleanpng.com/ under personal use license

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3. Introduction
Sean Carroll is a very outspoken promoter of a conjecture about quantum mechanics called the Many
Worlds Interpretation. In Sean Carroll’s book, Something Deeply Hidden – Quantum Worlds and the
Emergence of Spacetime, he claims that nobody understands quantum mechanics, citing the Aesop’s
fable of the fox who cannot reach a delicious clump of grapes overhead. Out of frustration the fox says
the grapes were probably sour and he never wanted them anyway. According to Carroll, the fox
represents “physicists” and the grapes represent “understanding quantum mechanics.” I think the
problem goes much deeper than not understanding quantum mechanics; in fact, I think a significant
number of physicists don’t fully understand probability at all, which is what quantum mechanics is
based on. The reason is because probability is often extremely counter-intuitive.
Take for example the famous “Monty Hall Problem,” which Marilyn vos Savant once published in her
weekly column in “Parade” magazine based on the TV game show “The Price is Right.” The grand
prize is a car hidden behind one of three closed doors with goats hidden behind the other two. The
contestant must choose one of the three doors. After choosing one of the doors, host Monty Hall opens
one of the other two doors, revealing a goat. He then asks the contestant if she would like to stick with
her first choice or switch to the other closed door. With a quantum-mechanical interpretation, the car
has a “wave function” that translates into probabilities of finding it behind each of the doors, shown
below as the blue waves over the doors with each door having a probability of ⅓.
When Monty opens Door No. 3 revealing a goat, the wave function “collapses” around Door No. 2, and
the area of the wave (its probability) over Door No. 2 suddenly doubles to ⅔ as shown below.
Attribution: Door images courtesy of Freepik Company; goat image courtesy of Getty Images
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4. When vos Savant correctly stated switching doors would double the chances of winning the car from ⅓
to ⅔, there were howls of protest from the readers of vos Savant’s column, some calling her stupid for
not realizing that the odds of winning the car are “obviously” the same for Door No. 1 and Door No. 2.
A few of them bragged about their advanced degrees in mathematics and physics, which only confirmed
what I suspected all along; i.e., even advanced degrees in mathematics or physics don’t guarantee
understanding conditional probabilities. It’s easy to show why revealing a goat behind Door No. 3
doubles the probability of finding the car behind Door No. 2: The a priori probability of finding the car
behind Door No. 2 Door No. 3 is P
∪ (C2)+P(C3)=⅔. Opening Door No. 3 to reveal a goat behind it,
G3, creates a new conditional probability P(C2 |G3)= ⅔ and “collapses” the original wave function.
The physics community considers “collapsing” a quantum wave function as a truly horrible thing to do,
naming it “the measurement problem.” After all, how can measuring an object possibly change its wave
function? Here’s what Carroll has to say about the measurement problem.
“What exactly a measurement is, and what happens when we measure something, and what this tells us about what’s really
happening behind the scenes: together, these questions constitute what’s called the measurement problem of quantum
mechanics. There is absolutely no consensus within physics or philosophy on how to solve the measurement problem,
although there are a number of promising ideas.”
Carroll concedes that no such measurement problem exists in classical mechanics. Well of course not.
When Monty Hall opens Door No. 3, this is a “measurement” that confirms there is a goat behind that
door, which necessarily changes the a priori probability P(C2) into a conditional probability P(C2 |G3)
from that measurement. But for some reason people cannot accept the fact that measuring a quantum
particle using a piece of lab equipment is in principle the same as opening a door and knowing what’s
behind it instead of guessing what it might be.
The Origin of Many Worlds
The symbol ψ stands for a wave function describing the quantum states of an isolated system. The
Schrödinger wave equation, ψ(x,t), is a special case that assigns complex numbers to ψ that evolves
deterministically through space and over time. For this case, the quantity |ψ(x,t)|2
is interpreted as the
probability of observing the particle in a specific region of space near x within a specific interval of
time near t. According to the Copenhagen interpretation of quantum mechanics, observing the particle
causes ψ(x,t) to “collapse.” This seems to makes sense because the old ψ(x,t) serves no purpose once
the particle has been pinned down to a particular place and time, so a new ψ(x,t) takes over. However,
many physicists consider ψ to be much more than just a mere mathematical expression; they believe ψ
to be a real physical object, and obviously real physical objects should not be destroyed by mere
observations or measurements.1
I believe the crux of the “measurement problem” stems from the
prevailing material reductionist paradigm in physics.
