1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
|
#!/usr/bin/env python3
import cmath
import math
import scipy.special as sp
from scipy import optimize as op
import random
import sys
# number of roots
nr_roots=4
# size of plot
plotsize_x=3
plotsize_y=3
# rescale plot (so roots are not too close together)
plotsize_scale_x=1
plotsize_scale_y=6
# numerical values
hbar=6.58e-16 # eV.s
m=9.11e-31 # kg
Vn=9 # eV
En=20e9 # V/m
V=1
E=En*hbar/(Vn**1.5*m**0.5)*math.sqrt(1.60e-19)
# sqrt with branch cut along iR_+
def sqrt_p(x):
r,p=cmath.polar(x)
if(p<cmath.pi/2):
return(cmath.rect(math.sqrt(r),p/2))
else:
return(cmath.rect(math.sqrt(r),(p-2*math.pi)/2))
# solution of (-\Delta+V-ip)phi=0
def phi(p,x,E,V):
return(sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[0])
# its derivative
def dphi(p,x,E,V):
return(cmath.exp(-1j*math.pi/3)*E**(1/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
# the function whose roots are to be computed and its derivative
def f(p):
return(sqrt_p(-1j*p)*phi(p,0,E,V)-dphi(p,0,E,V))
def df(p):
return(-1j/(2*sqrt_p(-1j*p))*phi(p,0,E,V)+sqrt_p(-1j*p)*dp_phi(p,0,E,V)-dp_dphi(p,0,E,V))
# derivatives of phi with respect to p
def dp_phi(p,x,E,V):
return(1j*cmath.exp(-1j*math.pi/3)*E**(-2/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
def dp_dphi(p,x,E,V):
return(-1j*(x-(V-1j*p)/E)*phi(p,x,E,V))
# check whether the root was already found
def check(root,roots):
# reject if the root doesn't fit on the plot
if(plotsize_scale_x*root.real<-plotsize_x or abs(plotsize_scale_y*root.imag)>plotsize_y):
return(False)
for x in roots:
if(abs(root-x)<1e-12):
return(False)
return(True)
# find roots
roots=[]
while len(roots)<nr_roots:
try:
root=op.newton(f, -random.random()*plotsize_x/plotsize_scale_x+(2*random.random()-1)*plotsize_y/plotsize_scale_y*1j, fprime=df)
if(check(root,roots)):
roots.append(root)
print("found "+str(len(roots))+" roots of "+str(nr_roots), file=sys.stderr)
except RuntimeError:
root=0
except:
break
# print result
for root in roots:
print("\\pole{(% .3f,% .3f)}" % (plotsize_scale_x*root.real,plotsize_scale_y*root.imag))
|
