# Mathematical models, the 'real world' and quantum mechanics

## Main Question or Discussion Point

A mathematical model may show that there are two possible states for something in the real world, let's say a coin, that may be either heads or tails; make a computer program to simulate coin flipping and you soon discover that the odds of heads or tails is 50/50. In the real world however, actually flipping a coin, you may discover that there is another state possible ... neither heads nor tails ... when it lands on it's edge, meaning that the odds are not QUITE 50/50. I personally have done this a few times in my life, flipped a coin and have it land and stop on it's edge. But a program with a simple head or tails outcome will never come up with a 'neither heads nor tails' result. I suppose that it could be argued that an 'ideal coin' would be infinitely thin and so could never come to rest on it's edge, so producing a guaranteed heads or tails always result. However, too thin and the edge would act like a knife blade and 'get stuck' in a surface onto which it was tossed if it landed edge first on it's first contact with the surface. A really wide edge would guarantee mostly a neither heads or tails result making the heads or tails result as unlikely as a normal coin coming to rest on it's edge. Between these to extremes of infinitely thin and extremely wide edges, I don't believe you ever really get 100% heads or tails result, free of possible edge landing results of coin flipping. So in the physical, macroscopic realm we will never be able to emulate that mathematical model of coin flipping 100%.

Quantum mechanics is all about probabilities. How does this situation come into play in quantum mechanics? Is there a 'cat isn't in the box this time' outcome? or 'the cat is still both dead and alive' or 'it's both a particle and wave' or 'there is neither a particle or wave there this time' possibilites? If there is no possibility for this 'none of the above conditions', then what is the fundamental reason why not?

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Nabeshin
Quantum mechanics is all about probabilities. How does this situation come into play in quantum mechanics? Is there a 'cat isn't in the box this time' outcome? or 'it's both a particle and wave' or 'there is neither a particle or wave there this time' possibilites? If there is no possibility for this 'none of the above conditions', then what is the fundamental reason why not?
Let me first just note that you're pushing the analogy a bit far here. While what you say about a coin is obviously true for a coin, quantum mechanics does not describe coins! Simply because the situations deal with probabilities does not mean that they are comparable. Also note that your final comment, about attempting to predict the outcome of the flip based on initial conditions may be possible in classical mechanics but is not in quantum mechanics. The existence of other variables we cannot measure which somehow "remove the probability" inherent in quantum mechanics is known as hidden variable theory. Not really important to this discussion, in my opinion, but just figured I'd mention it.

Onto the main point of the question, one can imagine, just to make up an example, a situation in which QM predicts that the spin of a particle is either up or down. Now your concern translates to something like "well what if it is neither up nor down?" Ultimately, this question comes down to observational evidence and the fact is that the states predicted by QM are precisely the states which are observed when we go out and perform an experiment. That is, when we do the above mentioned experiment, we find that the proportion of electrons in spin up plus the proportion in spin down is equal to one, i.e. there is no third state (Of course you can always try to say that the third state merely has a much lower probability of occurring so of course it did not show up in your experiment. This is weaseling your way out of the problem).

Someone may be able to provide a more theoretical answer, but likely this is to be in the context of QM and so if you think QM is not the whole story in the first place you are likely to be unsatisfied. Ultimately, it comes down to observational tests.

ZapperZ
Staff Emeritus
A mathematical model may show that there are two possible states for something in the real world, let's say a coin, that may be either heads or tails; make a computer program to simulate coin flipping and you soon discover that the odds of heads or tails is 50/50. In the real world however, actually flipping a coin, you may discover that there is another state possible ... neither heads nor tails ... when it lands on it's edge, meaning that the odds are not QUITE 50/50. I personally have done this a few times in my life, flipped a coin and have it land and stop on it's edge. But a program with a simple head or tails outcome will never come up with a 'neither heads nor tails' result. I suppose that it could be argued that an 'ideal coin' would be infinitely thin and so could never come to rest on it's edge, so producing a guaranteed heads or tails always result. However, too thin and the edge would act like a knife blade and 'get stuck' in a surface onto which it was tossed if it landed edge first on it's first contact with the surface. A really wide edge would guarantee mostly a neither heads or tails result making the heads or tails result as unlikely as a normal coin coming to rest on it's edge. Between these to extremes of infinitely thin and extremely wide edges, I don't believe you ever really get 100% heads or tails result, free of possible edge landing results of coin flipping. So in the physical, macroscopic realm we will never be able to emulate that mathematical model of coin flipping 100%.

Quantum mechanics is all about probabilities. How does this situation come into play in quantum mechanics? Is there a 'cat isn't in the box this time' outcome? or 'the cat is still both dead and alive' or 'it's both a particle and wave' or 'there is neither a particle or wave there this time' possibilites? If there is no possibility for this 'none of the above conditions', then what is the fundamental reason why not?
Please do a search on "quantum superpostion" and/or "Schrodinger Cat-states" in this forum. You need to first understand the principle of quantum superposition, experimental evidence of the existence of such principle in QM, and why this is different than what you think you've understood.

Zz.

A. Neumaier
2019 Award
In the real world however, actually flipping a coin, you may discover that there is another state possible ... neither heads nor tails ... when it lands on it's edge, [...]
If there is no possibility for this 'none of the above conditions', then what is the fundamental reason why not?
Textbook discussions are always idealizations. In real life, experimenters account for the non-ideal circumstances by introducing some efficiency factor. So you may have a spin in a mixed state R (a Hermitian 2x2 matrix). Ideally, the trace of R is 1 (the sum of the up and down probability, recorded in the two diagonal elements).

Taking into account imperfections, the trace of R is allowed to be less than 1. And the equations for R must account for dissipative effects - loss of information due to things not accounted for. This leads to Lindblad dynamics in place of the usual Schroedinger-von Neumann dynamics.

ZapperZ
Staff Emeritus
Maybe we are talking bout something along this line ...

http://quantummechanics.mchmultimedia.com/2010/quantum-mechanics/149/
Maybe we're not!

Why did you go from a basic question on quantum superposition to quantum entanglement? You can't go to the latter without understanding the former. You need to learn how to walk first before attempting to do a triple Axel on the skating ring.

Zz.

Well, Zapper, the dichotomous examples that I put in my original post, were there simply to illustrate binary situations that might be analogous to coin flipping that had some relation to coin flipping. I was in a hurry when I wrote it and if I had thought deeper on it I might have chosen a better example and might not have upset you. Sorry.

Actually I was hoping to see some debate on mathematical modeling, simulations and real world fit, etc, on a more abstract level. Your responses have however made me lose any further interest in pursuing the concept I was attempting to discuss. You have adequately proven my lack of knowledge in the field of QM. I think it's time for me to leave these forums, obviously they aren't for novices. I will leave you with this final link.

http://en.wikipedia.org/wiki/Mentor

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A. Neumaier
2019 Award
Actually I was hoping to see some debate on mathematical modeling, simulations and real world fit, etc, on a more abstract level. Your responses have however made me lose any further interest in pursuing the concept I was attempting to discuss. You have adequately proven my lack of knowledge in the field of QM. I think it's time for me to leave these forums, obviously they aren't for novices.
It doesn't make sense to discuss quantum mechanical modeling without some background.

If you are a novice lacking that background, the responses you get should encourage you to acquire enough of it so that the discussion of the questions you are interested makes sense, and to ask questions that help you get this background more quickly. Then you'd see that PF can be a great resource and deepen your understanding.