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Homework Statement
Question_________________________________________________________________________________
Find transmitted coefficient and reflected coefficient in case barrier potential E<V ?
determine.
##Ψ_{I} = Ae^{ikx}+Be^{-ikx}##
##Ψ_{II} = De^{βx}+Ee^{-βx}##
##Ψ_{III} = Ce^{ikx}##
Homework Equations
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##R = |\frac{J_{ref}}{J_{inc}}|##
##T = |\frac{J_{tran}}{J_{inc}}|##
and J is Probability current (https://en.wikipedia.org/wiki/Probability_current)
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For transmitted coefficient and reflected coefficient between ##Ψ_{I}## and ##Ψ_{III}##
where
incident wave = ## Ae^{ikx} ##
reflect wave = ## Be^{-ikx} ##
transmit wave = ## Ce^{ikx} ##
I can find a solution for this case.
https://en.wikipedia.org/wiki/Rectangular_potential_barrier#E_<_V0 << Here is the answer.
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The Attempt at a Solution
But my question is asked in the case transmitted coefficient and reflected coefficient between ##Ψ_{II}## and ##Ψ_{III}##.
I know
incident wave = ## De^{βx} ##
reflect wave = ## Ee^{-βx} ##
transmit wave = ## Ce^{ikx} ##
I checked it
##J = \frac{ħ}{2mi}(Ψ^* \frac{dΨ}{dx}-Ψ\frac{dΨ^*}{dx})##
I seen that incident wave and reflect wave are real function.
So ##J_{inc}=0## and ##J_{ref}=0##
Because ##Ψ^* \frac{dΨ}{dx}-Ψ\frac{dΨ^*}{dx}=Ψ\frac{dΨ}{dx}-Ψ\frac{dΨ}{dx}=0##
But transmit wave is complex function.
So ##J_{tran}=\frac{ħk}{m}|C|^2##
From Eq.
##R = |\frac{J_{ref}}{J_{inc}}|##
##T = |\frac{J_{tran}}{J_{inc}}|##
in this case ##R = \frac{0}{0}## and ##T = \frac{ \frac{ħk}{m}|C|^2}{0}##
what does mean?
##R = \frac{0}{0}## and ##T = \frac{ \frac{ħk}{m}|C|^2}{0}##
Or i miss something ? please re check my solution
