# Minimum energy of an electron trapped in a nucleus

We were asked to estimate the minimum kinetic energy of an electron trapped within a nucleus having diameter $$d$$.

My solution was this: find the momentum of the electron (via de Broglie relation) and use a very standard kinetic energy formula, like this (assuming minimum energy state has a wavelength of twice the nucleus' diameter):

$$p=h/\lambda=h/2d$$
$$E_K = p^2/2m_e = \frac{h^2}{8d^2m_e}$$

The professors marked this as incorrect, and instead gave this solution ($$E_0$$ is electron rest mass, 0.511 MeV):

$$p=h/\lambda=h/2d$$
$$E_K = [(pc)^2 + E_0^2]^{1/2} - E_o$$

These give drastically different results, but I'm just curious why the professor's approximation is physically more valid than my own.