The energy of a particle of mass m in a box having perfectly rigid walls is given by
E_{n}=\frac{n^{2}\hbar ^{2} }{8ma^{2}}, n = 1, 2, 3, . . . , (4.285)
where a is the size of the box.
(a) (i) For the electron in the box of size 10^{-10} m, we have
E_{n}=\frac{\hbar ^{2}c^{2} }{m_{e}c^{2}a^{2} } \frac{4\pi ^{2}n^{2} }{8} \equiv \frac{4\times 10^{4}(MeV fm)^{2} }{0.5Me V\times 10^{10}fm^{2} } \frac{\pi ^{2}}{2}n^{2}
=4\pi ^{2}n^{2}e V\simeq 39n^{2}eV. (4.286)
Hence E_{1}=39eV,E_{2}=156eV , and E_{3}= 315 eV .
(ii) For the sphere in the box of side 10 cm we have
E_{n}=\frac{(6.6\times10^{-34}J s )^{2} }{10^{-3}kg\times 10^{-2}m^{2}} n^{2}=43.6\times 10^{-63}n^{2}J (4.287)
Hence E_{1}=43.6\times 10^{-63}J,E_{2}=174.4\times 10^{-63}J , and E_{3}=392.4\times 10^{-63}J.
(b) The differences between the energy levels are
(E_{2}-E_{1})_{electron} =117eV, (E_{3}-E_{2}) _{electron} =195eV, (4.288)
(E_{2}-E_{1})_{sphere} =130\times 10^{-63}J, (E_{3}-E_{2})_{sphere} =218\times 10^{-63}J (4.289)
These results show that:
• The spacings between the energy levels of the electron are quite large; the levels are far apart from each other. Thus, the quantum effects are important.
• The energy levels of the sphere are practically indistinguishable; the spacings between the levels are negligible. The energy spectrum therefore forms a continuum; hence the quantum effects are not noticeable for the sphere.
(c) According to the uncertainty principle, the speed is proportional to \upsilon \sim \hbar /(ma). For the electron, the typical distances are atomic, a\simeq 10^{-10} m; hence
\upsilon \sim \frac{\hbar c}{mc^{2}a} c\sim \frac{200eV fm}{0.5MeV\times 10^{5} fm} c\simeq 4\times 10^{-3}c=1.2\times 10^{6} ms^{-1}, (4.290)
where c is the speed of light. The electron therefore moves quite fast; this is expected since we have confined the electron to move within a small region.
For the sphere, the typical distances are in the range of 1 cm:
\upsilon \sim \frac{\hbar }{ma} \sim \frac{6.6\times 10^{-34}Js}{10^{-3}kg\times 10^{-2}m} \simeq 6.6\times 10^{-29}ms^{-1} (4.291)
At this speed the sphere is practically at rest.