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## Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around ${3000}^{\circ}$K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of $2.726^\circ$K.

$\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.$

In this formula, $\nu$ is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and $I(\nu)$ is the energy density as a function of frequency $\nu$.

The other variables are: $T$ is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), $c$ is the speed of light in vacuum, $h$ is Planck’s constant, and $k$ is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against $\nu$) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures $T$. The horizontal axis labels the frequency $\nu$, and the vertical gives the energy density $I(\nu)$ per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency $\nu$ gives maximum density and give our answer in terms of the temperature $T$. All we do is compute the derivative of $I(\nu)$ with respect to $\nu$ and set it to zero and solve the resulting equation for $\nu$. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

$\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.$

(The equal sign here is an approximation!)

The $\nu_{\max}$ is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency $\nu_{\max}$ gets, and from Planck’s energy equation $E_{\max} = h\nu_{\max}$, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is $T = 2.726$ (degrees Kelvin), and working it out we get (approximately)

$\displaystyle\nu_{\max} = 160.2 \text{ GHz}$

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency $\nu_{\max}$ at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being $3.5^\circ$ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength $\lambda$ and frequency $\nu$ for electromagnetic waves we use the simple formula

$\displaystyle\lambda\nu = c$

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant $h = 6.6254 \times 10^{-27} \text{ erg sec}$

Boltzmann’s constant $k = 1.38049 \times 10^{-16} \text{ erg/K}$

Speed of light $c = 2.9979 \times 10^{10} \text{ cm/sec}$.

## Einstein summation convention

March 25, 2015 1 comment

Suppose you have a list of n numbers $A_1, A_2, A_3, \dots, A_n$.

Their sum $A_1 + A_2 + A_3 + \dots + A_n$ is often shorthanded using the Sigma notation like this

$\displaystyle\sum_{k=1}^n A_k$

which is read “sum of $A_k$ from k=1 to k=n.” This letter k that varies from 1 to n is called an ‘index’.

Vectors. You can think of a vector as an ordered list of numbers $A = (A_1, A_2, A_3, \dots, A_n) .$

If you have two vectors $A = (A_1, A_2, A_3, \dots, A_n)$ and $B = (B_1, B_2, B_3, \dots, B_n)$ their dot product is defined by multiplying their respective coordinates and adding the result:

$\displaystyle A\bullet B = A_1B_1 + A_2B_2 + A_3B_3 + \dots + A_nB_n.$

Using our summation notation, we can abbreviate this to

$\displaystyle A\bullet B = \sum_{k=1}^n A_k B_k.$

While working thru his general theory of relativity, Einstein noticed that whenever he was adding things like this, the same index k was repeated! (You can see the k appearing once in A and also in B.) So he thought, well in that case maybe we don’t need a Sigma notation! So remove it! The fact that we have a repeating index in a product expression would mean that a Sigma summation is implicitly understood. (Just don’t forget! And don’t eat fatty foods that can help you forget!)

With this idea, the Einstein summation convention would have us write the above dot product of vectors simply as

$\displaystyle A\bullet B = A_k B_k$

In his theory’s notation, it’s understood that the index k here would vary from 1 to 4, for the four dimensional space he was working with. That’s Einstein’s index notation where 1, 2, 3, are the indices for space coordinates (i.e., $A_1, A_2, A_3$), and k=4 for time (e.g., $A_4$). One could also write space-time coordinates using the vector $(x_1, x_2, x_3, t)$ where t is for time.

(Some authors have k go from 0 to 3 instead, with k=0 corresponding to time and the others to space coordinates.)

I used ‘k’ because it’s not gonna scare anyone, but Einstein actually uses Greek letters like $\mu, \nu, \dots$ instead of the k. The convention is that Greek index letters range over all 4 space-time coordinates, and Latin indices (like k, j, m,etc)  for the space coordinates only. So if we use $\mu$ instead of k the dot product of the two vectors would be

$\displaystyle A\bullet B = A_\mu B_\mu.$

So if we write $A_\mu B_\mu$ it means we understand that we’re summing these over the 4 indices of space-time. And if we write  $A_k B_k$ it means that we’re summing these over the 3 indices of space only. More specifically,

$\displaystyle A_\mu B_\mu = A_1B_1 + A_2B_2 + A_3B_3 + A_4B_4$

and

$\displaystyle A_k B_k = A_1B_1 + A_2B_2 + A_3B_3.$

There is one thing that I left out of this because I didn’t want to complicate the introduction and thereby scare readers! (I already may have! Shucks!) And that is, when you take the dot product of two vectors in Relativity, their indices are supposed to be such that one index is a subscript (‘at the bottom’) and the other repeating index is a superscript (‘at the top’).  So instead of writing our dot product as $A_\mu B_\mu$ it is written as

$\displaystyle A\bullet B = A_\mu B^\mu.$

(This gets us into covariant vectors, ones written with subscripts, and contravariant vectors, ones written with superscripts. But that is another topic!)

