# GRE Physics GR 1777 Problem Solution

## 014. Quantum Mechanics (Planck length), Dimensional Analysis

### Solution

If you don’t know the Planck length, that’s fine. This question asks that you know about the dimensional analysis.

$\hbar : [kg \cdot m^2 /s]$
$G : [m^3 / (kg \cdot s^2)]$
$c : [m/s]$

The unit of the length should be $m$. To find the proper expression for length, we construct the equation like this:

$\hbar^p \cdot G^q \cdot c^r : [kg \cdot m^2 /s]^p \cdot [m^3 / (kg \cdot s^2)]^q \cdot [m/s]^r = [m]$

The dimensions of this equation gives

$[kg]^{p-q} \cdot [m]^{2p+3q+r} \cdot [s]^{-p-2q-r} = [m]$

We can easily calculate their orders.

$p=\frac{1}{2}$
$q=\frac{1}{2}$
$r=-\frac{3}{2}$

Therefore, the Planck length is

$l_P = \sqrt{ \frac{\hbar G}{c^3}}$

### Alternative Solution

$l_P = \sqrt{ \frac{\hbar G}{c^3}}$