from scipy import *
from scipy import integrate, optimize
from pylab import *

def Schrod_deriv(y, r, l, E, Z):
    "Given y=[u,u'] returns dy/dr=[u',u'']"
    du2 = (l*(l+1)/r**2-2*Z/r-E)*y[0]
    return [y[1],du2]

def Shoot(E,R,l,Z):
    y0=[0.0,-1e-7]
    Rb=R[::-1]
    y = integrate.odeint(Schrod_deriv, y0, Rb, args=(l,E ,Z))[:,0]
    norm = integrate.simps(Rb, y**2)
    f0 = y[-1]/sqrt(norm)
    f1 = y[-2]/sqrt(norm)
    final = f0 + (f1-f0)*(0.0-Rb[-1])/(Rb[-2]-Rb[-1])
    #print 'f0=', f0, 'f1=', f1, 'extrapolated=', final
    return final
    
def FindBoundStates(R,l,Z,nmax,MaxSteps,E0,dE):
    Eb=[]  # bound states
    u0 = Shoot(E0,R,l,Z)
    for i in range(MaxSteps):
        E0 += dE
        u1 = Shoot(E0,R,l,Z)
        #print E0, u1, dE
        if u0*u1<0.0:
            Ebound = optimize.brentq(Shoot, E0-dE, E0, xtol=1e-16, args=(R,l,Z))
            Eb.append( Ebound )
            #print 'Found bound state at E=', Ebound
            if len(Eb)>=nmax or Ebound>0: break
        u0=u1
        dE = dE/1.091
    return Eb

def SolveSchroedinger(E,R,l,Z):
    Rb=R[::-1]
    y0=[0.0,-1e-7] # starting guess should be quite small
    y = integrate.odeint(Schrod_deriv, y0, Rb, args=(l,E ,Z))[:,0]
    norm = integrate.simps(Rb, y**2)
    return y[::-1]/sqrt(norm)
    

Z = 1.
MaxSteps=1000
nmax=5

R = logspace(-6,2.2,500)
E0 = -1.2*Z**2
dE = abs(E0)/12.
for l in range(3):
    bstates = FindBoundStates(R,l,Z,nmax-l,MaxSteps,E0,dE)
    E0 = bstates[0]  # higher l will not have lower energy, hence start from higher E
    dE = abs(E0)/12. # adjust steps
    print l, bstates
    for n,Ei in enumerate(bstates):
        u = SolveSchroedinger(Ei,R,l,Z)
        plot(R,u,label=('l=%s n=%s' % (l,n)))
legend(loc='best')
show()