How about we promote ourselves to Tensors? Fear not, let’s just treat it as a game with symbols! Well, tensors are just like vectors except that they can involved more than one index. For example, a vector such as in the above was written $A_\mu$, so it involves one index $\mu$. What if you have two indices? Well in that case we have a matrix which we can write $M_{\mu \nu}$. (Here, the two indices are sitting side by side like good friends and aren’t being multiplied! There’s an imaginary comma that’s supposed to separate them but it’s not conventional to insert a comma.)

The most important tensor in Relativity Theory is what is called the metric tensor written $\large g_{\mu\nu}$. It describes the distance structure (metric = distance) on a curved space-time. So much of the rest of the geometry of space, like its curvature, how to differentiate vector fields, curved motion of light and particles, shortest path in curved space between two points, etc, comes from this metric tensor $\large g_{\mu\nu}$.

The Einstein ‘gravitational tensor’ is one such tensor and is written $G_{\mu \nu}$. Tensors like those are called rank 2 tensors because they involve two different indices. Another good example of a rank 2 tensor is the energy-momentum tensor often written as $T_{\mu \nu}$. This tensor encodes the energy and matter distribution in spaces that dictate its geometry — the geometry (and curvature) being encoded in the Einstein tensor $G_{\mu \nu}$. (If you’ve read this far, you’re really getting into Relativity! And I’m very proud of you!)

You could have a tensors with 3, 4 or more indices, and the indices could be mixed subscripts and superscripts, like for example $D_{\mu \nu}^{\alpha \beta}$ and $F_{\tau}^\gamma$.

If you have tensors like this, with more than 1 or 2 indices, you can still form their dot products. For example for the tensors D and F, you can take any lower index of D (say you take $\nu$ and set it equal to an upper index of F — and add! So we get a new tensor when we do this dot product! You get

$\displaystyle D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu = D_{\mu 1}^{\alpha \beta} F_{\tau}^1 + D_{\mu 2}^{\alpha \beta} F_{\tau}^2 +D_{\mu 3}^{\alpha \beta} F_{\tau}^3 +D_{\mu 4}^{\alpha \beta} F_{\tau}^4$

where it is understood that since the index $\nu$ is repeated, you are summing over that index (from 1 to 4) (as I’ve written out on the right hand side). Notice that the indices that remain are $\mu, \alpha, \beta, \tau$. So this dot product gives rise to yet another tensor with these indices – let’s give the letter C:

$\displaystyle C_{\mu \tau}^{\alpha \beta} = D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu$.

This process where you pick two indices from tensors and add their products along that index is called ‘contraction‘ – even though it came out of doing a simple idea of dot product. Notice that in general when you contract tensors the result is not a number but is in fact another tensor. This process of contraction is very important in relativity and geometry, yet it’s based on a simple idea, extended to complicated objects like tensors. (In fact, you can call the original dot product of two vectors a contraction too, except it would be number in this case.)

Thank you!
$\Sigma\alpha\mu$

## Einstein’s Religious Philosophy

Here is a short, sweet, and quick summary of some of Albert Einstein’s philosophy and religious views which I thought were interesting enough to jot down while I have that material fresh in mind. (I thought it’s good to read all these various views of Einstein’s in one fell swoop to get a good mental image of his views.) These can be found in most biographies on Einstein, but I included references [1] and [2] below for definiteness. (Throughout this note, ‘he’ refers, of course, to Einstein.) Let’s begin!

