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.

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}}

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}}

Reference)
https://en.wikipedia.org/wiki/Planck_length

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