Sep-25-2020, 01:23 PM
Plotting of wave functions for an electrons quantum numbers (n,l ,ml) using spherical harmonics.
The first plot shows the probability of different distances which integral always is 1.
The first plot shows the probability of different distances which integral always is 1.
from scipy import * from pylab import * from numpy import * import sys import matplotlib.pyplot as plt from scipy import integrate R10 = lambda r: 4*exp(-2*r)*r**2 R20 = lambda r: 0.5*(1-r+r**2/4)*exp(-r)*r**2 R30 = lambda r: 1/(9**2*3)*((6-4*r+4*r**2/9)*exp(-r/3))**2*r**2 r = linspace(0,35,1000) r100 = R10(r) r200 = R20(r) r300 = R30(r) plt.plot(r, r100, label='|R10|²r²') plt.plot(r, r200, label='|R20|²r²') plt.plot(r, r300, label='|R30|²r²') plt.title('Probability density') plt.xlabel('aₒr') plt.ylabel('P(aₒr)') plt.legend() plt.show() I1 = integrate.quad(R10, 0, np.inf)[0] I2 = integrate.quad(R20, 0, np.inf)[0] I3 = integrate.quad(R30, 0, np.inf)[0] print('Normalization integral from 0 to infinity') print('<100|100> =', round(I1,8)) print('<200|200> =', round(I2,8)) print('<300|300> =', round(I3,8))Calculation of the expectation value for the electron-nucleous distance in unit of a Bohr radius.
from scipy import * from pylab import * from numpy import * import sys import matplotlib.pyplot as plt from scipy import integrate R10 = lambda r: 4*exp(-2*r)*r**3 R20 = lambda r: 0.5*(1-r+r**2/4)*exp(-r)*r**3 R30 = lambda r: 1/(9**2*3)*((6-4*r+4*r**2/9)*exp(-r/3))**2*r**3 r = linspace(0,35,1000) r100 = R10(r) r200 = R20(r) r300 = R30(r) I1 = integrate.quad(R10, 0, np.inf)[0] I2 = integrate.quad(R20, 0, np.inf)[0] I3 = integrate.quad(R30, 0, np.inf)[0] print('Integral from 0 to infinity') print('<100|r|100> =', round(I1,8),'aₒ') print('<200|r|200> =', round(I2,8),'aₒ') print('<300|r|300> =', round(I3,8),'aₒ')Expectation value for the potential energy
from scipy import * from pylab import * from numpy import * import sys import matplotlib.pyplot as plt from scipy import integrate R10 = lambda r: 4*exp(-2*r)*r R20 = lambda r: 0.5*(1-r+r**2/4)*exp(-r)*r R30 = lambda r: 1/(9**2*3)*((6-4*r+4*r**2/9)*exp(-r/3))**2*r r = linspace(0,20,1000) r100 = R10(r) r200 = R20(r) r300 = R30(r) plt.plot(r, r100, label='|R10|²r') plt.plot(r, r200, label='|R20|²r') plt.plot(r, r300, label='|R30|²r') plt.title('Expectation value for the potential energy') plt.xlabel('r') plt.ylabel('e²/4πεₒaₒ') plt.legend() plt.show() I1 = integrate.quad(R10, 0, np.inf)[0] I2 = integrate.quad(R20, 0, np.inf)[0] I3 = integrate.quad(R30, 0, np.inf)[0] print('Integral from 0 to infinity') print('<r100|Epot|r100> =', round(I1,8),'e²/4πεₒaₒ ≈', round(I1*27.2, 2), 'eV') print('<r200|Epot|r200> =', round(I2,8),'e²/4πεₒaₒ ≈', round(I2*27.2, 2), 'eV') print('<r300|Epot|r300> =', round(I3,8),'e²/4πεₒaₒ ≈', round(I3*27.2, 2), 'eV')