I] A particle of mass $m$ is confined to a narrow tube of length $L$. Find $(a)$ the wavelengths of the de Broglie waves which will resonate in the tube, (b) the corresponding particle momenta, and
(c) the corresponding energies. (d) Evaluate the energies for an electron in a tube with $L=0.50 \mathrm{~nm}$.
(a) The de Broglie waves will resonate with a node at each end of the tube because the ends are impervious. A few of the possible resonance forms are shown in Fig. $42-1$. They indicate that, for resonance, $L=\frac{1}{2} \lambda_{1}, 2\left(\frac{1}{2} \lambda_{2}\right), 3\left(\frac{1}{2} \lambda_{3}\right), \ldots, n\left(\frac{1}{2} \lambda_{n}\right), \ldots$ or
(b) Because the de Broglie wavelengths are $\lambda_{\mathrm{n}}=h / p_{n}$, the resonance momenta are
$$
p_{n}=\frac{n h}{2 L} \quad n=1,2,3, \ldots
$$ (c) As shown in Problem 42.15, $p^{2}=(2 m)(\mathrm{KE})$, and so
$$
(\mathrm{KE})_{n}=\frac{n^{2} h^{2}}{8 L^{2} m} \quad n=1,2,3, \ldots
$$
Notice that the particle can assume only certain discrete energies. The energies are quantized.
(d) With $m=9.1 \times 10^{-31} \mathrm{~kg}$ and $L=5.0 \times 10^{-10} \mathrm{~m}$, substitution yields
$$
(\mathrm{KE})_{n}=2.4 \times 10^{-19} n^{2} \mathrm{~J}=1.5 n^{2} \mathrm{eV}
$$