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?