- #1
mnky9800n
- 4
- 0
Homework Statement
Consider the potential [tex]V=V_0 d[\delta(x-a)+\delta(x-a)][/tex]. Find the transmission probability of the potential for a particle of mass m and wave number [tex]\sqrt{\frac{2mE}{\hbar^2}}[/tex] incident on the potential. Discuss the behavior when [tex]ka~\pi/2[/tex].
Homework Equations
Schrodinger's Equation
[tex]H\psi=E\psi[/tex]
The Attempt at a Solution
The solutions to SE for each region are as follows:
region 1: a < x
[tex]\psi_1 = e^{ikx}+Re^{-ikx}[/tex]
region 2: -a < x < a
[tex]\psi_2 = Ae^{ikx}+Be^{ikx}[/tex]
region 3: a < x
[tex]\psi_3 = Te^{ikx}[/tex]
Because the [tex]\delta[/tex]-function means the wave function is continuous but the derivative is not at x = a we can say:
[tex]\psi_1(-a) = \psi_2(-a)[/tex]
[tex]\psi_2(a) = \psi_3(a)[/tex]
therefore:
[tex]1+Re^{2ika} = A+Be^{2ika}[/tex]
[tex]A+Be^{-2ika} = T[/tex]
This is where I get stuck. I have no idea where to go from here.
Last edited: