(a) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid walls potential of length 10 fm, estimate the minimum kinetic energy of the neutron.
(b) Estimate the minimum kinetic energy of an electron bound within the nucleus described in (a). Can an electron be confined in a nucleus? Explain.
The energy of a particle of mass m in a one-dimensional box potential having perfectly rigid walls is given by
E_{n}=\frac{\pi ^{2}\hbar ^{2}}{2ma^{2}} n^{2}, n = 1, 2, 3, …, (4.315)
where a is the size of the box.
(a) Assuming the neutron to be nonrelativistic (i.e., its energy E\ll m_{n}c^{2}), the lowest energy the neutron can have in a box of size a = 10 fm is
E_{\min } =\frac{\pi ^{2}\hbar ^{2}}{2m_{n}a^{2}} =\frac{\pi ^{2}(\hbar ^{2}c^{2})}{2(m_{n}c^{2})a^{2}} \simeq 2.04 MeV, (4.316)
where we have used the fact that the rest mass energy of a neutron is m_{n}c^{2}\simeq 939.57MeV and \hbar c\simeq 197.3MeV fm. Indeed, we see that E_{\min } \ll m_{n} c^{2}.
(b) The minimum energy of a (nonrelativistic) electron moving in a box of size a = 10 fm is given by
E_{\min }=\frac{\pi ^{2}\hbar ^{2}}{2m_{e}a^{2}} =\frac{\pi ^{2}(\hbar ^{2}c^{2})}{2(m_{e}c^{2})a^{2}} \simeq 3755.45MeV (4.317)
The rest mass energy of an electron is m_{e}c^{2}\simeq 0.511 MeV , so this electron is ultra-relativistic since E_{\min }\gg m_{e}c^{2}. It implies that an electron with this energy cannot be confined within such a nucleus.