Quantum Mechanics - Planck Length

The characteristic distance at which quantum gravitational effects are significant, the Planck length can be determined from a suitable combination of the physical constants G, ħ, and c. Which of the following correctly gives the Planck length?

A. Għc
B. Għ²c³
C. G²ħc
D. G^½ħ²c
E. (Għ/c³)^½

(GR9677 #29)

Solution:

Check units:

Let's l = Planck length
 l = G^x ħ^y c^z

G unit = m³ kg⁻¹ s⁻²
ħ unit = Js = Nms = kg m² s⁻¹
c unit = m s⁻¹
l unit = m

m = (m³ kg⁻¹ s⁻²)^x  (kg m² s⁻¹)^y (m s⁻¹)^z

For m:
1 = 3x + 2y + z

For kg:
0 = − x + y
x = y

For s:
0 = − 2x − y − z
0 = − 3x − z

z = − 3x

1 = 3x + 2y + z 

1 = 3x + 2x − 3x = 2x
x = y = 1/2
z = − 3x = − 3/2

l = G^x ħ^y c^z 
= G^1/2 ħ^1/2 c^−3/2
= (Għ/c³)^½

Answer: E