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 [tex]\hbar[/tex], which has units of L²M T^-1. If [tex]\hbar[/tex] were zero, quantum effects would not exist, so in the limit that [tex]\hbar \rightarrow 0[/tex], the length scale where quantum effects becomes important must also go to zero. So our length scale must depend on [tex]\hbar[/tex] to a positive power.

Can you see how to combine [tex]\hbar[/tex] and [tex]G[/tex] to eliminate the mass scale? How about how to combine powers of [tex]c[/tex] 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 [tex]G[/tex] and [tex]\hbar[/tex] 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?

 

[rwisz]

Quantum effects become important when dealing with very small particles such as atoms and subatomic particles. At this scale, classical mechanics fails to accurately describe the behavior of these particles and quantum mechanics must be used.

In terms of gravitational calculations, quantum effects become important at the Planck length scale, which is approximately 10^-35 meters. At this scale, the effects of gravity on particles are significant and cannot be ignored. This is where the theories of quantum mechanics and general relativity, which govern the behavior of particles and gravity respectively, must be combined to accurately describe the behavior of matter. At larger length scales, such as those we encounter in everyday life, the effects of gravity can be accurately described using classical mechanics.