Ian Jauslin
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authorIan Jauslin <jauslin@ias.edu>2019-05-22 22:59:03 -0400
committerIan Jauslin <jauslin@ias.edu>2019-05-22 22:59:03 -0400
commitdf7449e4a29ec8d3182cf7b2aebcb86f7ac596c2 (patch)
tree182c5c85f57264b38c60a02ec268e5c00d937b34 /figs/contour.fig/contour.py
As presented at VirginiaTech on 2019-05-24HEADv1.0master
Diffstat (limited to 'figs/contour.fig/contour.py')
-rw-r--r--figs/contour.fig/contour.py82
1 files changed, 82 insertions, 0 deletions
diff --git a/figs/contour.fig/contour.py b/figs/contour.fig/contour.py
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+++ b/figs/contour.fig/contour.py
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+#!/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))