Undergrad Is the Planck Length the Ultimate Limit of Measurement?

[nomadreid]

If I understand correctly, the Planck length, or maybe a tenth of it, is considered the point at which quantum effects mess up any attempt to apply our present physical laws, and so one could not even in theory make a dependable measurement of something smaller. However, I was wondering what would be wrong with the following simpler argument; it must be wrong somewhere because my answer is too big.

Suppose we had some slit of width d . To measure it would require a photon with a wavelength of d/2 or smaller, that is, an Energy of at least 2hc/d, the equivalent rest mass of 2h/(cd). If we try to get the photon into the slit of width d, d will have to be bigger than the Schwarzschild radius r = 2GM/c² = 4Gh/(c³d) . That is, d>4Gh/(c³d), or d>2√(Gh/c³) , but that is a factor of 2√(2π) too big, as the Planck length is √(Għ/c³). What is wrong? Thanks.

 

[Vanadium 50]

What is wrong?

You're reading into terms of order one when you only have an order of magnitude estimate.

 

[nomadreid]

Thanks for the reply, Vanadium 50. I ask forgiveness for my own density (hopefully less than that needed to collapse into a black hole), but I am afraid that I don't quite understand your comment. What are you saying that I am estimating?

 

[Nugatory]

If I understand correctly, the Planck length, or maybe a tenth of it, is considered the point at which quantum effects mess up any attempt to apply our present physical laws...

Not "at which quantum effects mess up..." but "around which, give or take a factor of ten or thereabouts, quantum effects must mess up..."

The statement isn't precise enough to worry about a factor of $2\sqrt{\pi}$

 

[nomadreid]

Thanks, Nugatory. Ah. So the Planck length is a rather fuzzy border?

 

[Vanadium 50]

The Planck length is very well defined. The size where you have to worry about quantum gravity is fuzzy - just like the size where you have to worry about everything else. Are you making a 10% measurement? A 1% measurement? A 0.00000001% measurement?