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New derivation of pi links quantum physics and pure math

113 points by bpolania 11y ago

dang 11y ago
Url changed from http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jCp http://phys.org/news/2015-11-derivation-pi-links-quantum-phy..., which points to this.
fengwick3 11y ago
I don't know enough about quantum physics, but from what I understand, this pi seems to have arose as an artifact of the variational method used, rather than quantum mechanics itself.
- Create 11y ago
  be floored ^1. https://en.wikipedia.org/wiki/Buffon%27s_needle https://en.wikipedia.org/wiki/Buffon%27s_needle ^1 and don't fall for phy PR
  - Create 11y ago
    Dear Downvoter: can you tell when The Cat makes it through the crack?
- coldtea 11y ago
  It's quantum mechanics that enable the "variational method used" to give this result -- it wont appear in all calculations done with the same method in other domains.
keithpeter 11y ago
Extended quote from original article... "When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels." Rings bells from celestial mechanics. You can end up with things like a series solution converging on an elliptical or circular orbit when you apply series methods to simple (or 'simple' in this case - it is quantum mechanics!) systems. I'll see if I can hack my way though their paper (linked to OA, downloadable in full text).
Garlef 11y ago
What's so special about this ? I was always under the impression that quantum physics is allready 99% pure math ;)
- rurban 11y ago
  Esp. when they appear to be excited to see a physical phenomenon derived from the length of circle. I mean, really?
- nileshtrivedi 11y ago
  I felt the same. For example, gravity/electromagnetism force depends on distance, hence naturally lead to circular/spherical contours, which would involve Pi.
gus_massa 11y ago
Pi appears in many many many quantum calculations, so the title of the press article is very bad. The title of the research article is much better "Quantum mechanical derivation of the Wallis formula for pi" because as far as I know the Wallis formula has never appear before in a QM calculation.
4ad 11y ago
Of course there is nothing special about this result. The hyped "science" articles describing this are longer than the paper! Lubos Motl wrote about it here: http://motls.blogspot.co.at/2015/11/pi-found-in-hydrogen-atom.html http://motls.blogspot.co.at/2015/11/pi-found-in-hydrogen-ato...
johncolanduoni 11y ago
Somebody should tell these guys about hbar. Just kidding, it is interesting that this series showed up in QM. Terrible article title though...
murbard2 11y ago
Many series that calculate pi are very simple, as a result, it can randomly appear in physical problems. See for instance http://ics.org.ru/doc?pdf=440&dir=e http://ics.org.ru/doc?pdf=440&dir=e
- sawwit 11y ago
  That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappear in different contexts with high probability.
  - murbard2 11y ago
    More precisely it would be the algorithmic probability. Not only are there short programs that compute mathematical constants, but there are many such programs, so the total mass is even higher.
    - sawwit 11y ago
      I'm not sure what you mean. Do you mean that a mathematical constant has multiple different shortest programs?
      - murbard2 11y ago
        Not necessarily shortest, but it has many short programs. The algorithmic probability of a sequence x is the sum over the set of all prefix-free programs u that calculate x of 2^-len(u). Think of the shortest description length as a MAP, while the algorithmic probability integrates over the full prior. A constant which has many short programs can thus have a greater algorithmic probability than another constant with a slightly lower Kolmogorov complexity. To put it back in context, it's possible that a very short program computes some constant, but it's unlikely to be an important mathematical constants. What's particular about mathematical constants is that they keep appearing in many different situations.
- rr56 11y ago
a3n 11y ago
Maybe because, not being a mathematician, and certainly not a quantum mechanic, I understand pi and circles as being lock-step mutually defining; from what I've read, this derivation of pi doesn't seem to be described as anything more than an interesting coincidence, and to call it a "link" is reaching.
stared 11y ago
The game of mafia (aka Werewolf) is linked to PI (my paper: http://arxiv.org/abs/1009.1031 http://arxiv.org/abs/1009.1031, for a summary: http://boardgamegeek.com/thread/1157558/polish-paper-werewolf http://boardgamegeek.com/thread/1157558/polish-paper-werewol...), also via the Wallis formula. But so what? Pi appears in physics roughly as often as 2 in finance. Even to describe the simples quantum states (ground states of the harmonic potential) you use Pi (as a normalization factor).

