Exploring Quantum Physics - Final Exam Part 2 Question 3

Question:

Despite the very long description - question 3 by itself is trivial. In some sense, it is just a hint for the upcoming questions. To a bare minimum, the question listed a bunch of quantity and asked which one has to unit of energy. So all we have to do is to match dimensions.

Solution:

Energy unit
= Force times Distance
= Mass times Acceleration times Distance
= M(LT^-2)(L)
= ML²T^-2

kr = Energy
Therefore k has a unit of Force MLT^-2

Planck's constant has unit of Joule Second = Energy Second = ML²T^-1

Finally $\mu$ has unit of mass.

So $\frac{\hbar^2 k^2}{\mu}$ has unit $\frac{(ml^2t^{-1})^2(mlt^{-2})^2}{m} = m^3l^6t^{-6}$

Therefore we see $\left(\frac{\hbar^2 k^2}{\mu}\right)^{1/3}$ has the unit of energy.