Copper Bars and Quantum fluctuations

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Copper Bars and Quantum fluctuations

Postby hyksos on October 6th, 2017, 12:06 pm 

I recently watched a lecture where a professor was talking about precision of floating point numbers in a computer. The topic of round-off error and precision came up. There was a desk at the front of the room , and the lecturer made an analogy with imagining how you would represent the "true length" of the desk. As if to nod-and-wink to physicists in the room, he then made a very short digression :

"I won't get into whether that is even physically possible in practice."

In this thread, I will try to approach this question in the deepest possible way. Does it even make sense to talk about the "actual length" of a desk in a room?

To ease visualization and discussion, we will replace the desk with a tiny bar of copper. The bar will be in the shape of a perfect cylinder, with the diameter being roughly the size of the ink cartridge in a ballpoint pen. The length of the cylinder will be nearly 1 centimeter (at least measured crudely with a ruler). I will further stipulate that the crystalline structure of the cylinder is perfectly machined, so that the molecular grid is a perfect lattice with no imperfections or impurities. For the rest of this article, I will be discussing how or even if, we would assign a true length to this tiny copper bar.

The far ends of the bar are determined by the locations of two copper molecules -- one sitting the farthest left extent of the bar, and the other at the farthest right end. If the bar is at room temperature, we expect that those ending molecules would be shaking around from thermal fluctuations within the bar. (see phonons )

Going back to the decimal length, we would consider the farthest righthand digits of the significand. Given in meters, we have almost exactly 1 centimeter.

1.000001018283839 x 10-2 m

As time progresses in our lab we continue to measure the length of the copper bar using the finest laser measuring device. Due to thermal fluctuations we expect that the last three digits of this number will change over time, bouncing between various values seemingly at random.

1.000001018283653 x 10-2 m
1.000001018283113 x 10-2 m
1.000001018283608 x 10-2 m
1.000001018283345 x 10-2 m
1.000001018283915 x 10-2 m
1.000001018283821 x 10-2 m

We could cool the copper bar cryogenically, getting its temperature down to (perhaps) milli-kelvins. We repeat our measurements as before and then note that while the far end molecules fluctuate in position, the measurements tend to gravitate around a 'true' value that is the center of a gaussian distribution.

As long as there is any heat in the bar at all, we will never get a true measurement of its length, so we can try harder. We could decrease the bar's temperature down to nanokelvin, say 50 nK. We expect that the true "end" of the bar would be the widest that an electron would fly out in its orbit around a single nucleus of copper atom. We expect that this must be a particular, point-like location in space. Nanokelvin temperatures allows us to gain additional stable digits of precision, but we are left with the problem of the true length.

1.000001018283674815675915 x 10-2 m
1.000001018283674815675821 x 10-2 m

There are still fluctuating values on the far righthand digits of the significand, even after we have removed all thermal heat from the bar. Worse, these fluctuations don't follow a gaussian at all. They are not a product of error in our measuring device, nor are they the product of heat energy. So what is that fluctuating?

At these distances we must relent on achieving our original mission. We could never know the "true length" of this copper bar, because electrons cannot have a position beyond what would be allowed by the Heisenberg Uncertainty Principle. The wiggling decimal digits at the end of the measurement are the product of quantum fluctuations, which persist in ultra-cold solids , well after all the wiggling due to heat energy is removed. We could never remove these quantum fluctuations, because they are an intrinsic property of matter itself.

... and that dovetails with this thread : viewtopic.php?f=2&t=33307
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Re: Lengthy Discussion

Postby Faradave on October 7th, 2017, 12:43 pm 

I don't disagree. I think we must recognize a problem in the definition of "length". Our units may be defined in terms of the exact distance that light travels in a vacuum in a particular amount of time. However, it does no good to specify duration more precisely than such duration can be physically measured.

Further, how far light travels is at the mercy of physical measures of the locations of emission and absorption. Physical detectors all have the same problems you noted of the copper bar itself. This in turn, reflects on the definitions so derived.
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