1. Einstein began to appreciate and identify more with his Jewish heritage in later life (as he approached 50).
2. He had profound faith in the order and discernible laws in the universe, which he said was the extent to which he calls himself ‘religious.’
3. God had no choice but to create the universe in the way He did.
4. He believed in something larger than himself, in a greater mind.
5. He called nationalism an infantile disease.
6. He received instruction in the Bible and Talmud. He is a Jew, but one who is also enthralled by “the luminous figure of the Nazarene.”
7. He believed Jesus was a real historical figure and that Jesus’ personality pulsates in every word in the Gospels.
8. He was not an atheist, but a kind of “deist.”
9. He did not like atheists quoting him in support of atheism.
10. He believed in an impersonal God, who is not concerned with human action.
11. His belief in an impersonal God was not disingenuous in order to cover up an underlying ‘atheism’.
12. He was neither theist nor atheist.
13. He did not believe in free will. He was a causal determinist. (Not even God has free will! 🙂 )
14. Though he did not believe in free will, nevertheless he said “I am compelled to act as if free will existed.”
15. He liked Baruch Spinoza’s treatment of the soul and body as one.
16. He did not believe in immortality.
17. He believed that the imagination was more important than knowledge.
18. He believed in a superior mind that reveals itself in world of experience, which he says represents his conception of God.
19. He believed in a “cosmic religious feeling” which he says “is the strongest and noblest motive for scientific research.”
20. “Science without religion is lame, religion without science is blind.”

There you have it, without commentary! 😉

### References.

[1] Albert Einstein, Ideas and Opinions.

[2] Walter Isaacson, Einstein: His Life and Universe. (See especially chapter 17.)

======================================

## Einstein and Pantheism

Albert Einstein’s views on religion and on the nature and existence of God has always generated interest, and they continue to. In this short note I point out that he expressed different views on whether he subscribed to “pantheism.” It is well-known that he said, for instance, that he does not believe in a personal God that one prays to, and that he rather believes in “Spinoza’s God” (in some “pantheistic” form). Here are two passages from Albert Einstein where he expressed contrary views on whether he is a “pantheist”.

I’m not an atheist and I don’t think I can call myself a pantheist. We are in the position of a little child entering a huge library filled with books in many different languages.…The child dimly suspects a mysterious order in the arrangement of the books but doesn’t know what it is. That, it seems to me, is the attitude of even the most intelligent human being toward God.” (Quoted in Encyclopedia Britannica article on Einstein.)

Here, Einstein says that he does not think he can call himself a pantheist. However, in his book Ideas and Opinions he said that his conception of God may be described as “pantheistic” (in the sense of Spinoza’s):

This firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of experience, represents my conception of God. In common parlance this may be described as “pantheistic” (Spinoza)” (Ideas and Opinions, section titled ‘On Scientific Truth’ – also quoted in Wikipedia with references)

It is true that Einsteins expressed his views on religion and God in various ways, but I thought that the fact that he appeared to identify and not identify with “pantheism” at various stages of his life is interesting. Setting aside that label, however, I think that his general conceptions of God in both these quotes — the mysterious order in the books, a universe already written in given languages, a superior mind that reveals itself in such ways — are fairly consistent, and also consistent with other sentiments he expressed elsewhere.

## Matter and Antimatter don’t always annihilate

It is often said that when matter and antimatter come into contact they’ll annihilate each other, usually with the release of powerful energy (photons).

Though in essence true, the statement is not exactly correct (and so can be misleading).

For example, if a proton comes into contact with a positron they will not annihilate. (If you recall, the positron is the antiparticle of the electron.) But if a positron comes into contact with an electron then, yes, they will annihilate (yielding a photon). (Maybe they will not instantaneously annihilate, since they could for the minutest moment combine to form positronium, a particle they form as they dance together around their center of mass – and then they annihilate into a photon.)

The annihilation would occur between particles that are conjugate to each other — that is, they have to be of the same type but “opposite.” So you could have a whole bunch of protons come into contact with antimatter particles of other non-protons and there will not be mutual annihilation between the proton and these other antiparticles.

Another example. The meson particles are represented in the quark model by a quark-antiquark pair. Like this: $p\bar q$. Here p and q could be any of the 6 known quarks $u, d, c, b, t, s$ and the $\bar q$ stands for the antiquark of $q$. If we go by the loose logic that “matter and antimatter annihilate” then no mesons can exist since $p$ and $\bar q$ will instantly destroy one another.

For example, the pion particle $\pi^+$ has quark content $u\bar d$ consisting of an up-quark u and the anti-particle of the down quark. They don’t annihilate even though they’re together (in some mysterious fashion!) for a short while before it decays into other particles. For example, it is possible to have the decay reaction

$\pi^+ \to \mu^+ + \nu_\mu$

(which is not the same as annihilation into photons) of the pion into a muon and a neutrino.