One of the “promising ideas” that Carroll mentions is the one that Hugh Everett III came up with in
1956 while working on his PhD thesis at Princeton University, becoming known as the Many Worlds
Interpretation or MWI.2
Much has been written about it in the literature. Like the fox in the tale of the
sour grapes, believers in MWI avoid the measurement problem by asserting that the wave function
never really collapses; instead, all possible quantum measurements are actualized by branching into
separate non-communicating parallel “worlds” (see the illustration on the following page).
1 Hmm … a physical object having complex values? Another problem is that ψ has dimensions of 1/√Length, which is
clearly a non-physical quantity. Nevertheless, some physicists still consider ψ to be the only physical object in the
universe, and furthermore the universe is an isolated system comprised of a single universal wave function.
2 John Wheeler was his thesis advisor. A copy of the “long thesis” published in 1973 is available here:
https://calisphere.org/item/ark:/81235/d8kp7v50b/
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5. Alice makes a series of measurements of electron superposed spins states (1/√2 |U+1/√2 |D), where U
is “spin up” and D is “spin down.” Here, the probabilities of U and D are both 50%. According to the
“pop sci” version of MWI, the universe splits into two branches each time Alice measures spin: One
universe where a U spin electron has occurred and the other where a D spin electron has occurred.
Meanwhile, Alice splits into two versions of herself, one measuring U and the other measuring D.
After a second measurement, the two universes split again into four non-communicating parallel copies,
as shown on the left side of the diagram. Although none of the Alice clones can communicate with
other parallel copies, each of them has memories of upstream Alices from whom she had been cloned,
so each of the four Alices remembers performing two spin measurements. The results of those two
measurements are shown by the sets of U and D letters next to four Alices’ images in the left-hand
portion of the illustration above.
It’s All in the Mind
A lot of other people, including me, have trouble believing that entire universes would physically split
into two parts as the result of simple actions performed by a single human living on a small planet
revolving around a medium-size, nondescript, white star in a commonly-occurring spiral galaxy among
trillions of other galaxies. Then I came across a paper written by Max Tegmark with a very different
slant on MWI.3
While reading it, I had an “aha moment” that made me understand why so many
notable scientists, presumably including the great John A. Wheeler, embraced MWI.
If I read Tegmark’s argument correctly, then in a nutshell Alice is integrated into the superposition she is
measuring. So when she measures the spin of an electron, she observes both U and D at the same time,
but her conscious recollections split between them. Meanwhile, the universal wave function goes along
its merry way as if nothing happened. This is a very radical version of Wheeler’s participatory
universe, where everything we call “physical reality” is literally exists only in the mind of the observer,
who in this case is Alice. Therefore, I’m calling this the Many Alices Interpretation (MAI).
One consequence of MAI is that randomness and probability are just imaginary concepts of the mind.
Determinism still “rules” because ψ continues to evolve over time, undeterred by what humans might
believe is really happening. Let’s see how this plays out referring to the illustration above.
3 The Interpretation of Quantum Mechanics: Many Worlds or Many Words?
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6. Alice makes two more spin measurements as illustrated under the heading “After 4 Measurements.”
Alice has now split into 16 versions of herself, each with a specific memory of four measurements as
shown by 16 sets of four U and D letters to the right of her images. If the Alices could confer with each
other, they would quickly see there were 64 total measurements with 32 of them being U and 32 of them
being D. They would thus conclude that the probability of U and D are both exactly 50%.
