Hydrogen Orbitals (Python Notebook)

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Oct 4, 2020
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- Free Particle (Python Notebook)
- Particle in a Finite Potential Box (Python Notebook)

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Source: https://github.com/DalInar/schroding...eckpoint.ipynb

(Notes: Different version of scikit-image is used. The right version should be https://pypi.org/project/scikit-image/0.12.3/#files)

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import numpyimport mathimport matplotlib.pyplot as pltfrom matplotlib.widgets import Slider, Button, RadioButtonsimport scipy.specialfrom scipy.special import sph_harmfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cmfrom matplotlib.colors import ListedColormapimport skimage​def hydrogen_wf(n,l,m,X,Y,Z):    R = numpy.sqrt(X**2+Y**2+Z**2)    Theta = numpy.arccos(Z/R)    Phi = numpy.arctan2(Y,X)        rho = 2.*R/n    s_harm=sph_harm(m, l, Phi, Theta)    l_poly = scipy.special.genlaguerre(n-l-1,2*l+1)(rho)        prefactor = numpy.sqrt((2./n)**3*math.factorial(n-l-1)/(2.*n*math.factorial(n+l)))    wf = prefactor*numpy.exp(-rho/2.)*rho**l*s_harm*l_poly    wf = numpy.nan_to_num(wf)    return wf  dz=0.5zmin=-10zmax=10x = numpy.arange(zmin,zmax,dz)y = numpy.arange(zmin,zmax,dz)z = numpy.arange(zmin,zmax,dz)X,Y,Z = numpy.meshgrid(x,y,z) #X, Y, Z are 3d arrays that tell us the values of x, y, and z at every point in space​#Change these to change which orbital to plotn=4l=2m=0​data = hydrogen_wf(n,l,m,X,Y,Z)data = abs(data)**2​R = numpy.sqrt(X**2+Y**2+Z**2)​fig, ax = plt.subplots()plt.subplots_adjust(left=0.15, bottom=0.15)im = plt.imshow(data[int((0-zmin)/dz),:,:], vmin=0, vmax = numpy.max(data), extent=[zmin,zmax,zmin,zmax])plt.colorbar()sli = Slider(plt.axes([0.25, 0.025, 0.65, 0.03]), "Y", z[0], z[len(z)-1], valinit=0)ax.set_title("Hydrogen Orbital xz Slice (y="+str("%.2f"%sli.val)+"): n="+str(n)+", l="+str(l)+", m="+str(m))​def update(val):    index = int((sli.val-zmin) / dz)    im.set_data(data[index,:,:])    ax.set_title("Hydrogen Orbital xz Slice (y="+str("%.2f"%sli.val)+"): n="+str(n)+", l="+str(l)+", m="+str(m))       sli.on_changed(update)plt.show()

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import numpyimport mathimport matplotlib.pyplot as pltfrom matplotlib.widgets import Slider, Button, RadioButtonsimport scipy.specialfrom scipy.special import sph_harmfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cmfrom matplotlib.colors import ListedColormapimport skimagefrom skimage import measure​def hydrogen_wf(n,l,m,X,Y,Z):    R = numpy.sqrt(X**2+Y**2+Z**2)    Theta = numpy.arccos(Z/R)    Phi = numpy.arctan2(Y,X)        rho = 2.*R/n    s_harm=sph_harm(m, l, Phi, Theta)    l_poly = scipy.special.genlaguerre(n-l-1,2*l+1)(rho)        prefactor = numpy.sqrt((2./n)**3*math.factorial(n-l-1)/(2.*n*math.factorial(n+l)))    wf = prefactor*numpy.exp(-rho/2.)*rho**l*s_harm*l_poly    wf = numpy.nan_to_num(wf)    return wf  dz=0.5zmin=-10zmax=10x = numpy.arange(zmin,zmax,dz)y = numpy.arange(zmin,zmax,dz)z = numpy.arange(zmin,zmax,dz)X,Y,Z = numpy.meshgrid(x,y,z) #X, Y, Z are 3d arrays that tell us the values of x, y, and z at every point in spacen=4l=2m=0data = hydrogen_wf(n,l,m,X,Y,Z)data = abs(data)**2R = numpy.sqrt(X**2+Y**2+Z**2)​fig = plt.figure()ax = fig.add_subplot(111, projection='3d')ax.set_xlim([0,len(x)])ax.set_ylim([0,len(y)])ax.set_zlim([0,len(z)])max_val = numpy.max(data)​verts, faces, _, _ = measure.marching_cubes(data, max_val/2, spacing = (1,1,1))result=ax.plot_trisurf(verts[:,0], verts[:,1], faces, verts[:,2], cmap ='Spectral', lw=0)​sli = Slider(plt.axes([0.25, 0.025, 0.65, 0.03]), "iso", 0, max_val, valinit=max_val/2)ax.set_title("Hydrogen Orbital Isosurface ("+str("%.5f"%sli.val)+"): n="+str(n)+", l="+str(l)+", m="+str(m))​def update(val):    ax.clear()    verts, faces = measure.marching_cubes(data, sli.val, spacing = (1,1,1))    result = ax.plot_trisurf(verts[:,0], verts[:,1], faces, verts[:,2], cmap ='Spectral', lw=0)    ax.set_xlim([0,len(x)])    ax.set_ylim([0,len(y)])    ax.set_zlim([0,len(z)])    ax.set_title("Hydrogen Orbital Isosurface ("+str("%.5f"%sli.val)+"): n="+str(n)+", l="+str(l)+", m="+str(m))           sli.on_changed(update)plt.show()

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import numpyimport mathimport matplotlib.pyplot as pltimport scipy.specialfrom scipy.special import sph_harm​def hydrogen_wf(n,l,m,X,Y,Z):    R = numpy.sqrt(X**2+Y**2+Z**2)    Theta = numpy.arccos(Z/R)    Phi = numpy.arctan2(Y,X)        rho = 2.*R/n    s_harm=sph_harm(m, l, Phi, Theta)    l_poly = scipy.special.genlaguerre(n-l-1,2*l+1)(rho)        prefactor = numpy.sqrt((2./n)**3*math.factorial(n-l-1)/(2.*n*math.factorial(n+l)))    wf = prefactor*numpy.exp(-rho/2.)*rho**l*s_harm*l_poly    wf = numpy.nan_to_num(wf)    return wf  dz=0.5zmin=-10zmax=10x = numpy.arange(zmin,zmax,dz)y = numpy.arange(zmin,zmax,dz)z = numpy.arange(zmin,zmax,dz)X,Y,Z = numpy.meshgrid(x,y,z) #X, Y, Z are 3d arrays that tell us the values of x, y, and z at every point in spacen=3l=1m=0data = hydrogen_wf(n,l,m,X,Y,Z)data = abs(data)**2R = numpy.sqrt(X**2+Y**2+Z**2)​plt.figure()plt.plot(z, data[int(len(z)/2),int(len(z)/2),:])plt.title("$|\psi_{nlm}|^2$(x=0,y=0,z): n="+str(n)+", l="+str(l)+", m="+str(m))plt.xlabel('z')plt.ylabel("$|\psi_{nlm}|^2$")plt.show()

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