Question 4.P.10: Consider the box potential V(x) = {0, 0<xa, ∞, elsewhere....

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

$=4\pi ^{2}n^{2}e V\simeq 39n^{2}eV$.                    (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.