The problem is this:
"Consider an electron confined in a region of nuclear dimensions (about 5 fm). Find its minimum possible kinetic energy in MeV. Treat this problem as one-dimensional, and use the relativistic relation between E and p."
Consider an electron confined in a region of nuclear dimensions (about 5 fm). Find its minimum possible kinetic energy in MeV. Treat this problem as one-dimensional, and use the relativistic relation between E and p.
From Heisenberg's uncertainty relation for position and momentum $\Delta x \Delta p\geq\frac{ℏ}{2}$, $\Delta pc= \frac{ℏc}{2\Delta x}=39.4MeV$$\Delta pc= \frac{ℏc}{2\Delta x}=39.4\text{ MeV}$ where $2\Delta x=5fm$$2\Delta x=5\text{ fm}$ is the width of the region. Plugging this into the relativistic energy-momentum relation $E^2=(mc^2)^2+(pc)^2$, $E=39.4MeV$$E=39.4\text{ MeV}$ which is the correct answer.
However, in my book there is also an equation for the zero-point energy which is defined as the lowest possible kinetic energy for a quantum particle confined in a region (one-dimensional) of width $a$ and is given by $\langle K \rangle= \frac{ℏ^2}{2ma^2}$ and the answer I get here is about $1.52 GeV$$1.52\text{ GeV}$.
Why are/should these answers be so different?
Also the questions says "The large value you will find is a strong argument against the presence of electrons inside nuclei, since no known mechanism could contain an electron with this much energy."
How is this argument made exactly?
