As far as I can tell, the area of the tip of the pencil doesn't factor into any of his calculations. But why doesn't it? Is he assuming it's a zero area point like we do in geometry?
Taken to an extreme, if the tip were 1 cm² and flat, then it wouldn't ever tip over, just as a coffee cup doesn't tip over.
Not only that, but if the tip is a single point then will the pencil
ever be vertical? That would imply that the pencil's center of mass is
perfectly aligned with the tip relative to the gravitational field
despite both being located in a continuum. Can any mechanism or
natural process bring about an intended zero probability event?
There are similar discussions to be had about arbiters, whose usual
conclusion is that metastability is inevitably resolved by thermal
noise. The article addresses that issue by postulating a vacuum and a
very low absolute temperature, but how would the pencil be shielded
from the gravitational effects of distant bodies in motion? Does this
experiment have to take place in an empty universe?
Very fun! It would be interesting to do this with actual quantum mechanics (ie solving the schrodinger equation for an inverted quadratic potential). The usual bound states of the harmonic oscillator can be analytically continued into resonances with a complex energy, whose imaginary part should give the rate of decay in the situation the article discusses. Haven't checked the calculation but I would imagine this gives the same result. https://arxiv.org/abs/2012.09875 looks like it has the details.
Doesn't the uncertainty principle just set a lower limit for how well a perfectly skilled pencil balancer would be able to balance it though? As in, if the experiment is repeated enough times, by pure chance \theta_0 \dot\theta_0 could at times be smaller than \frac{3 \hbar }{2 ml^2} ?
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[ 4.6 ms ] story [ 13.6 ms ] threadTaken to an extreme, if the tip were 1 cm² and flat, then it wouldn't ever tip over, just as a coffee cup doesn't tip over.
There are similar discussions to be had about arbiters, whose usual conclusion is that metastability is inevitably resolved by thermal noise. The article addresses that issue by postulating a vacuum and a very low absolute temperature, but how would the pencil be shielded from the gravitational effects of distant bodies in motion? Does this experiment have to take place in an empty universe?