However, the equation is just one line. — TheMadFool

It's expressed in rules R1 and R2.

What I see is the problem how, math is a language, a perfectly sensible expression (equation of quantum superposition) in math when translated into another language (natural languages like English), most who do so end up with a contradiction? I can't wrap my head around that, sir/madam, as the case may be. — TheMadFool

Popular science writing is both a blessing and a curse...

I'm not satisfied with your answer. Thank you for taking the trouble to explain it though. G'day.

The following rules apply to a quantum flip:

R1. quantum flip(heads) = heads + tails
R2. quantum flip(tails) = heads - tails

Applying this to the earlier experiment:
1. prepare: heads
2. quantum flip(heads) = heads + tails
3. quantum flip(heads + tails) = quantum flip(heads) + quantum flip(tails) = (heads + tails) + (heads - tails) = heads + heads
4. measure: heads — Andrew M

I don't understand. Can you please write that in the usual bra–ket notation?
Especially the minus sign in R2 is strange.

I'm not satisfied with your answer. Thank you for taking the trouble to explain it though. G'day. — TheMadFool

You're welcome! What in particular were you not satisfied with?

Can you please write that in the usual bra–ket notation?
Especially the minus sign in R2 is strange. — SolarWind

Sure. The minus sign represents a 180 degree phase shift which differentiates the two superposition states. Adding them results in destructive interference for the tails component.

The coin (or particle) can be represented by a qubit where heads (or spin-up) is defined as:

$\hspace{10 mm}|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

and tails (or spin-down) is defined as:

$\hspace{10 mm}|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

The quantum coin flipper is implemented by a Hadamard (H) operation:

$\hspace{10 mm}H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

1. The coin is prepared in a heads state:

$\hspace{10 mm}|0\rangle$

2. Perform first quantum coin flip:

$\hspace{10 mm}H|0\rangle = \frac{1}{\sqrt2}(|0\rangle + |1\rangle)$

3. Perform second quantum coin flip:

$\hspace{10 mm}\begin{aligned}H(\frac{1}{\sqrt2}(|0\rangle + |1\rangle)) &= \frac{1}{\sqrt2}(\frac{1}{\sqrt2}(|0\rangle + |1\rangle) + \frac{1}{\sqrt2}(|0\rangle - |1\rangle)) \\ &= \frac{1}{2}|0\rangle + \frac{1}{2}|1\rangle + \frac{1}{2}|0\rangle - \frac{1}{2}|1\rangle \\ &= |0\rangle\end{aligned}$

4. Perform measurement:

$\hspace{10 mm}|0\rangle$

You're welcome! What in particular were you not satisfied with? — Andrew M

It just doesn't feel right to me.

Here's an equation/formula, a rather simple one, F = ma (Newton).

In English F = ma can be rendered as the magnitude of a force F acting on a mass m that imparts an acceleration a is equal to the product of m and a.

Do something similar with the equation for superposition (Schrödinger's?).

Do something similar with the equation for superposition (Schrödinger's?). — TheMadFool

It seems to me that you're thinking of a superposition as a kind of law (like F=ma). It's not. It's just a particular kind of state that a quantum system can be in.

In classical mechanics, the definite position of a system can be calculated. The classical expectation is that it will be in that definite position whether or not it is measured.

In quantum mechanics (using the Schrödinger equation), the wave function for a system is calculated. The wave function is then used to calculate the probability of finding the system at any definite position. For example, if the wave function has amplitude (i.e., height or depth) at positions' 2 and 10 and those amplitudes are equal, then there is a 50% probability of finding the system at position 2 and a 50% probability of finding the system at position 10. This combination of potentially measurable positions prior to measurement is termed a superposition.

It seems to me that you're thinking of a superposition as a kind of law (like F=ma). It's not. It's just a particular kind of state that a quantum system can be in. — Andrew M

Even if we take Schrödinger's equation as just that and not a law which I think it is, we should be able to come up with a one line description of it.

Take this equation: $y = 2x + 7$. In English it would be y is equal to twice x increased by 7.

we should be able to come up with a one line description of it.