113 points by bpolania 11y ago

dang 11y ago

Url changed from http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jCp http://phys.org/news/2015-11-derivation-pi-links-quantum-phy..., which points to this.

fengwick3 11y ago

I don't know enough about quantum physics, but from what I understand, this pi seems to have arose as an artifact of the variational method used, rather than quantum mechanics itself.

Create 11y ago
be floored ^1. https://en.wikipedia.org/wiki/Buffon%27s_needle https://en.wikipedia.org/wiki/Buffon%27s_needle ^1 and don't fall for phy PR
- Create 11y ago
  Dear Downvoter: can you tell when The Cat makes it through the crack?
coldtea 11y ago
It's quantum mechanics that enable the "variational method used" to give this result -- it wont appear in all calculations done with the same method in other domains.

keithpeter 11y ago

Extended quote from original article... "When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels." Rings bells from celestial mechanics. You can end up with things like a series solution converging on an elliptical or circular orbit when you apply series methods to simple (or 'simple' in this case - it is quantum mechanics!) systems. I'll see if I can hack my way though their paper (linked to OA, downloadable in full text).

Garlef 11y ago

What's so special about this ? I was always under the impression that quantum physics is allready 99% pure math ;)

rurban 11y ago
Esp. when they appear to be excited to see a physical phenomenon derived from the length of circle. I mean, really?
nileshtrivedi 11y ago
I felt the same. For example, gravity/electromagnetism force depends on distance, hence naturally lead to circular/spherical contours, which would involve Pi.

gus_massa 11y ago

Pi appears in many many many quantum calculations, so the title of the press article is very bad. The title of the research article is much better "Quantum mechanical derivation of the Wallis formula for pi" because as far as I know the Wallis formula has never appear before in a QM calculation.

4ad 11y ago

Of course there is nothing special about this result. The hyped "science" articles describing this are longer than the paper! Lubos Motl wrote about it here: http://motls.blogspot.co.at/2015/11/pi-found-in-hydrogen-atom.html http://motls.blogspot.co.at/2015/11/pi-found-in-hydrogen-ato...

johncolanduoni 11y ago

Somebody should tell these guys about hbar. Just kidding, it is interesting that this series showed up in QM. Terrible article title though...

murbard2 11y ago

Many series that calculate pi are very simple, as a result, it can randomly appear in physical problems. See for instance http://ics.org.ru/doc?pdf=440&dir=e http://ics.org.ru/doc?pdf=440&dir=e

sawwit 11y ago
That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappear in different contexts with high probability.
- murbard2 11y ago
  More precisely it would be the algorithmic probability. Not only are there short programs that compute mathematical constants, but there are many such programs, so the total mass is even higher.
  - sawwit 11y ago
    I'm not sure what you mean. Do you mean that a mathematical constant has multiple different shortest programs?
    - murbard2 11y ago
      Not necessarily shortest, but it has many short programs. The algorithmic probability of a sequence x is the sum over the set of all prefix-free programs u that calculate x of 2^-len(u). Think of the shortest description length as a MAP, while the algorithmic probability integrates over the full prior. A constant which has many short programs can thus have a greater algorithmic probability than another constant with a slightly lower Kolmogorov complexity. To put it back in context, it's possible that a very short program computes some constant, but it's unlikely to be an important mathematical constants. What's particular about mathematical constants is that they keep appearing in many different situations.
rr56 11y ago

a3n 11y ago

Maybe because, not being a mathematician, and certainly not a quantum mechanic, I understand pi and circles as being lock-step mutually defining; from what I've read, this derivation of pi doesn't seem to be described as anything more than an interesting coincidence, and to call it a "link" is reaching.

stared 11y ago

The game of mafia (aka Werewolf) is linked to PI (my paper: http://arxiv.org/abs/1009.1031 http://arxiv.org/abs/1009.1031, for a summary: http://boardgamegeek.com/thread/1157558/polish-paper-werewolf http://boardgamegeek.com/thread/1157558/polish-paper-werewol...), also via the Wallis formula. But so what? Pi appears in physics roughly as often as 2 in finance. Even to describe the simples quantum states (ground states of the harmonic potential) you use Pi (as a normalization factor).