Now if we consider quarkonium, i.e. a quark and its antiquark together, such as for instance $\pi^0 = u\bar u$ or $\eta = d\bar d$, so that you have a quark and its own antiquark, then they do annihilate. But, before they do so they’re together in a bound system giving life to the $\pi^0, \eta$ particles for a very very short while (typically around $10^{-23}$ seconds). They have a small chance to form a particle before they annihilate. It is indeed amazing to think how such Lilliputian time reactions are part of how the world is structured. Simply awesome! 😉

PS. The word “annihilate” usually has to do when photon energy particles are the result of the interaction, not simply as a result of when a particle decays into other particles.

Sources:

(1) Bruce Schumm, Deep Down Things. See Chapter 5, “Patterns in Nature,” of this wonderful book. 🙂

(2) David Griffiths, Introduction to Elementary Particles. See Chapter 2. This is an excellent textbook but much more advanced with lots of Mathematics!

## Richard Feynman on Erwin Schrödinger

I thought it is interesting to see what the great Nobel Laureate physicist Richard Feynman said about Erwin Schrödinger’s attempts to discover the famous Schrödinger equation in quantum mechanics (see quote below). It has been my experience in reading physics that this sort of “heuristic” reasoning is part of doing physics. It is a very creative (sometimes not logical!) art with mathematics in attempting to understand the physical world. Dirac did it too when he obtained his Dirac equation for the electron by pretending he could take the square root of the Klein-Gordon operator (which is second order in time). Creativity is a very big part of physics.

“When Schrödinger first wrote it down [his equation],
he gave a kind of derivation based on some heuristic
arguments and some brilliant intuitive guesses. Some
of the arguments he used were even false, but that does
not matter; the only important thing is that the ultimate
equation gives a correct description of nature.”
— Richard P. Feynman
(Source: The Feynman Lectures on Physics, Vol. III, Chapter 16, 1965.)

## How to see Schrödinger’s Cat as half-alive

Schrödinger’s Cat is a thought experiment used to illustrate the weirdness of quantum mechanics. Namely, unlike classical Newtonian mechanics where a particle can only be in one state at a given time, quantum theory says that it can be in two or more states ‘at the same time’. The latter is often referred to as ‘superposition’ of states.

If D refers to the state that the cat is dead, and if A is the state that the cat is alive, then classically we can only have either A or D at any given time — we cannot have both states or neither.

In quantum theory, however, we can not only have states A and D but can also have many more states, such a for example this combination or superposition state:

ψ = 0.8A + 0.6D.

Notice that the coefficient numbers 0.8 and 0.6 here have their squares adding up to exactly 1:

(0.8)2 + (0.6)2 = 0.64 + 0.36 = 1.

This is because these squares, (0.8)2 and (0.6)2, refer to the probabilities associated to states A and D, respectively. Interpretation: there is a 64% chance that this mixed state ψ will collapse to the state A (cat is alive) and a 36% chance it will collapse to D (cat is dead) — if one proceeds to measure or find out the status of the cat were it to be in a quantum mechanical state described by ψ.

Realistically, it is hard to comprehend that a cat can be described by such a state. Well, maybe it isn’t that hard if we had some information about the probability of decay of the radioactive substance inside the box.

Nevertheless, I thought to share another related experiment where one could better ‘see’ and appreciate superposition states like ψ above. The great physicist Richard Feynman did a great job illustrating this with his use of the Stern-Gerlach experiment (which I will tell you about). (See chapters 5 and 6 of Volume III of the Feynman Lectures on Physics.)

In this experiment we have a magnetic field with north/south poles as shown. Then you have a beam of spin-half particles coming out of a furnace heading toward the magnetic field. The result is that the beam of particles will split into two beams. What essentially happened is that the magnetic field made a ‘measurement’ of the spins and some of them turned into spin-up particles (the upper half of the split beam) and the others into spin-down (the bottom beam). So the incoming beam from the furnace is like the cat being in the superposition state ψ and magnetic field is the agent that determined — measured! — whether the cat is alive (upper beam) or dead (lower beam). (Often in physics books they use the Dirac notation |↑〉for spin-up state and |↓〉 for spin-down.)

In a way, you can now see the initial beam emanating from the furnace as being in a superposition state.

Ok, so the superposition state of the initial beam has now collapsed the state of each particle into two specific states: spin-up state (upper beam) and the spin-down state (lower beam). Does this mean that these states are no longer superposition states?

Yes and No! They are no longer in superposition if the split beams enter another magnetic field that points in the same direction as the original one. If you pass the upper beam into a second identical magnetic field, it will remain an upper beam — and the same with the lower beam. The magnetic field ‘made a decision’ and it’s going to stick with it! 🙂

That is why we call these states (upper and lower beams) ‘eigenstates’ of the original magnetic field. They are no longer mixed superposition states — the cat is either dead or alive as far as this field is concerned and not in any ‘in between fuzzy’ states.