Take this equation: y=2x+7. In English it would be y is equal to twice x increased by 7. — TheMadFool

Perhaps this is what you're looking for?

Perhaps this is what you're looking for? — Andrew M

:up: Yes and No.

The link you provided to a science forum does lead me to the exact same question I asked but it still seems deeply immersed in mathematical concepts that don't have corresponding, matching ideas that are part of normal, non-technical discourse. The question then is, is math invented or discovered?

For example, if the wave function has amplitude (i.e., height or depth) at positions' 2 and 10 and those amplitudes are equal, then there is a 50% probability of finding the system at position 2 and a 50% probability of finding the system at position 10. This combination of potentially measurable positions prior to measurement is termed a superposition. — Andrew M

Such a wavefunction does not exist. Dirac deltas are not eigenfunctions. They are distributions.

The question then is, is math invented or discovered? — TheMadFool

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world.

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world. — GraveItty

Invented and derived. :chin:

Perhaps we forgot what we had invented. This happens for real I believe. Andrew Wiles (mathematician who proved Fermat's Last Theorem) must've all but forgotten his proof.

Don't take derivation litterally, as if physical structures posses mathematical structures. A circular structure doesn't possess a math formula for a circle. The circles on top of my soup can induce (maybe that's a better word than derivation) a math formula for a circle. It's not an inherent formula of the circle. At least, not for me. A Platonist would disagree, although he places the formula in an extra worldly heavenly realm. Some place it in the objects themselves. People like Deutsch and Texmark, who, especially the former, try to base QM on a formal math basis. Without any knowledge of what the math describes I think that's a fruitless effort. David Bohm's hidden variables, though the nature of these variables stays obscure (but hey, isn't the Nature of reality obscure?, come closer to a physical reality, explaining the probalistic nature of QM. It's a pitty the people at the Copenhagen conference gave the fiat to the probability interpretation! How different the standard texts on QM would have looked!

Don't take derivation litterally, as if physical structures posses mathematical structures — GraveItty

Why would physical structures not possess mathematical structures. When I look at an ordinary die, I see a cube and when I look at the sun/moon, I see a sphere.

Why would physical structures not possess mathematical structures. When I look at an ordinary die, I see a cube and when I look at the sun/moon, I see a sphere. — TheMadFool

Of course. No doubt about that. It is their mathematical formulation and their abstraction in the mathematical realm (a mathematical sphere is a different sphere as a physical one) that they don't possess. There are in fact
no structures in the physical world corresponding exactly to mathematical structures unless prepared in a very precise way as to accommodate a precise math form. A temporally finite piece of music (its sound pattern, that is), an arbitrary pulse of sound, or an infinite periodic pattern, can be written, but only approximated as an infinite sum of sine waves of sound with appropriate coefficients (a procedure like the one used in the epicycle approach to the motion of celestial bodies). There corresponds no exact mathematical form in the realm of math. Only an approximation. If the piece or pulse become too extended then the pattern can't be approximated by math. Though the pattern is actually there. Only single sine soundwaves and finite combinations of them have exact correspondences in the mathematical kingdom. You can question if an approximation corresponds to the exact pattern of the piece of music. There is no exact or even approximate mathematical structure of the musical sound pattern if the piece is too long. So eventhough there is a physical pattern, there is corresponding pattern in math. Of course you can transpose the non-functional, non-approachable sound pattern of the piece of music to the realm of math as a form. But this form can't be expressed as a function (other sets of base-functions could be chosen, but this doesn't change the argument). There is in fact not much you do with the transposed form. You can construct tangent lines (or planes and volumes if the sound pattern is 2- or 3-dimensional. But that's about it. The pattern is non-reducible.

So what's my point? My point is that forms in the mathematical realm owe their existence to the physical reality.

It is their mathematical formulation and their abstraction in the mathematical realm (a mathematical sphere is a different sphere as a physical one) that they don't possess. — GraveItty

Why? Are you trying to say that, for instance, Brad Pitt, Albert Einstein, Abraham Lincoln, (basically men) are not men?