Unfortunately, the Alices cannot communicate among themselves so some of them do see a 2/2 split,
while others see 0/4, 1/3, 3/1 and 4/0 splits which would leave them rather unconvinced that U and D
probabilities are even close to 50%. But the interesting thing is that the 16 Alices themselves have been
grouped according to a binomial probability distribution with a U success rate of 50% as shown on the
far right of the figure. As the number of measurements increases into double and triple digits, the
binomial grouping of Alices becomes more and more concentrated near the 50% level. Thus, according
to MAI, “Alice” (whoever she really is) is more “likely” to be among those near the center of the
distribution, and she is most “likely” to believe the U probability is 50%.
Math Problems
The trouble with scenario is that as long as each Alice bifurcates into two parts, her clones will always
be grouped in binomial distributions with U=D=50%. For the general case (√α |U + √β |D), where
the weighting factors α and β are not equal, we start running into some mathematical problems. The
bifurcation strategy of splitting Alice into two parts per measurement won’t result in binomial
distributions with the correct U success rate equal to α. If α=k/n and β=(n-k)/n, where k and n are
integers, and Alice splits into n branches for each measurement she makes, and U is assigned to k
branches, and D is assigned to (n-k) branches, then and only then does a binomial distribution of Alices
with a U success rate α emerge. That’s quite a tall order, and I would expect someone who tries to
explain how the wave function handles all this math might have to do a great deal of hand waving.4
Another Way of Splitting Alice
There is an alternative that would allow a simple bifurcation technique to work with all values of α and
β. Instead of assigning an integer number of U and D in opposite branches, fractional values could be
assigned based on the values of α and β, resulting in fractional slices of Alice. This might present
problems for Alice living in fractional bodies, but at least the mathematics would work out in the end.
Looking at the results in the far right of the figure on Page 3, the table below shows the results when
α = 0.75 and β = 0.25. The first column describes the possible outcomes of four observations, the
second column shows the numbers of branches that match those descriptions, and the third column
shows the probabilities that Alice will measure the outcomes described in the first column.
Descriptions of
Measurements
Numbers of
Branches
Probabilities of Making
Measurements
0U & 4D 1 1  0.750
 0.254
= 0.00390625
1U & 3D 4 4  0.751
 0.253
= 0.04687500
2U & 2D 6 6  0.752
 0.252
= 0.21093750
3U & 1D 4 4  0.753
 0.251
= 0.42187500
4U & 0D 1 1  0.754
 0.250
= 0.31640625
4 Pun intended.
-4-

7. Notice the sum of probabilities in the third column equals one, which is always a nice thing when
working with probabilities. We can also think of this is as the sum of Alice herself, combining all 16 of
her fractional doppelgangers into one whole Alice. Most importantly, the probabilities shown in the
third column exactly match the binomial probability distribution with the U success rate equal to 75%.
Alice is most likely to find herself in one of the four branches matching the 3U & 1D outcome, in the
row highlighted in yellow, which corresponds to measuring U three out of four times, or a success
probability of 75%. So it seems like the strategy of bifurcating Alice into fractional slices works
mathematically, so this might be the way MWI would work according to MAI.
Denial, Anger, Bargaining, Depression and Acceptance
Initially, Everett’s interpretation was not well received. The physics community reacted in horror that a
universe could physically split into infinite parallel versions of itself, and there were many angry
arguments and papers published over it. Over time physicists came to accept Everett’s interpretation
just because it seemed the only way to avoid the dreaded wave function collapse. Max Tegmark’s
paper, cited in Footnote 3, mentions an informal (albeit unscientific) survey among 48 scientists who
attended at a quantum mechanics workshop held at UMBC in 1997 where they were asked which
interpretation of quantum mechanics they preferred. The results were: Copenhagen 13, MWI 8, Bohm
4, Consistent Histories 4, Modified Dynamics 1, and none of the above or undecided 18. Copenhagen
was the winner, but MWI was a close second. The participants had their own reasons for voting the
ways they did, and although I don’t know what their reasons were, I’ll offer some guesses.
