Phy5645/Free particle SE problem

(a) The plane wave, $\psi =e^{ikz},\!$ does not depend on $x\!$ or $y\!$. Therefore, the Schrödinger equation becomes $\left({\frac {d^{2}}{dz^{2}}}+k^{2}\right)\psi =0\!$. We may easily see that this is a solution to the equation:

${\frac {d^{2}}{dz^{2}}}\left(e^{ikz}\right)+k^{2}e^{ikz}=0.$

(b) In spherical coordinates, the Laplacian is given by

$\nabla ^{2}=\partial _{r}^{2}+{\frac {2}{r}}\partial _{r}+{\frac {1}{r^{2}}}\partial _{\theta }^{2}+{\frac {\cot \theta }{r^{2}}}\partial _{\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}\partial _{\phi }^{2}.$

The spherical wave $\psi ={\frac {e^{ikr}}{r}}\!$ does not depend on $\theta \!$ or $\phi \!$. Therefore, the Schrödinger equation becomes

$\left({\frac {d^{2}}{dr^{2}}}+{\frac {2}{r}}{\frac {d}{dr}}+k^{2}\right)\psi =0$

$\Rightarrow {\frac {d^{2}}{dr^{2}}}\left({\frac {e^{ikr}}{r}}\right)+{\frac {2}{r}}{\frac {d}{dr}}\left({\frac {e^{ikr}}{r}}\right)+k^{2}{\frac {e^{ikr}}{r}}={\frac {2e^{ikr}}{r^{3}}}-{\frac {2ike^{ikr}}{r^{2}}}+{\frac {2}{r}}\left(-{\frac {e^{ikr}}{r^{2}}}+{\frac {ike^{ikr}}{r}}\right)=0.$

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