クローニッヒ・ペニーのモデル
ħ 2 2 m a A = ∑ n = − ∞ ∞ 1 α 2 − ( k + 2 π n a ) 2 {\displaystyle {\frac {\hbar ^{2}}{2m}}{\frac {a}{A}}=\sum _{n=-\infty }^{\infty }{\frac {1}{\alpha ^{2}-(k+{\frac {2\pi n}{a}})^{2}}}}
ここで部分分数分解を用いて式を変形すると、
ħ 2 2 m a A = ∑ n = − ∞ ∞ 1 α 2 − ( k + 2 π n a ) 2 = − 1 2 α ∑ n = − ∞ ∞ [ 1 ( k + 2 π n a ) − α − 1 ( k + 2 π n a ) + α ] = − a 4 α ∑ n = − ∞ ∞ [ 1 π n + k a 2 − α a 2 − 1 π n + k a 2 + α a 2 ] = − a 4 α [ ∑ n = − ∞ ∞ 1 π n + k a 2 − α a 2 − ∑ n = − ∞ ∞ 1 π n + k a 2 + α a 2 ] {\displaystyle {\begin{aligned}{\frac {\hbar ^{2}}{2m}}{\frac {a}{A}}&=\sum _{n=-\infty }^{\infty }{\frac {1}{\alpha ^{2}-(k+{\frac {2\pi n}{a}})^{2}}}\\&=-{\frac {1}{2\alpha }}\sum _{n=-\infty }^{\infty }\left[{\frac {1}{(k+{\frac {2\pi n}{a}})-\alpha }}-{\frac {1}{(k+{\frac {2\pi n}{a}})+\alpha }}\right]\\&=-{\frac {a}{4\alpha }}\sum _{n=-\infty }^{\infty }\left[{\frac {1}{\pi n+{\frac {ka}{2}}-{\frac {\alpha a}{2}}}}-{\frac {1}{\pi n+{\frac {ka}{2}}+{\frac {\alpha a}{2}}}}\right]\\&=-{\frac {a}{4\alpha }}\left[\sum _{n=-\infty }^{\infty }{\frac {1}{\pi n+{\frac {ka}{2}}-{\frac {\alpha a}{2}}}}-\sum _{n=-\infty }^{\infty }{\frac {1}{\pi n+{\frac {ka}{2}}+{\frac {\alpha a}{2}}}}\right]\end{aligned}}}
ψ ( 0 ) = e i k ⋅ 0 u ( 0 ) = e i k L u ( L ) = ψ ( L ) {\displaystyle \psi (0)=e^{ik\cdot 0}u(0)=e^{ikL}u(L)=\psi (L)}
⇒ u ( 0 ) = e i k L u ( L ) = e i k L u ( N a ) {\displaystyle \Rightarrow u(0)=e^{ikL}u(L)=e^{ikL}u(Na)}
⇒ e i k L = 1 {\displaystyle \Rightarrow e^{ikL}=1}
⇒ k L = 2 π n {\displaystyle \Rightarrow kL=2\pi n}
⇒ k = 2 π L n ( n = 0 , ± 1 , ... , ± N 2 ) {\displaystyle \Rightarrow k={2\pi \over L}n\qquad \left(n=0,\pm 1,\dots ,\pm {N \over 2}\right)}