The Motivation for Many Worlds
The main motivations for MWI are 1) it gets rid of the “ugly” wave function collapse in the
Copenhagen interpretation, and 2) it banishes randomness from the universe because as we have seen,
all possible quantum measurements are 100% actualized in the model. Consequently, probability is all
in the mind and the universe itself is completely deterministic, which appeals to some folks who just
don’t like uncertainty. Einstein was one of them because he never accepted the possibility that God
plays dice with the universe, and he probably would have strongly supported Everett’s thesis if he had
lived long enough to see it (Einstein died in 1955). Let’s examine both of these motivations.
Some mathematicians and physicists believe the wave function is ontological. As revealed by the title
of Max Tegmark’s Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, he
believes mathematics is the ultimate reality and that every consistent mathematical system must
physically exist as a “universe” in a Level IV multiverse of all possibilities. If the wave function is
mathematically consistent, it must therefore physically exist as an entire universe. Anyone who
believes this is true would consider the collapse of the wave function “ugly.” But is this true?
In my view, the wave function is no more ontological than binomial, Poisson or Gaussian probability
functions. Its value is entirely due to the fact that it works as a predictive model, but it doesn’t “direct”
anything. Its usefulness as a predictive model disappears as soon as its predictions are realized. The
same thing happens to the wave function when a measurement is made. Knowledge removes some
uncertainty and changes all future probabilities. The wave function doesn’t entirely “collapse” but it
must change in order to conform to those probabilities.
This brings up the second point. Those who don’t fully understand probability obviously would want to
banish it from reality, and MWI supposedly does that. In the “pop sci” version of MWI, both branches
in the bifurcation are fully realized, so probability appears to vanish. But this assumes all
measurements are coin-toss measurements producing equal numbers of heads as tails. In other words,
there is an underlying 50% probability assumed in the model but this is never explicitly stated. As we
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8. see when the U and D probabilities are different, the simple bifurcation model doesn’t work unless the
supposedly non-existent probabilities are incorporated into the branching mechanisms. This only
appears to eliminate uncertainty because the deterministic wave function supposedly controls the
branching mechanism, but this is just a sleight of hand that hides uncertainty within the wave function.
The Role of Consciousness
Prior to the quantum revolution that began in the 1920s, physicists viewed the universe as what James
Jeans called a Great Machine. As quantum mechanics developed, it became an accepted fact that
consciousness is not only capable of modifying the Machine through the exercise of free will, but it also
plays an indispensable role in creating it. There is a debate that continues to the present say as to how
much a role it plays. As Einstein quipped, “Does the Moon disappear when I look away from it and
reappear when I look back?” Many physicists today would answer “Yes” to that question. In their
view, consciousness is absolutely essential for turning a possibility into an actuality through
observation; i.e., consciousness collapses the wave function. Carried to an extreme, this leads to the
Schrödinger’s cat thought experiment where everything and everybody are superposed in a giant wave
function that is constantly growing and then collapsing when it interacts with consciousness.5
According to MWI, Alice doesn’t play any role whatsoever because every quantum possibility exists as
an actuality in some parallel world. Her observations have absolutely no effect on the wave function,
which evolves forever according to its own rules without a beginning or end. The many worlds are
radically deterministic but completely out of control, like Jeans’ Great Machine running amok.
I find it odd that Wheeler would approve of MWI because he’s at the forefront of the participatory
universe hypothesis, but presumably he did approve of it because he was Everett’s thesis advisor. On
the other hand, Wheeler might have preferred the MAI modification even more, which places Alice at
the center of creation (at least inside her own mind). Everything that can happen does happen in some
parallel world, but the only world that exists for her is the one she’s currently witnessing; and in some
strange way this might be true. There’s a famous Daoist koan:
“Once Zhuangzi dreamt he was a butterfly, a butterfly flitting gaily about. He knew
nothing about Zhuangzi. Then suddenly he awoke and he was at once solidly and
unmistakably himself, Zhuangzi. But he didn’t know whether he was a man who
dreamt he was a butterfly or was a butterfly dreaming he was a man.”
