Hydrogen Atom Probability Density

1. Introduction

In this computational laboratory you will take advantage of one of Mathematica's unique plotting tools to visualize the probability of the electron in a hydrogen atom. A density plot shows the value of a function of two variables where colors are associated with the value of the function. You will use this visualization technique to develop intuition about the hydrogen atom probability.

2. Necessary Commands and Definitions

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

2.1 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;

2.2 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 2.1. 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]

2.3 Define the Hydrogen Atom Wave Function.

The hydrogen atom wave functions are defined here and parameters chosen to determine specific wave functions (i.e., the principle quantum number, the angular momentum, and its z-component). Note the radial part is not properly normalized to make the scaling the plots easier.

Rnl[n_,l_,r_] := (r^l)*Exp[-r/2]*LaguerreL[n-l-1, 2*l + 1,r];
Ylm[l_, m_, theta_, phi_] := SphericalHarmonicY[l, m, theta, phi];

2.4 Define the Density Plot.

The density plot is applied inside the Mathematica  Manipulate command to allow easy modifications of the parameters by the user. The parameters of the plot are set by clicking on the buttons or moving the sliders. The code is adapted from Reference [1] below. Here are some notes on running the calculation.

1. Depending on your choice of the principle quantum number n, these calculations are compute-intensive, i.e. they take a long time. The slides may not be very responsive so the best way to use them is to click at a point on the slider’s scale, see that the cell backet is highlighted as the calculation runs,  and after a few seconds the result will be displayed.

2. The vertical scale slider is a logarithmic scale; changing the value by one changes the scale by a factor of ten. Once you set the scale above a certain maximum, the plot will not change.

Manipulate[If[l≥n,l=n-1];
If[m>l,m=l];
DensityPlot[Module[{r=Norm[{x,0,z}]},
(* factor 0f 4((πr)^2) in included to make scaling the plots easier. *)4 π r^2 (Rnl[n,l,r])^2 Re[Ylm[l,m,ArcCos[z/r],ArcTan[x,0]]*Conjugate[Ylm[l,m,ArcCos[z/r],ArcTan[x,0]]]]
 ](* end of Module *),
{x,-side,side},{z,-side,side},Mesh->False,Frame->False,PlotPoints->Round[resolution],ColorFunctionScaling->True,ColorFunction->"SunsetColors",PlotRange->{0,10^scale}],{{n,8,"principal quantum number"},1,8,1,ControlType->Setter},{{l,3,"angular momentum"},Range[0,n-1],ControlType->SetterBar},{{m,1,"magnetic quantum number"},Range[0,l],ControlType->SetterBar},{{side,40,"Length of a side"},1,200,ControlType->Slider[Dynamic[side]]},{{scale,16,"Vertical Scale"},1,30,ControlType->Slider[Dynamic[scale]]},
{{resolution,40,"Resolution"},30,100,ControlType->Slider[Dynamic[scale]]}
]

3. Analysis

1. Clearly state in standard notation the hydrogen atom wave function. Describe the meaning of each quantum number.

2. For n=2, what are allowed values of l  and m ?

3. Generate a plot of the radial probability density P(r) (integrated over the angular part) for n=2, l=1, m=1, and r< 20. Make sure P(r) is properly normalized.

4. Generate a plot of the angular probability density n=2, l=1, m=1 as a function of θ for the full range of θ.

5. Consider the density plot for the n=3, l=2, m=0 state. What do the bright regions represent? What, if anything, does this density plot tell you about the trajectories of the electrons in the hydrogen atom?

6. Choose two sets of quantum numbers n, l, m in the ranges 3<n<8, 1<l<n-1, and m>0. For each of your choices of quantum numbers make a density plot of the hydrogen probability density using the code in Part 2.4. Briefly describe each one. Where is the electron most likely to found? Where is the least likely place to find it? 

7. If you increase or decrease the value of the orbital angular momentum l while keeping the principle quantum number n and m fixed, what happens to the number of high probability density regions in your plot? Your answer should be in the language of spherical coordinates. Print out enough plots to support your assertions.

[1] Code adapted from E.Zeleny, “Hydrogen Orbital Densities”, http://demonstrations.wolfram.com/HydrogenOrbitalDensities/, Wolfram Demonstrations Project, Published: June 20, 2012.