def solve_ode(self, l):
"""Solves the Schrodinger equation for a given l value using the Numerov method
in the ode module.
Arguments:
l: angular momentum number
Returns:
points: set of points containing the solution to the Schrodinger equation
"""
if self.verbose:
print("Calculating points...")
# Set up initial conditions
# We can't start at r=0, because the centrifugal term will diverge. Instead,
# we use the central difference formula for the second derivative and X(0)=0
# to write an (worse) approximation for the third point.
points = np.zeros_like(self.rgrid)
points[1] = 1/self.stepsize # This is an arbitrary -- it sets the normalization.
points[2] = points[1] * (2 - self.schrod_eqn(self.rgrid[1],l)*(self.stepsize**2))
points = ode.numerov(self.schrod_eqn, x_grid = self.rgrid, y_grid=points,
start_index=1, end_index = self.xn,
verbose=self.verbose, l=l)
return points
def solve_ode(self, l):
"""Solves the Schrodinger equation for a given l value using the Numerov method
in the ode module.
Arguments:
l: angular momentum number
Returns:
points: set of points containing the solution to the Schrodinger equation
"""
if self.verbose:
print("Calculating points...")
# Set up initial conditions
# We can't start at r=0, because the centrifugal term will diverge. Instead,
# we use the central difference formula for the second derivative and X(0)=0
# to write an (worse) approximation for the third point.
points = np.zeros_like(self.rgrid)
points[
1] = 1 / self.stepsize # This is an arbitrary -- it sets the normalization.
points[2] = points[1] * (2 - self.schrod_eqn(self.rgrid[1], l) *
(self.stepsize**2))
points = ode.numerov(self.schrod_eqn,
x_grid=self.rgrid,
y_grid=points,
start_index=1,
end_index=self.xn,
verbose=self.verbose,
l=l)
return points
def solve_ode(self, E, direction=None, endpoint=None):
"""Solves the Schrodinger equation using the Numerov method.
Arguments:
E: energy
direction: which way the solver works
endpoint: sets a particular stopping point
Returns:
points: solution to differential equation
"""
if endpoint is None:
endpoint = self.turnpoint_index
if direction is None:
# If a direction isn't specified, recurse downward and integrate
# left and right sides.
l_points = self.solve_ode(E, direction='right', endpoint=endpoint)
r_points = self.solve_ode(E, direction='left', endpoint=endpoint)
points = np.zeros_like(self.xgrid)
points[:endpoint] = l_points[:endpoint]
points[endpoint:] = r_points[endpoint:] * (l_points[endpoint] /
r_points[endpoint])
points = self.normalize(points)
return points
if self.verbose:
print("Calculating points...")
points = ode.numerov(self.schrod_eqn,
i0=0,
i_slope=-1,
x_grid=self.xgrid,
direction=direction,
end_index=endpoint,
verbose=self.verbose,
E=E)
# Convert back from y(x) to chi(x)
return points * np.sqrt(self.rgrid)
def solve_ode(self, E, direction=None, endpoint=None):
"""Solves the Schrodinger equation using the Numerov method.
Arguments:
E: energy
direction: which way the solver works
endpoint: sets a particular stopping point
Returns:
points: solution to differential equation
"""
if endpoint is None:
endpoint = self.turnpoint_index
if direction is None:
# If a direction isn't specified, recurse downward and integrate
# left and right sides.
l_points = self.solve_ode(E, direction='right', endpoint=endpoint)
r_points = self.solve_ode(E, direction='left', endpoint=endpoint)
points = np.zeros_like(self.xgrid)
points[:endpoint] = l_points[:endpoint]
points[endpoint:] = r_points[endpoint:] * (l_points[endpoint]/r_points[endpoint])
points = self.normalize(points)
return points
if self.verbose:
print("Calculating points...")
points = ode.numerov(self.schrod_eqn, i0 = 0, i_slope = -1,
x_grid = self.xgrid,
direction = direction,
end_index = endpoint,verbose=self.verbose, E=E)
# Convert back from y(x) to chi(x)
return points*np.sqrt(self.rgrid)
import ode
import numpy as np
import matplotlib.pylab as pylab
grid = np.linspace(0,2*np.pi,100)
def g(x):
return np.ones_like(x)
res = ode.numerov(g,0.,1.,grid,end_index=50)
pylab.plot(grid,res)
#pylab.plot(grid,np.sin(grid))
#pylab.show()
res = ode.numerov(g,0.,1.,grid,end_index=50,direction='left')
pylab.plot(grid,res)
#pylab.plot(grid,np.sin(grid))
pylab.show()
def wf(M, xm): # find w.f. and deriv at xm
c = (h*h)/6.
wfup, nup = ode.numerov(f, [0,.1], M, xL, h) # 1 step past xm
dup = ((1+c*f(xm+h))*wfup[-1] - (1+c*f(xm-h))*wfup[-3])/(h+h)
return wfup, dup/wfup[-2]
def wf(M, xm): # find w.f. and deriv at xm
c = (h * h) / 6.
wfup, nup = ode.numerov(f, [0, .1], M, xL, h) # 1 step past xm
dup = ((1 + c * f(xm + h)) * wfup[-1] -
(1 + c * f(xm - h)) * wfup[-3]) / (h + h)
return wfup, dup / wfup[-2]
import ode
import numpy as np
import matplotlib.pylab as pylab
grid = np.linspace(0, 2 * np.pi, 100)
def g(x):
return np.ones_like(x)
res = ode.numerov(g, 0., 1., grid, end_index=50)
pylab.plot(grid, res)
#pylab.plot(grid,np.sin(grid))
#pylab.show()
res = ode.numerov(g, 0., 1., grid, end_index=50, direction='left')
pylab.plot(grid, res)
#pylab.plot(grid,np.sin(grid))
pylab.show()
import ode
import numpy as np
import matplotlib.pylab as pylab
grid = np.linspace(1,10,100)
def g(x):
return 1/x
res = ode.numerov(g,0.,1.,grid)
pylab.plot(grid,res)
#pylab.plot(grid,np.sin(grid))
#pylab.show()
res = ode.numerov(g,-1.62827,-0.319947,grid,direction='left')
pylab.plot(grid,res)
#pylab.plot(grid,np.sin(grid))
pylab.show()
import ode
import numpy as np
import matplotlib.pylab as pylab
grid = np.linspace(1, 10, 100)
def g(x):
return 1 / x
res = ode.numerov(g, 0., 1., grid)
pylab.plot(grid, res)
#pylab.plot(grid,np.sin(grid))
#pylab.show()
res = ode.numerov(g, -1.62827, -0.319947, grid, direction='left')
pylab.plot(grid, res)
#pylab.plot(grid,np.sin(grid))
pylab.show()