# Using (Potential) Energy to estimate velocity?

I'm reading Cohen's Concepts of nuclear physics. In the first chapter, he calculates a value for Electrostatic energy to insert a proton inside a nucleus, using coulomb potential energy formula $$E_{c}=\frac{Z e^{2}}{4 \pi \epsilon_{0} R}$$ where R is the typical radius for A=120 nucleus. This comes out to be 13 MeV.

He then says that though this is an underestimation, the order of magnitude is correct. And for this energy, the nucleons have non-relativistic velocities (0.15c).

My question is simple - This is electrostatic potential energy, how could this be used to estimate the velocity of nucleons. For that, shouldn't the kinetic energy of nucleons be used?

• 13 MeV = 3 * 10^21 Hz. If angular momentum is h bar then the radius is 1.83 fm and the velocity is 0.1205 c – R. Emery Dec 24 '20 at 18:09
• Just realized that I forgot to add that 13 MeV is typical for gamma rays hence the conversion to Hz – R. Emery Dec 24 '20 at 21:53

I believe he is referring to the velocity that the proton would need at a distance, in order to reach the surface of the nucleus.

• How/why would that be equal to the velocity of protons inside the nucleus – aneet kumar Dec 24 '20 at 13:45
• @aneetkumar It is not. That would require knowing the nature of the strong nuclear force and the corresponding potential, which are not calculable exactly. That is why the author is only doing an estimate using the electrostatic potential energy. – Yejus Dec 24 '20 at 14:07

The name of the theorem escapes me, but the statement is that for classical 1D motion under potential $$V\sim x^n$$

$$\langle T\rangle = \frac{n}{2}\langle V\rangle$$

Telling us that knowing the expectation of V is enough to know about T.

Assuming the ground state. We know second order perturbations to the hamiltonian shift the ground state to lower E, (the p^4 term is symmetric on the ground state, so first order vanishes) which is how we know that we underestimate T with this estimation.

Edit: the name is the Virial Theorem