Zhuangzi didn’t remember awakening as a butterfly who dreamt of being a man, but he wonders if he
could be dreaming that dream in the present. Alice has much more of a disadvantage than Zhuangzi
because she can’t even remember dreaming about any of her other selves.
Thoughts Regarding Bell’s Theorem
According to the Copenhagen interpretation of quantum mechanics, the outcome of measuring a particle
in a superposed state is completely random. Some physicists, notably Einstein, believed the universe is
both deterministic and local, and they could not bring themselves to believe that true stochasticity is
possible in a deterministic universe. Others, notably Bohr, argued that quantum randomness is real, and
he and Einstein engaged in a series of arguments over this question. In 1935, Einstein and two other
colleagues of his published a paper they were convinced would prove Bohr was wrong.6
According to
the Copenhagen interpretation, two systems in a state of quantum entanglement and separated by a large
5 Schrödinger himself didn’t believe any of it. He invented this to show the absurdity of such an idea, but it backfired.
Many physicists have extended the absurdity by claiming that it isn’t just the cat in an alive/dead quantum superposition,
but the human observer, Wigner, is also in a superposed quantum state until he looks inside the box and observes the cat,
and so is Wigner’s friend until Wigner informs his friend how the observation turned out, and on and on and on …
6 Boris Podolsky and Nathan Rosen were co-authors; hence, the paper came to be known as the EPR paradox.
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9. distance seemed to behave as a single entity, where a measurement of one property of one system will
instantaneously influence a measurement of a complementary property of the other. Einstein et al
argued that instantaneous communication between systems is impossible; therefore, the systems must
contain “hidden variables” that preordain what those properties are before they are measured.
In 1964, John S. Bell published a paper entitled, “On the Einstein Podolsky Rosen Paradox,” with a
theorem that he proved would show whether Einstein was correct based on an experiment. In 1980, the
first of a series of Bell-like experiments were performed that confirmed that at least one of the
following two statements must be false: a) communication cannot travel faster than light, or b) there
exist hidden variables that determine the outcomes of quantum measurements. In other words, physics
had to abandon either locality or realism. A majority of physicists decided to keep locality and abandon
realism. Bell had been in the realist camp along with Einstein. As the results of the Bell-like
experiments came in, he said that there would be one possible loophole that would save realism, namely
if there were no statistical independence among Alice, her spin detector and the electrons she measures.
This strict dependency is called superdeterminism (SD), which Bell believed might be possible because
Bell-like experiments could not falsify it. One consequence of SD is that intentional agency goes out
the window, and so Bell said he felt it was “unlikely” we live in an SD world.7
Concluding Remarks
What does MWI or MAI accomplish that the other interpretations do not? Not much in my opinion. It
gets rid of the wave function collapse, which lacks a physical mechanism that explains it and is deemed
“ugly.” Instead, MWI splits the universe into myriad parts, which also lacks a physical mechanism and
is just as “ugly.” Proponents of Everett’s interpretation claim a loophole exists, similar to the SD
loophole, allowing MWI to escape falsification by Bell’s theorem; however, they say the MWI loophole
is different than the SD loophole, and they are adamant that MWI is not remotely the same as SD. To
me, this looks like a distinction without a difference because one of the motivations behind MWI was to
impose a strict form of determinism on the universe to get rid of randomness. But as we saw earlier
MWI doesn’t even accomplish this goal because although a deterministically-evolving wave function
causes the branching, |ψ|2
determines the underlying probabilities of the branching process.
The truth of the matter is a wave function is not a physical thing, but rather an abstract mathematical
object. Physical things are represented by real numbers, whereas ψ is represented by complex numbers.
Probabilities, being real numbers between 0 and 1, are the only “real” things that |ψ(x,t)|2
represents.
But the main fallacy behind MWI is thinking that probabilities “control” outcomes; but probabilities
don’t control anything. Instead, they only predict likelihoods of future events based on their frequencies
observed in the past. The fact that the Schrödinger equation does a masterful job in predicting the
probabilities of future measurements doesn’t mean it controls those measurements any more than a
binomial function controls coin tosses.
