GRE Physics GR1777 Problem 014 Solution

GRE Physics GR 1777 Problem Solution

014. Quantum Mechanics (Planck length), Dimensional Analysis


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

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

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

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

The dimensions of this equation gives

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

We can easily calculate their orders.

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

Therefore, the Planck length is

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

Alternative Solution

If you already know about the Planck length, then you have nothing to worry about.

The Planck length is
$latex l_P = \\sqrt{ \\frac{\\hbar G}{c^3}} $




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