(a) From Equation 5.71, \cos (q a)=\cos (k a) \Rightarrow q=\pm k=\pm \frac{\sqrt{2 m E}}{\hbar} ; \quad E=\frac{\hbar^{2} q^{2}}{2 m} \Rightarrow \frac{d E}{d q}=\frac{\hbar^{2} q}{m} .
\cos (q a)=\cos (k a)+\frac{m \alpha}{\hbar^{2} k} \sin (k a) (5.71).
\frac{1}{N a} G_{\text {free }}(E)=\frac{1}{\pi} \frac{m}{\hbar^{2}|q|}=\frac{m \hbar}{\pi \hbar^{2} \sqrt{2 m E}}=\frac{1}{\pi \hbar} \sqrt{\frac{m}{2 E}} .
(b) -a \sin (q a)=\left\{-a \sin (k a)+\frac{m \alpha}{\hbar^{2}}\left[-\frac{1}{k^{2}} \sin (k a)+\frac{a}{k} \cos (k a)\right]\right\} \frac{d k}{d q} , Equation (5.65): \frac{d k}{d q}=\frac{\sqrt{2 m}}{2 \hbar \sqrt{E}} \frac{d E}{d q} .
k \equiv \frac{\sqrt{2 m E}}{\hbar} (5.65).
\sin (q a)=\left\{\sin (k a)+\frac{m \alpha}{\hbar^{2} k}\left[\frac{1}{k a} \sin (k a)-\cos (k a)\right]\right\} \frac{m}{\hbar^{2} k} \frac{d E}{d q} .
\frac{d E}{d q}=\frac{\sin (q a)}{\sin (k a)+\frac{m \alpha}{\hbar^{2} k}\left(\frac{1}{k a} \sin (k a)-\cos (k a)\right)} \frac{\hbar^{2} k}{m} .
\sin (q a)=\sqrt{1-\cos ^{2}(q a)}=\sqrt{1-\left[\cos (k a)+\frac{m \alpha}{\hbar^{2} k} \sin (k a)\right]^{2}} .
=\sqrt{1-\cos ^{2}(k a)-\frac{2 m \alpha}{\hbar^{2} k} \cos (k a) \sin (k a)-\left(\frac{m \alpha}{\hbar^{2} k}\right)^{2} \sin ^{2}(k a)} .
=\sin (k a) \sqrt{1-\frac{2 m \alpha}{\hbar^{2} k} \cot (k a)-\left(\frac{m \alpha}{\hbar^{2} k}\right)^{2}} .
\frac{1}{N a} G(E)=\frac{1}{\pi}\left(\frac{m}{\hbar^{2} k}\right) \frac{\left|\sin (k a)+\frac{m \alpha}{\hbar^{2} k}\left(\frac{1}{k a} \sin (k a)-\cos (k a)\right)\right|}{\sin (k a) \sqrt{1-\frac{2 m \alpha}{\hbar^{2} k} \cot (k a)-\left(\frac{m \alpha}{\hbar^{2} k}\right)^{2}}} .
= \frac{m}{\pi \hbar^{2} k} \frac{\left|1+\frac{m \alpha}{\hbar^{2} k}\left[\frac{1}{k a}-\cot (k a)\right]\right|}{\sqrt{1-\frac{2 m \alpha}{\hbar^{2} k} \cot (k a)-\left(\frac{m \alpha}{\hbar^{2} k}\right)^{2}}} .
(c) With m=\hbar=a=1 , the expressions in (a) and (b) reduce to
\alpha=0: \quad \frac{1}{\pi \sqrt{2 E}} ; \alpha=1: \frac{1}{\pi \sqrt{2 E}}\left[\frac{1+\frac{1}{2 E}-\frac{1}{\sqrt{2 E}} \cot (\sqrt{2 E})}{\sqrt{1-\frac{1}{2 E}-\frac{2}{\sqrt{2 E}} \cot (\sqrt{2 E})}}\right] .
In the graph, the lower curve is for α = 0, the upper one (with the breaks) is for α = 1.
\operatorname{Plot}\left[\frac{1}{\pi \sqrt{2 x }},\{ x , 0,50\}, \text { PlotRange } \rightarrow\{0, .5\}\right] .
\operatorname{Plot}\left[\frac{1}{\pi \sqrt{2 x}}\left(1+\frac{1}{2 x}-\frac{1}{\sqrt{2 x}} \operatorname{Cot}[\sqrt{2 x}]\right) / \sqrt{1-\frac{1}{2 x}-\frac{2}{\sqrt{2 x}} \cot [\sqrt{2 x}]}\right. ,
\{ x , 0,50\}, \text { PlotRange } \rightarrow\{0, .5\}] .
Show[%3, %4].
