GRE GR 1777, Problem 014 Solution

Physics GRE GR 1777

  1. Quantum Mechanics (Planck length) + Dimensional analysis

Solution 1) 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.
Therefore, the Planck length is
l_P = \sqrt{ \frac{\hbar G}{c^3}}
The answer is (A).
Solution 2) If you already know about the Planck length, then you have nothing to worry about.
The Planck length is
l_P = \sqrt{ \frac{\hbar G}{c^3}}

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