Problem 19.88 Challenge

a) จาก \(E_{n}=\frac{n^{2}h^{2}}{8mL^{2}}\)

\(\Delta E_{\gamma}= (n_{f}^{2} - n_{i}^{2})\frac{h^{2}}{8mL^{2}}\)

\(hf= (n_{f}^{2} - n_{i}^{2})\frac{h^{2}}{8mL^{2}}\)

\(\frac{hc}{\lambda}= (n_{f}^{2} - n_{i}^{2})\frac{h^{2}}{8mL^{2}}\)

\(\frac{hc}{\lambda}= (4^{2} - 1^{2})\frac{h^{2}}{8mL^{2}}\)

\(L =\sqrt{\frac{15h\lambda}{8mc}}\;\blacksquare\)

b) จาก \(ΔE_{\gamma}= (n_{f}^{2} - n_{i}^{2})\frac{h^{2}}{8mL^{2}}\)

\(-hf'= (n_{f}^{2} - n_{i}^{2})\frac{h^{2}}{8mL^{2}}\)

\(-\frac{hc}{\lambda '}= (2^{2} - 4^{2})\frac{h^{2}}{8m}(\frac{8mc}{15h\lambda})\)

\(\lambda '=\frac{5\lambda}{4}\;\blacksquare\)

c) จาก \(p_{n}=\frac{nh}{2L}\)

\(mv_{n}=\frac{nh}{2L}\)

\(v_{n}=\frac{nh}{2mL}\)

\(v_{\min}=\frac{h}{2mL}; n = 1\)

\(v_{\min}=\frac{h}{2m}(\sqrt{\frac{8mc}{15h\lambda}})\)

\(v_{\min}=\sqrt{\frac{2hc}{15m\lambda}}\;\blacksquare\)