Phy5645/Free particle SE problem

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Submitted by team 1

Question: A free particle Schrodinger Equation

Time-independent Schrodinger equation for a free particle 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 \frac{1}{2m} \left( \frac{\hbar}{i} \frac{\partial}{\partial \mathbf{r}} \right)^2 \psi \left(\mathbf{r} \right) = E \psi\left(\mathbf{r} \right) }

It is customary to write 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 E = \frac{\hbar^2 k^2}{2m} \!} to simplify 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 \left( \nabla^2 + k^2 \right) \psi \left( \mathbf{r} \right) = 0. }

Show that (a) a 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\left(\mathbf{r} \right) = e^{ikz} \!} , and (b) a 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\left(\mathbf{r} \right) = \frac{e^{ikr}}{r} \! } where 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 r = \sqrt{x^2 + y^2 + z^2} \! } , satisfy the equation. (In either case, the wave length of the solution 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 \lambda = \frac{2\pi}{k} \!} and the momentum by de Broghie's relation 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 p = \hbar k \! } . )

Answer:

(a) 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 Schrodinger 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( \partial_z^2 + k^2 \right) \psi = 0 \!} . Obviously this is a solution to the equation of

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{\partial^2}{\partial z^2} \left( e^{ikz} \right) + k^2 e^{ikz} = 0. }

(b) In polar coordinates, the Laplacian can be rewritten as

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{\cos\theta}{r^2 \sin\theta} \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{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 Schrodinger 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( \partial_r^2 + \frac{2}{r} \partial_r + 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{\partial^2}{\partial r^2} \left( \frac{e^{ikr}}{r} \right) + \frac{2}{r} \frac{\partial}{\partial r} \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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