Only an approximation. — GraveItty

I see, a Platonic point of view as far as I can tell. Did Plato ever consider the imagination? Did he not realize, I'm sure he must've, that, in a certain sense, perfection exists only imagination? How does he tell apart imagination and the world of forms? Imagination = World of forms? :chin:

My point is that forms in the mathematical realm owe their existence to the physical reality. — GraveItty

Why can't it be the other way round? The physical seems to be obeying mathematical forms.

Are you trying to say that, for instance, Brad Pitt, Albert Einstein, Abraham Lincoln, (basically men) are not men? — TheMadFool

I'm not sure about Brad, and coming to think about neither about Einstein. Of course they belong to mankind. Though coming to think about it... No, seriously, that is not what I am trying to say. They are men without a mathematical equivalent somewhere because their mathematical equivalent doesn't exist, as it doesn't for the soundwave pattern. That is, not expressable in mathematical terms. For example, a sinus function can be expressed graphically as a wavy line in the plane. But what about a line that can't be functionally expressed? It simply doesn't exist in the mathematical realm. Nevertheless, I can draw the line. It exists physically.

Why can't it be the other way round? The physical seems to be obeying mathematical forms. — TheMadFool

This question is exactly the reason I argued like I did and I answered it already. If a physical form (the drawn or imagined line) can't be expressed as a math formula, there is no counterpart of the form in the math space.

I see, a Platonic point of view as far as I can tell. Did — TheMadFool

Not at all. Plato imagined a rmathematical heaven, like the math formalists. The place of math is simply the mind. Like intuitionists think. Approximations don't have nothing to do with being a Platonist or not, although Plato indeed thinks that real physical forms are approximations of the mathematical ones (which are not the same as physical forms).

They are men without a mathematical equivalent somewhere because their mathematical equivalent doesn't exist — GraveItty

How do you know that? What's nonmathematical about man?

This question is exactly the reason I argued like I did and I answered it already. If a physical form (the drawn or imagined line) can't be expressed as a math formula, there is no counterpart of the form in the math space. — GraveItty

But there were times, I believe, when many mathematical topics today were once never thought to have been mathematical. You seem math-literate, I'm sure you can think of such an instance. How about the mathematical turn to natural philosophy flagged off by Copernicus/Galileo/Newton?

The place of math is simply the mind — GraveItty

You don't know that.

Here's some food for thought: Is the theory of a mathematical universe an illusion, a bewitchment by language (re: Wittgenstein)? Not everything in math, to my knowledge, is about numbers/shapes (arithmetic/geometry) but when someone claims "it's all math" he means to say that arithmetic/geometry is involved.

Thus what we have here is a discipline/field (math) whose expansionist behavior is gobbling up other fields/disciplines but the catch is, its (math's) definition is also being altered to factor this in until what we have today, as I'm led to believe, math is a study of patterns. I'm sure Thales, Pythagoras, Archimedes would raise pertinent objections to this point of view. I dunno!

What's nonmathematical about man? — TheMadFool

They can't be represented by a formal math system, whatever the system. Just like the line that can't be represented by a math expression has no correlate in the mathematical world, a man thinking about math expressions and pondering upon non-math-expressable or non computable forms like a random but non-random line, will have no correlate too.

Math is just a language and certainly not the language of Nature, as is so commonly claimed by physicists. And like all languages it has limitations. It's also called a universal language by many scientists (and the parrots, quasi intelligently, reciting their words). BS! It's a constraining and confining net thrown on physical reality, thereby darkening many facts, though in some situations it fits perfectly and with reasonable effectiveness.

The question then is, is math invented or discovered? — TheMadFool

Math is derived from structures in the physical world. — GraveItty

Originally perhaps. Not so much these days. Abstractions and generalizations abound. After many years I have concluded that math is both created and discovered. When I define a new (but tiny) math object, that is more creative than discovered. Afterwards one can discover what follows from such a definition. But generally speaking most mathematicians pay little attention to the question.

