When do quantum effects become important?

[andyfry]

At what length scale do quantum effects become important in gravitational calculations??

 

[fzero]

You can figure this out from a scaling argument. The gravitational constant, G, has dimensions of (length)³(mass)^-1(time)^-2 or, in shorthand, L³M^-1T^-2. The speed of light, c, is the universal conversion between time and distance, since it has units of LT^-1.

Finally, quantum effects are characterized by Planck's constant h, or more commonly used \hbar, which has units of L²M T^-1. If \hbar were zero, quantum effects would not exist, so in the limit that \hbar \rightarrow 0, the length scale where quantum effects becomes important must also go to zero. So our length scale must depend on \hbar to a positive power.

Can you see how to combine \hbar and G to eliminate the mass scale? How about how to combine powers of c with the result to convert time to length units?

 

[andyfry]

Yeah, this was what I was trying to do! Thought i had solved it correctly and was just looking for confirmation. However I've now noticed I got a few dimensions confused. I'll try again...
Multiplying G and \hbar gives dimensions of L⁵T^-3.
c^-3 will have dimensions L^-3T^-3 (I believe?)
So multiplying by this gives dimensions L²
Looking at the indices (10^-34*10^-11(10⁸)^-3) gives a magnitute of 10^-69 for L².
So quantum effects must be taken into account below 1*10^-35m!
I hope that was right... :S Dimensional analysis is pretty new to me!

 

[andyfry]

Pretty sure this is correct.
Would be great if someone could just confirm please?