Ok, that addresses the “Yes” part of the answer. Now for the “No” part.

Let’s suppose we have a different magnetic field, one just like the original one but perpendicular in direction to it. (So it’s like you’ve rotated the original field by 90 degrees; you can rotate by a different angle as well.)

In this case if you pass the original upper beam (that was split by the first magnetic field) into the second perpendicular field, this upper beam will split into two beams! So with respect to the second field the upper beam is now in a superposition state!

Essential Principle: the notion of superposition (in quantum theory) is always with respect to a variable that is being measured. In our case, that variable is the magnetic field. (And here we have two magnetic fields, hence we have two different variables — non-commuting variables as we say in quantum theory.)

Therefore, what was an eigenstate (collapsed state) for the first field (the upper beam) is no longer an eigenstate (no longer ‘collapsed’) for the second (perpendicular) field. (So if a wavefunction is collapsed, it can collapse again and again!)

The Schrodinger Cat experiment could possibly be better understood not as one single experiment, but as a stream of many many boxes with cats in them. This view might better relate to the fact that we have beam of particles each of which is being ‘measured’ by the field to determine its spin status (as being up or down).

Best wishes, Sam

Reference. R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. III (Quantum Mechanics), chapters 5 and 6.

Postscript. It occurred to me to add a uniquely quantum mechanical feature that is contrary to classical physics thinking.   The Stern-Gerlach experiment is a good place to see this feature.

We noted that when the spin-half particles emerge from the furnace and into the magnetic field, they split into upper and lower beams. Classically, one might think that before entering the field the particles already had their spins either up or down before a measurement takes place (i.e., before entering the magnetic field) — just as one might say that the earth has a certain velocity as it moves around the sun before we measure it. Quantum theory does not see it that way. In the predominant Copenhagen Interpretation of quantum theory, one cannot say that the particle spins were already a mix of up or down spins before entering the field. Reason we cannot say this is that if we had rotated the field at an angle (say at right angles to the original), the beams would still split into two, but not the same two beams as before! So we cannot say that the particles were already in a mix of those that had spins in one direction or the other. That is one of the strange features of quantum theory, but wonderfully illustrated by Stern-Gerlack.

Mathematics. In vector space theory there is a neat way to illustrate this quantum phenomenon by means of bases’. For example, the vectors (1,0) and (0,1) form a basis for 2D Euclidean space R2. So any vector (x,y) can be expressed (uniquely!) as a superposition of them; thus,

(x,y) = x(1,0) + y(0,1).

So this would be, to make an analogy, like how the beam of particles, described by (x,y), can be split into to beams — described by the basis vectors (1,0) and (0,1).

However, there are a myriad of other bases. For example, (2,1) and (1,1) also form a basis for R2. A general vector (x,y) can still be expressed in terms of a superposition of these two:

(x,y) = a(2,1) + b(1,1)

for some constants a and b (which are easy to solve in terms of x, y). So this other basis could, by analogy, represent a magnetic field that is at an angle with respect to the original — and its associated beams (2,1) and (1,1) (it’s eigenstates!) would be different because of their different directions. As a matter of fact, we can see here that these eigenstates (collapsed states), represented by (2,1) and (1,1), are actual (uncollapsed) superpositions of the former two, namely (1,0) and (0,1). And vice versa!

Analogy: let’s suppose the vector (5,3) represents the particle states coming out of the furnace. Let’s think of the basis vectors (1,0) and (0,1) represent the spin-up and spin-down beams, respectively, as they enter the first magnetic field, and let the other basis vectors (2,1) and (1,1) represent the spin-up and spin-down beams when they enter the second rotated (maybe perpendicular) magnetic field. Then the particle state (5,3) is a superposition in each of these bases!

(5,3) = 5(1,0) + 3(0,1),

(5,3) = 2(2,1) + 1(1,1).

So it is now quite conceivable that the initial mixed state of particles as they exit the furnace can in fact split in any number of ways as they enter any magnetic field! I.e., it’s not as though they were initially all either (1,0),(0,1) or (2,1),(1,1), but (5,3) could be a simultaneous combination of each — and in fact (5,3) can be combination (superposition) in an infinite number of bases.

Indeed, it now looks like this strange feature of quantum theory can be described naturally from a mathematical perspective! Vector Space bases furnish a great example!

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