Transmission and Reflection Coefficients

1. Nov 14, 2006

stunner5000pt

Consider the step potential defined by
V(x) = 0 if x < 0
Vb > 0 if x=> 0

a) Compute te relfection and tranmission coefficients for a particle of energy E > Vb approaching from th left

For x < 0
Schrodinger equaion since V = 0
$$-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} = E \psi(x)$$

$$\frac{\partial^2 \psi}{\partial x^2} = -\frac{2mE}{\hbar^2} \psi(x)$$

define $$k_{1}^2 = \frac{2mE}{\hbar^2}$$

$$\frac{\partial^2 \psi}{\partial x^2} = -k^2 \psi(x)$$

$$\psi(x) = A \cos(k_{1}x) + B \sin(k_{1} x)$$

for x=> 0
$$-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} = (E-V_{B}) \psi(x)$$
$$k_{2}^2 = \frac{2m}{\hbar^2} (E - V_{B})$$
$$\frac{\partial^2 \psi}{\partial x^2} = k_{2}^2 \psi(x)$$
$$\psi_(x) = C \exp(ik_{2} x) + D\exp(-ik_{2}x)$$
but we only care about the one with the negative exponential
hence C = 0
$$\psi_(x) = D\exp(-ik_{2}x)$$

are these solutions correct? Im just afraid i may have made a stupid mistake by switching the negative sign somewhere :(

ok to proceed i apply the boundary conditions s.t.

$$\psi_{I}(0) = \psi_{T}(0)$$
$$\psi_{I}'(0) = \psi_{T}'(0)$$

we calculate probability current density j

$$j_{trans} = \frac{\hbar k_{2}}{m} |D|^2$$
$$j _{inc} = \frac{\hbar k_{1}}{m} |A|^2$$
$$j_{refl} = \frac{\hbar k_{1}}{m} |B|^2$$

and transmission coefficient is calculated like this

$$T = \left| \frac{j_{trans}}{j_{inc}} \right|$$
$$R = \left| \frac{j_{refl}}{j_{inc}} \right|$$

2. Nov 15, 2006

nrqed

You want a wave traveling to the right so you must keep the C wavefunction and set D =0.

Patrick

3. Jul 4, 2010

FunkyDwarf

Sorry to resurrect a long dead thread, but if i wanted to impose the condition D = 0, ie incoming waves only from one direction on standing wave computed numerically, how would i go about doing that? I know Fourier analysis can give me the momentum components, but thats about it. Would i have to do it in the time dependent case?

4. Jul 5, 2010

kuruman

Say you have a wave incident from the left. The wavefunction would be

ψI = A*Exp(ik1x)+B*Exp(-ik1x) in region I (x ≤ 0)
ψII = C*Exp(ik2x) in region II (x ≥ 0)

You have an incident and reflected wave in region I (constants A and B respectively) and a transmitted wave in region II. Of course, you have to impose the proper boundary conditions at x=0.

For a lovely simulation of this and potential barriers in general, you may wish to go to