Phy5645/Square Wave Potential Problem

Case 1: 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>0}

let 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 k = \frac{\sqrt{2mE}}{\hbar}}

then

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 \psi}{dx^2}= -k^2 \psi }

whose general solution is:

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(x) = A sin(kx) +B cos(kx) , (0<x<a) }

by Bloch's theorem , the wave function in the cell immediately to the left of the origin:

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(x) = e^{-i\kappa a} \left(A sin(k(x+a)) + B cos(k(x+a)) \right) , ~(0<x<a) }

at 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=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 ~\psi} must be continuous across; so:

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 B = e^{-i\kappa a} \left(A sin(k a) + B cos( k a) \right)}

and the derivative of the wave function suffers a discontinuity proportional the "strength" of the delta function:

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 ka - e^{-i\kappa a}\left( A cos(k a) + B Sin(ka) \right) = \frac{-2m \alpha}{\hbar^2} B }

therefore

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 A sin(ka) = \left(e^{i\kappa a} - cos (ka) \right) B}

the derivative suffers from a discontinuity proportional to the strength of the delta function:

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 \rho A - e^{-i\kappa a}\rho\left(A cosh(\rho a) + B sinh(\rho a) \right) = \frac{2m \alpha}{\hbar^2} }

which implies

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(e^{i\kappa a}-cos(ka) \right)\left(1- e^{-i\kappa a}cos(ka)\right) + e^{-i\kappa a}sin^2(ka) = \frac{-2m\alpha}{\hbar^2 k} sin(ka) }

finally

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 cos(\kappa a) = cos(ka) + \frac{m \alpha}{\hbar^2 k}sin(ka)}

Case 2: 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<0} and 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 ~0<x<a}

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{-\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E \psi }

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 \psi}{dx^2} = \rho^2 \psi }

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 \rho = \frac{\sqrt{-2mE}}{\hbar} }

the general solution is:

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(x) = A sinh(\rho x) + B cosh(\rho x) } for 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 0<x<a\!}

by Bloch's theorem the solution 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 -a<x<0\!} is

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(x) = e^{-i\kappa a}\left( A sinh(\rho(x+a)) B \cosh(\rho(x+a)) \right) }

for 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(x)\!} to be continuous at 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=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 B = e^{-i\kappa a} \left( A sinh(\rho a) + B \cosh(\rho a) \right) }

which implies

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 A sinh(\rho a ) = B \left( e^{i\kappa a} - cosh(\rho a) \right) }

which implies

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 A \left( 1- e^{-i\kappa a} cosh(\rho a) \right) = B\left( \frac{2 m \alpha}{\hbar^2 \rho} + e^{-i \kappa a}sinh(\rho a) \right)}

by substitution:

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^{i \kappa } - cosh(\rho a) )(1- e^{-i \kappa a} cosh(\rho a)) = \frac{ 2m \alpha}{\hbar^2 \rho} sinh(\rho a)+ e^{-i \kappa a}sinh^2(\rho a) }

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^{i\kappa a} - 2 cosh{\rho a} + e^{-i\kappa a} cosh^2(\rho a) - e^{-i \kappa a} sinh^2(\rho a) = \frac{ 2m \alpha}{\hbar^2 \rho} sinh(\rho a) }

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 ^{i\kappa a}+e^{-i\kappa a} = 2 cosh(\rho a) + \frac{2m \alpha}{\hbar^2\rho}sinh(\rho a) }

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 cos(\kappa a) = cosh(\rho a) + \frac{m \alpha}{\hbar^2 \rho} sinh(\rho a) }

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