Scattering from a Finite
Potential Barrier
The potential barrier that we want to consider is given by:
0
if x < 0
V(x) =
V0 if x > 0
Let L=1 and V0=5 and the energy of the particle be E=3. The calculations are done in reduced
mass, atomic units. So mass = 1 and hbar=1.
In[1]:=
L = 1; V0 = 5; Energy = 3;
V@x_D := If@x > 0, V0, 0D
Plot@V@xD, 8x, - 1, 1<D
5
4
3
Out[3]=
2
1
-1.0
-0.5
0.5
1.0
We label the regions I, II from left to right. Both the regions are regions of constant potential, and
therefore in the interior of any region, using translational invariance, we can say that the solutions to
the Time Independent Schrodinger Equation would be corresponding plain wave solutions:
L =.; V0 =.; Energy =.; k1 = Sqrt@2 EnergyD;
Ψ1@x_D := A1 Exp@I k1 xD + B1 Exp@- I k1 xD
k2 = Sqrt@2 HEnergy - V0LD;
Ψ2@x_D := A2 Exp@I k2 xD + B2 Exp@- I k2 xD
B2 = 0;
We want the components as Fourier components, for easy time evolution.
In[62]:=
k1 =.;
k2 = SqrtAk12 - 2 V0E;
Ψ1@x_D := A1 Exp@I k1 xD + B1 Exp@- I k1 xD
Ψ2@x_D := A2 Exp@I k2 xD + B2 Exp@- I k2 xD
Since there is no reflection in region II, we can set B2=0. Now we impose the criteria of continuity at
x=0
2
scattering.nb
In[66]:=
Out[66]=
sol = Simplify@Solve@8Ψ1@0D Š Ψ2@0D, Ψ1 '@0D Š Ψ2 '@0D<, 8B1, A2<DD
::B1 ®
A1 Jk1 k1 +
k12 - 2 V0 N
k12 - 2 V0
2 A1 k1
, A2 ®
k1 +
k12 - 2 V0
>>
The eigenfunction is given by:
In[67]:=
Ψ@x_D := HHeavisideTheta@- xD Ψ1@xD . sol@@1DD@@1DDL +
HHeavisideTheta@xD Ψ2@xD . sol@@1DD@@2DDL
So,
In[85]:=
Out[85]=
Jk1 Jk1 +
4 k1
Out[86]=
In[88]:=
Out[88]=
2
. sol@@1DD
A1
k2 A2 2
T =
. sol@@1DD
k1 A1
B1
R =
Jk1 +
k12 - 2 V0 N
2
k12 - 2 V0 N
2
k12 - 2 V0
k12 - 2 V0 N
Simplify@R + TD
1
2
syms k1 k2 A1 B1 A2 R T V0 x
k2 = sqrt(k1^2 − 2*V0)
%define the wavefunctions
psi1 = symfun(A1*exp(i*k1*x)+B1*exp(−i*k1*x),x)
psi2 = symfun(A2*exp(i*k2*x),x)
%calculate the derivatives
d1 = diff(psi1,x)
d2 = diff(psi2,x)
%solve the simultaneous equations imposed by the condition of continuity
s=solve(psi1(0)==psi2(0),d1(0)==d2(0),B1,A2)
s.B1
s.A2
R = simplify((s.B1/A1)^2)
T = simplify(k2/k1 * (s.A2/A1)^2)
simplify(R+T)
scattering -- Sage
1 of 1
http://localhost:8080/home/admin/2/print
scattering
var('k1 k2 A1 B1 A2 R T V0 x')
k2 = sqrt(k1^2 - 2*V0)
define the wavefunctions
psi1 = A1*exp(i*k1*x)+B1*exp(-i*k1*x)
psi2 = A2*exp(i*k2*x)
s=solve([psi1(x=0)==psi2(x=0),psi1.diff(x)(x=0)==psi2.diff(x)(x=0)],
[B1,A2])
R = (s[0][0].right()/A1)**2
T = k2/k1 * (s[0][1].right()/A1)**2
R.full_simplify()
(2*k1^4 - 4*V0*k1^2 + V0^2 - 2*sqrt(k1^2 - 2*V0)*(k1^3 V0*k1))/V0^2
T.full_simplify()
-2*(k1^4 - 2*V0*k1^2 - sqrt(k1^2 - 2*V0)*(k1^3 - V0*k1))/V0^2
(R+T).full_simplify()
1
09/25/2013 04:55 PM