The Hydrogen Atom Orbitals
Giulio Dujany



Solutions of Schrodinger's equation for the Hydrogen atom can be factorize in a function of the distance of the electron from the nucleus (r) and a function of the direction (ϑ, ϕ), due to the spherical symmetry of the problem.
The function of the direction is a spherical harmonic, an eigenfunction of the square and of the third component of the orbital angular momentum operator (L and L_{z} respectively). Y_{l m} (ϑ,ϕ) is labeled by two quantum numbers: l, related to the eigenvalue of L and m, related to the eigenvalue of L_{z} . l and m can assume integer values: 0≤ l ≤ n1 (where n is the first quantum number related to the eigenvalue of energy) and l ≤ m ≤ l. ϑ is the colatitude and ϕ is the azimuth angle of the spherical coordinate system (0≤ϑ≤π, 0≤ϕ<2π).
Maple already knows spherical harmonics so we use the already implemented functions requiring that they are written using elementary functions.

(2.1) 
For example:

(2.2) 
We check that they are correctly normalized. This is necessary because of the probabilistic interpretation of quantum mechanics: the square of the wave function is the probability density to find the electron. Thus Y_{l m}(ϑ,ϕ) is the probability density to find the electron in the direction identified by ϑ and ϕ. This integral being 1 means that it is sure to find the electron in some direction.

(2.3) 
The plots of the squares of spherical harmonics are symmetric for rotation around zaxis (this is because spherical harmonics are eigenfunctions of L_{z}). In order to have also a dependence from ϕ we define OrbY making a change of basis in the linear space spanned by spherical harmonics. These functions are no longer eigenfunctions of L_{z } but their plots are more usually shown in books.

(2.4) 
Of course the probabilistic interpretation still stands.

(2.5) 




Radial functions make use of generalized Laguerre polynomials, r is the distance of the electron from the nucleus
and r 2 [0,+∞).

(3.1) 

(3.2) 
For example:

(3.3) 
a_{0} is the Bohr radius and Maple already knows its definition and value:

(3.4) 

(3.5) 

(3.6) 

(3.7) 





(4.1) 
For example:

(4.2) 
We check that they are normalized. As we work in spherical coordinates the integrand is ψ_{n l m}(r,ϑ,ϕ) r sin(ϑ).

(4.3) 
We also define wave functions using OrbY instead of Y to have the dependence from ϕ also in their square.

(4.4) 
For example:

(4.5) 
We verify normalization:

(4.6) 




Integrating in spherical coordinates we must add the Jacobian determinant of the change of variables. Thus the probability density to find the electron at a given distance from the nucleus is not Χ_{n l}(r) but Χ_{n l}(r) r. We define R_{n l}(r) = r Χ_{n l}(r) whose square is the correct radial probability density.

(5.1) 
For example:

(5.2) 
We verify normalization:

(5.3) 

(5.4) 




Radial functions are define for r = 0..∞ but, after a given value of r, functions go asymptotically to zero. For this reason in our plots the range of r will go from 0 to the value of r for which the surface under the graph of R_{n l}(r) reaches 0.999.
For R_{n l}(r) this means to plot the 99.9% of probability to find the electron. For Χ_{n l}(r) instead there is not such probabilistic interpretation but it is only a trick to cut off the tail from the plot.

(6.1) 




Plotting Y_{l m}(ϑ,ϕ) we can see that it is symmetric under rotation around the zaxis.
The plot of OrbY_{l m}(ϑ,ϕ) instead has a dependence also from ϕ.


8 
Isoprobability surfaces



We want to plot the surface containing 90% of probability to find the electron.
We therefore evaluate, using radial probability, the value rmax of r for which the cumulative probability reaches 0.9

(8.1) 
Using an implicit plot we obtain the surface of all the points whose probability is the cube of the radial probability at rmax.
We can use Orb instead of ψ to have the dependence from ϕ.




In this section the content of previous paragraphs is automated building interactive components.
In the first combo box it is possible to choose between showing an eigenvalue of L_{z} or a function whose square depends also by ϕ as defined in the previous paragraphs (Orbital).
We can then select the three quantum numbers (automatically the values of l and m are only the possible ones according to the choice of n). m is the third quantum number related to the eigenvalue of L_{z }only selecting the option "Eigenvalue Lz", conversely in "Orbital" it is has a different meaning but it is only a label. The upper limit of n has no physical constraints and can be changed adding values in the corresponding combo box.
n = l = m =
Equation: ψ=
Χ_{n l}(r) R_{n l}(r) = r Χ_{n l}(r)
Radial Plots:
Y_{l m}(ϑ,ϕ)
Angular Plot:
ψ_{n l m}(r,ϑ,ϕ)
Isoprobability surface:




• 
Harrington D. A., Orbitals Package Examples, Maplesoft Application Center, 2007.

• 
Hercules D., Density of Probability of an Electron near the Nucleus, Maplesoft Application Center, 2010.

• 
Rossetti C., Rudimenti di meccanica quantistica, Levrotto & Bella, Torino, 2008.

• 
Shankar R., Principles of Quantum Mechanics, Plenum Press, New York, 1980.

• 
Watson G. N., Whittaker E. T., A Course of Modern Analysis, Cambridge University Press, 1927.


