The Zeeman Effect in Hydrogen

1. Introduction

We have developed a model of the hydrogen atom to explain the observed, n=2, energy levels of the hydrogen atom and we now want to understand the behavior of those states when placed in a magnetic field. In this laboratory we will compare our predictions with the observations of W.E.Lamb and R.C.Retherford. Lamb received the Nobel Prize in 1955 for the work we show here. We will build on our experience to include the spin-orbit and relativistic corrections to the n=2 state in hydrogen and develop the theory of the interaction of the hydrogen atom with an external magnetic field. We will make a prediction of the energy splitting of the n=2 state in an applied, external magnetic field. We will then determine the magnetic field dependence of the photons that are absorbed for three transitions between the split states.

2. Constants and Commands

2.1 Entering Matrices

Consider the 4x4 matrix shown here. To enter this matrix in Mathematica use the following command. The quantity b is some, yet-to-be-determined parameter.

Vmat = { {  0,2*b, 0, 0},
         {2*b,  0, 0, 0},
         {  0,  0, b, 0},
         {  0,  0, 0,-b}}

The command below will neatly display the matrix Vmat.

TableForm[Vmat]

2.2 Getting Eigenvalues

To get the eigenvalues of Vmat use the following command.

Eigenvalues[Vmat]

2.3 Getting Eigenvectors

To get the eigenvectors of Vmat use the following command.

Eigenvectors[Vmat]

2.4 Plotting Sets of Eigenvalues

In this laboratory we will explore the dependence of the hydrogen atom eigenvalues for the n=2 state on an applied magnetic field. To demonstrate how one generates a plot of the eigenvalues as a function of the applied, external field, consider the commands shown below. You will have to modify this code to perform the laboratory.

(* Define ranges, stepsizes, etc. *)

bmax = 5.0; (* maximum value of b. *)
step  = 0.01; (* stepsize in b. *)
nsteps = (bmax/step); (* number of steps. *)
dimension = 4; (* dimension of the matrix. *)
b = 0; (* initial value of b. *)

(* Get the initial set of eigenvalues and make a table
of b values and eigenvalues.  The matrix Vmat was 
defined above. *)

energies=Eigenvalues[Vmat];
PlotTable1 = Table[{b,energies[[i]]},{i,1,4}];

(* Loop over the value of b here. In each iteration
 create another table of b values and eigenvalues 
 and add that result to the existing table. *)

Do[ 
b = j*step;
Vmat={{      0,2*b,0,  0},
            {2*b,     0,0,  0},
            {      0,     0,b,  0},
            {      0,     0,0,-b}};
energies=Eigenvalues[Vmat];
PlotTable2 = Table[{b,energies[[i]]},{i,1,4}];
PlotTable1 =  Join[PlotTable1,PlotTable2],
{j,1,nsteps}
]

(* Plot the result. *)

ListPlot[PlotTable1,
PlotRange->{{-0.0001,5.0},{-12.0,12.0}},
Frame->True,
FrameLabel->{"b","Eigenvalue","Plotting Demo",""},
GridLines->Automatic,
ImageSize->72*4.5,
AspectRatio->1.0,
PlotStyle->RGBColor[1,0,0]
]

3. Procedure

1. To understand the Zeeman effect, consider an electron bound in a hydrogen atom when an external magnetic field is applied that points in the z-direction. What is the contribution of the electron to the magnetic moment of the hydrogen atom? What is the change in energy (Subscript[ΔE, zee])of the hydrogen atom when the external field is applied in terms of (Subscript[L, z]), (Subscript[S, z]), and any other constants?

2. Generate a matrix version of (Subscript[ΔE, zee]) for the basis states | n l (Subscript[m, l])s (Subscript[m, s])> for  n=2. You should have only numerical entries in the final matrix.

3. Generate the matrices for the spin-orbit and relativistic corrections. 

4. Figure 1 below shows the dependence of the eigenvalues in the external magnetic field. The goal in this section is to reproduce this figure. Do this by adding the three matrices from Parts 2-3 and getting the eigenvalues as a function of applied field (see Section 2.4 for an example of the commands you need).

5. Once you have agreement between your calculation and Figure 1 consider transitions between the state labelled α and those labelled a,b, and c as shown by the arrows in Figure 1. What is the general expression for the frequency of the photons absorbed in these transitions? Make a plot of this frequency for each transition as a function of the applied field.

6. In 1947, W.E.Lamb and R.C.Retherford published the following results for the frequency of the photons for the transitions discussed in Part 5. Add these data to the plot of your prediction of those frequencies you made in Part 5.

LambData = {{0.05667,8559.19},{0.03872,8997.48},{0.03590,9083.12},{0.03077,9125.94},{0.02180,9345.09},{0.01487,9486.15},{0.26846,8987.41},{0.23795,9090.68},{0.21590,9138.54},{0.15385,9360.20},{0.10897,9491.18},{0.12000,11539.04},{0.14436,11836.27},{0.17590,12236.78}}

7. How is the data similar to your prediction? How is it different? If it is different, how can you account for the difference?