What is the period of bloch funstion under the condition ka=(2l+ 1)\pi ?
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The functions which are of interest in the theory of the hydrogen atom are $$ f_{n}(x)=x^{i+t} e^{-x \cdot 2 \pi} L_{n-t-1}^{2 t+1}\left(\frac{x}{n}\right) $$ where $n$ and $/$ are integers with $0 \leq l \leq n-1$. (Note that here $k=2 l+1$, and we have replaced $n$ by $n-l-1 ;$ in this problem $L_{2}^{3}$, say, means $l=1, n=4$.) For $l=1$, show that $$ \begin{gathered} f_{2}(x)=x^{2} e^{-x i 4}, \quad f_{3}(x)=x^{2} e^{-x, 6}\left(4-\frac{x}{3}\right) \\ f_{4}(x)=x^{2} e^{-x i 8}\left(10-\frac{5 x}{4}+\frac{x^{2}}{32}\right) \end{gathered} $$
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An electron is confined inside a sphere of radius $a$ by a potential $P(r)=0$ for $r \leq a$ and $P(r)=\infty$ for $r>a .$ Show that the eigenfunctions for states with zero angular momentum are given by $\psi(r)=$ const $(\sin k r) / k r .$ Determine the relation between $k$ and the energy, and show that the energy eigenvalues are $n^{2} h^{2} / 8 m a^{2}$.
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