I need to calculate the transmission coefficient of a Potential V=v 0 (x) I trie

Question

I need to calculate the transmission coefficient of a Potential V=v0(x)

I tried this one with an Ansatz of a raising and a fallingexponential function on both sites. Needing Continuity in Psi andPsi' it is giving me a transmission coefficient of 1. Isthere anybody who can confirm this? it does not sound veryrealistic to me.

Explanation / Answer

The Schroedinger equation is -''+ (x) = k2 >>>equation A where = (2m/hbar2)V0 , k =(2mE)/hbar , E>0 (scattering state) For x < 0 and x > 0, the delta function is zero and thegeneral solution is = Aeikx + Be-ikx For a particle coming in from the left side, solution is (for x< 0) = Aeikx + Be-ikx (firstterm is incident wave, second term is the reflected wave) For x > 0, only the transmitted part exists = Ceikx Continuity of wave function at x=0 gives A + B = C >>> equation B ' is not continuous since V(x) is infinite at x=0. But we canintegrate equation A from - to + where is asmall number and then take the limit -> 0, -'(0) + (0) = 0 where '(0) is the change in ' across x=0. we cancalculate this from the solutions above. '(0) = ikC - ikA+ikB = ik(C-A+B) = (0) =C From above 2 continuity conditions C = ik(C-A+B) = ik(C-A+C-A) = 2ik(C-A) C = -2ikA/(-2ik) The transmission coefficient is T = |C/A|2 = |-2ik/(-2ik)|2 =4k2/(2 + 4k2)= 1/(1 +(mV02/2Ehbar2) hope this helps!

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