This is how I think about quantum measurements: A polarized light beam passes through a polarizing
filter oriented 45 to the direction of the polarized light, and Maxwell’s equations say the filter changes
the amplitudes of the E and B waves by 1/√2, thus cutting the power of the beam in half. Nature
operates on a “need to know basis,” so as long as 50% of the photons make it through in agreement with
Maxwell’s equations, we don’t need to know in advance which specific photons will make it and which
ones won’t. Therefore, Nature censors those details and won’t reveal them to us until the photons are
measured. This is also known as the Copenhagen interpretation of quantum mechanics.
7 Whether SD is likely or unlikely has nothing to do with how we feel about it. Establishing SD’s likelihood is impossible
because the sample size of observable universes is 1. Furthermore, there’s no way to falsify SD experimentally. So we
either live in an SD universe or we don’t, and statistically speaking the likelihood is either 100% or 0%.
-7-

10. Appendix A – How Classical Objects Emerge from the Law of Averages
There seems to be a conceptual stumbling block in the scientific community of understanding how the
world of tables and chairs emerges from the realm of quantum particles. After all, tables and chairs are
made of quantum particles, so shouldn’t tables and chairs be quantum objects also? This leads to the
notion that quantum wave functions rule both elementary particles and the world of tables and chairs.
This in turn led to the many worlds interpretation. However, the answer to the classical vs quantum
conundrum is quite simple: It’s just because of the law of averages and the Central Limit Theorem.
Suppose a single quantum particle is confined within a thin tube. Let the square of the magnitude of the
Schrödinger wave function |ψ(x)|2
be a bi-modal probability distribution function (pdf) with two peak
probabilities at x=-1 and x=1, depicted as the variably-shaded area below. The darker blue colors
depict larger probabilities, highlighting the particle’s wave property. Superposition is also evident by
the peak probabilities near x=-1 and x=1. The equation for |ψ(x)|2
and its mean, μ, and standard
deviation, σ, are shown above the figure.
Now suppose there is a large population of non-interacting elementary particles inside the tube, all
having this same wave function. Instead of looking at the position, x, of a single particle, let’s look at
the average positions of samples of n particles, Xn. Here the Central Limit Theorem kicks in, where the
probability distribution of the averages of the sample groups, f (Xn), converges into a normal
distribution having a mean equal to the population’s mean and a standard deviation equal to the
population’s standard deviation divided by √n. In the plot below, |ψ(x)|2
for a single particle is shown in
blue and the distribution f (Xn) of the averages many samples of n=30 is shown in red.8
By “zooming out” from a single particle and looking at the averages of many sample groups of n=30,
the two superposition peaks of |ψ(x)|2
disappear with hardly a trace of particles for x<-1 or x>1. The
Xn values bunch up in the middle, so quantum uncertainty diminishes and “classical” behavior begins to
emerge. As n increases, the red curve’s peak will grow taller and narrower around x=0, making
quantum behavior vanish entirely. In other words, classical behavior emerges naturally by increasing
the number of particles we measure (or see) at the same time. This is just due to the law of averages.
8 It must be stressed that the distribution of the sample averages is not the same as the average of the sample distributions.
-8-
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
|ψ(x)|2
f (Xn)

11. We can see how quickly quantum superposition can evolve into classical behavior using the following
simple example with discrete probabilities. Let a particle have two quantum states, x1 = -1 and x2 = +1,
and the wave function ψ(x) of the particle be a superposition of x1 and x2 both having a 50% probability.
