Phy5645/Free particle SE problem

(a) The plane wave, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = e^{ikz},\!} does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\!} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!} . Therefore, the Schrödinger equation becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{d^2}{dz^2} + k^2 \right) \psi = 0 \!} . We may easily see that this is a solution to the equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2}{dz^2} \left( e^{ikz} \right) + k^2 e^{ikz} = 0. }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = \frac{e^{ikr}}{r} \! } does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \!} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \!} . Therefore, the Schrödinger equation becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + k^2 \right) \psi = 0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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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