Understanding Hydrogen Atom Probability Density Plots

1. Introduction

In this computational laboratory you will continue our expedition to visualize the probability of the electron in a hydrogen atom. You might remember from other courses that hydrogen orbitals are often pictured like the ones shown in the figure below which shows a wave function plot of the hydrogen 2p (n=2, l=1) m=±1 state with the colors representing different phases. We have used the standard notation for the angular momentum quantum numbers so l=0, 1, 2, 3 correspond to the letters s, p, d, f. In this laboratory we will study the connection between our development of the hydrogen atom wave functions and the representation here.

You will first consider a compelling three-dimensional visualization of the electron probability density that touches on early presentations of this topic. We will then explore those visualizations more deeply using our current understanding of the hydrogen atom.

2. Necessary Commands and Definitions

The functions and algorithms needed to complete this laboratory are contained in this section.

2.1 Three Dimensional Plot

We will be making three dimensional plots of functions that depend on two variables, i.e. the spherical harmonics (Subscript[Y, l]^m)(θ,ϕ). The command below demonstrates how you do this in Mathematica.

Plot3D[Sin[theta]*Cos[phi],{theta,0,π},{phi,0,2*π},
AxesLabel->{"θ (rad)","ϕ (rad)","Function"},
BaseStyle->FontSize->16,
ImageSize->6.5*72]

2.2 Input Parameters

The user parameters are described here and defined in the cell below.

                  n - principle quantum number
                  l - angular momentum quantum number
                  m - z-component of the angular momentum

n = 4;
l = 2;
m = 1;

2.3 Using Laguerre polynomials and spherical harmonics.

The solution to the hydrogen atom Schroedinger equation consists of Laguerre polynomials and spherical harmonics. The commands in Mathematica to use those functions are shown below. The values of the n, l, and m are set in part 2.2. Notice the spherical harmonic can be complex.

LaguerreL[n-l-1, 2*l + 1,2.3]  (* for r=2.3 *)

SphericalHarmonicY[l, m, 0.2, 0.3] (* for θ = 0.2 radians, ϕ = 0.3 radians. *)

To calculate probability densities or probabilities we  take the square of the magnitude as shown here and then pluck off the real part (the imaginary part should be zero).

test=SphericalHarmonicY[l, m, 0.2, 0.3]*Conjugate[SphericalHarmonicY[l, m, 0.2, 0.3]]
Re[test]

2.4 Define the Hydrogen Atom Wave Function.

The hydrogen atom wave functions are defined as

(Subscript[Ψ, nlm])(r,θ,ϕ) = (Subscript[R, nl])(r)(Subscript[Y, l]^m)(θ,ϕ) =(Sqrt[((((2/(Subscript[na, 0]))))^3)(( (n-l-1)!)/(2((n[(n+l)!])^3)))]) (((e^(-r/(Subscript[na, 0])))(((2r)/(Subscript[na, 0]))))^l)(n+l)!(Subscript[ℒ, n-l-1]^(2l+1)) (Subscript[Y, l]^m)(θ,ϕ)

where n is the principle quantum number, l is the angular momentum quantum number, m is the projection of the angular momentum quantum number on the z axis, and (Subscript[ℒ, n-l-1]^(2l+1)) is the Laguerre polynomial. The Bohr radius (Subscript[a, 0]) = 0.528 Å sets the length scale.

2.5 Textbook Representations of Hydrogen Orbitals.

Download the Mathematica notebook entitled VisualizingAtomicOrbitals.nb [1] from the course website. Follow the guidance in that notebook to select different orbitals, rotate the view, resize the image, and add coordinate axes to the image.

3. Analysis

1. Using the notebook VisualizingAtomicOrbitals.nb select an image of the 2(Subscript[p, x]) state (it should look similar to the figure above). Turn on the axes and labels on the plot. For any ϕ angle, what is the most likely polar (θ) angular range to find the electron? Explain your answer. For θ=(90^o) what is the most probable azimuthal (ϕ) angle? Explain.

2. The hydrogen orbitals you investigated in part 1 only show the angular part of the wave function and rely on a particular form of that angular part. For the 2(Subscript[p, x]) state the angular part (Subscript[Ψ, nlm]^angular) of the wave function is the following.

(Subscript[Ψ, 21±1]^angular)(θ,ϕ)  = (1/(Sqrt[2]))((Subscript[Y, 1]^1)(θ,ϕ) + (Subscript[Y, 1]^(-1))(θ,ϕ)) 

Calculate the probability density of (Subscript[Ψ, 21±1]^angular)(θ,ϕ)  using the Plot3D command. Is the result consistent with what you see in part 1? Explain.

3. The hydrogen atom angular wave functions we have used are the individual spherical harmonics (Subscript[Y, l]^m)(θ,ϕ). Calculate the probability density of (Subscript[Y, 1]^1)(θ,ϕ) and (Subscript[Y, 1]^(-1))(θ,ϕ) separately. Compare these two new probability densities. How are they similar? How are the different? Compare these results with part 2.  How are the probability densities in parts 2 and 3 the same? How are they different? Explain.

4. Which of the representations of the hydrogen atom probability density for the n=2, l=1, m=±1 state are correct? Explain.

[1]  Guenther Gsaller, “Visualizing Atomic Orbitals”, http://demonstrations.wolfram.com/VisualizingAtomicOrbitals/,  Wolfram Demonstrations Project,  Published: June 12, 2007