The statistics for those probabilities are μ=0 and σ=1. Suppose there are two particles both having the
same statistics. Taking the averages of the two particles, there are four combinations:
Particle(A), x = -1 ; Particle(B), x =-1 ; X2 =-1
Particle(A), x = -1 ; Particle(B), x=+1 ; X2 = 0
Particle(A), x = +1 ; Particle(B), x=-1 ; X2 = 0
Particle(A), x = +1 ; Particle(B), x=+1 ; X2 =+1
Xn =2k/n – 1, for k=0, 1 … n, and f (Xn)=a binomial distribution p(k/n)=
(n
k)0.5n
n=2, k=0, X2 =-1: f (X2)=p(0/2)=0.25
n=2, k=1, X2 =0: f (X2)=p(1/2)=0.50
n=2, k=2, X2 =+1: f (X2)=p(2/2)=0.25
According to the Central Limit Theorem, the distribution function f (Xn) should converge into a normal
distribution N (Xn) whose standard deviation equals the population’s standard deviation divided by √n.
For the case n=2, the statistics of N (X2) are μ=0, σ=0.707. The plots below show f (X2) values by
blue 𝐱 symbols and N (X2) discrete probability values by red  symbols.9
The fit between f (Xn) and N (Xn) isn’t bad for n=2, but it becomes much better with n=8:
We can see that even when there are very few particles, averaging quickly rubs out the quantum
superposition x=1, and the discrete probabilities approximate a normal distribution curve.
9 It should be noted that N (Xn) is a continuous probability density function, so it must be converted into a probability
distribution by multiplying all of the discrete N (Xn) values by some constant so that Σ N (Xn)= 1.
-9-

12. Appendix B – What Is a Measurement and Why Does It Happen?
In his book Something Deeply Hidden, Sean Carroll says there is no consensus among scientists and
philosophers about what really happens when a quantum measurement is made; so let me humbly offer
a suggestion from a retired engineer and amateur scientist.
My suggestion is that a quantum measurement involving a collapse of the wave function occurs when a
quantum wave function energetically interacts with a measuring device. There are two requirement for
a wave function collapse: 1) the wave function must exhibit sufficient energy uncertainty between the
superposed energy states when interacting with the measuring device, 2) the measuring device must
have sufficient capacity to absorb the entropy, S, from the collapsed wave function. A single quantum
particle cannot measure another quantum particle because it doesn’t meet those requirements. Thus,
when two electrons collide, they do interact quantum mechanically but not in a way that can collapse a
wave function into a single eigenvalue. Detecting or measuring a quantum particle by collapsing its
wave function typically requires a large laboratory device, and sometimes the device is gigantic, with an
example being CERN’s ATLAS detector you can view here: https://atlas.cern/Discover/Detector
In order for ATLAS to detect any elementary particle, it must interact energetically with it, either
through electromagnetism for electrically-charged particles or by absorbing the kinetic energy from
collisions with electrically-neutral particles. To measure an electron’s spin, a small laboratory magnetic
device will suffice. The negatively-charged electron has a magnetic moment, μ, pointing in the opposite
direction of the electron’s inherent spin, ŝ. An electron in a superposed spin state can be considered as
two partial electrons with opposite spins, shown as two ghostly images on the left below.
When the electron interacts with an external magnetic field, B, there is an energy gap, Δe=2μB,
between the energy of interaction of B with -μ (the |U state) and +μ (the |D state). The magnetic field
of the lab equipment determines the directions |U and |D. In the absence of a magnetic field, the
electron itself would have no idea what directions “up” or “down” might be.
The decoherence is due to Heisenberg’s uncertainty principle with respect to energy and time, given by
the formula Δt½ħ/Δe, where Δt is commonly interpreted as the minimum time it takes to measure the
energy of a system, e, within a Δe error tolerance or uncertainty. An alternative interpretation is that Δt
is the approximate duration uncertainty can persist while Δe exists between |U and |D. Decoherence
occurs after a delay of Δt, superposition is irreversibly lost, and a definite U or D measurement
emerges. Planck’s constant, ħ, has a very small value, so magnetic decoherence typically occurs after
about one billionth of a second.
The entropy of superposition is Sψ =-kB (αLogα+βLogβ) per the Gibbs equation. When ψŝ collapses,
Sψ  0 and then the second law of thermodynamics requires the missing entropy to be transferred to the
environment through the measuring device. Thus, the measuring device must have a sufficient
thermodynamic capacity to absorb the wave function’s missing entropy (see Condition 2 above).
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