There corresponds no exact mathematical form in the realm of math. — GraveItty

I recently defined an "LFT form" and "attractor form" for linear fractional transformations. These are "exact" mathematical forms. However, "form" can take on other meanings in math. I'm not talking about Platonic forms.

So what's my point? My point is that forms in the mathematical realm owe their existence to the physical reality. — GraveItty

Some do.

The scope of mathematics today is astounding, with an unknown number of areas of investigation. Each day about 250 new papers arrive in ArXiv.org, to be roughly categorized in about 35 general areas. Awhile ago I challenged Alexandre to find out how many math articles exist on Wikipedia.

It's a constraining and confining net thrown on physical reality, thereby darkening many facts — GraveItty

Possibly. Mathematicians working in pure math ask, Why throw a net?

Such a wavefunction does not exist. Dirac deltas are not eigenfunctions. They are distributions. — GraveItty

Yes, I was envisaging a wave function that just approximates those statistics. Also, welcome to the forum!

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world. — GraveItty

And what is structure? I notice you give a nod to Aristotle in another post, who defined things in terms of matter and form.

Originally perhaps. Not so much these days. Abstractions and generalizations abound. After many years I have concluded that math is both created and discovered. When I define a new (but tiny) math object, that is more creative than discovered. Afterwards one can discover what follows from such a definition. — jgill

I can remember that Mandelbrot said in an interview that his realization of the set named after him (stemming from the iterations in the complex field of the simple zexp2+k function, giving rise to these colored pictures on which you can zoom in ad infinity, or ad nauseam) felt more like a discovery than an invention. Well, if that's the case for him then why not. I just don't think this is so. Math can't be discovered (say I indeed, but not holding it for an absolute truth). The field of math is of course very rich. I don't think it is. And of course many mathematicians don't throw it on the physical universe. You can find mathematical models (say the different quantum field models) that have no counterpart in physics. These models obviously don't have a physical counterpart, but they describe the same physical structures with non-existing parameters. All math corresponds to physical structures, though they are abstracting them to a high level. The spin 1/2 of the farming can be represented by a spinor or a vector traveling twice around a Mōbius band, but this isn't what an electron truly is. Neither is it a mathematical point or a collection of charges on a 4d curved Planck-sized structure seemingly 3d in our 3d universe (though the last comes close). Even "the great Feynman" said that the cause of the electrons spin is a mystery, though we'll described by physics. How the hell can an electron 's spin rotate once if you rotate the electrons twice (which is closely connected to anti-symmetric wavefunctions), a fact I have seen beautifully illustrated by rotating one of two interfering electron fields once, which resulted in an inversion of the interference pattern. Somehow the anti symmetry and the spin half are connected. Of course they are connected by math (like spin 1 vectors and symmetric wavefunctions are connected) but what happens in reality remains a mystery. I have a gut feeling about that though. But this is not the place to get into that. A Mōbius band is a form that can be realized in the physical world. Certain properties of it can be described by math. But if we make the band sufficiently erratic (while keeping its overall form) math can't describe anymore (like the erratic line I mentioned earlier, without a functional form can't be described by a mathematical formula and as such doesn't exist in the mathematical world. You can of course measure the line's distance to a n origin but it still can't be described by a formula, though you could of course claim that it can be described by an infinity of real numbers, though not being calcuable). However it may be be, for me math is a human invention and it can reasonably be applied to the physical world. It would be far more unreasonable if it couldn't. It certainly is not an inherent property of Nature but can sometimes describe it very well. Again, the simple example of a physical line form, say a long sewing thread, can't be calculated. Measured yes.

a fact I have seen beautifully illustrated by rotating one of two interfering electron fields once, which resulted in an inversion of the interference pattern. — GraveItty

So we've settled on waves now? Interference happens when the behavior is waves. Is this right?

So we've settled on waves now? Interference happens when the behavior is waves. Is this right? — Caldwell

If you ask so, then we can settle on them. Fasten your seatbelts though for some heavy destructive interference can be expected. Which basically answers your second question.