Spin-1/2 Particles in a 

Homogeneous Magnetic Field

Maria Daghofer,  Wolfgang von der Linden
\(Institut\ fuer\ Theoretische\ Physik\)
Technische Universitaet Graz
A - 8010 Graz, 
    Austria
Projekt Multimediale Lehre des
Zentralen Informatikdienstes der TU Graz

2001 - 12 - 13
Null

Parameters

These are three values for<(Subscript[S, x])>,\(feel\ free\ to\ choose\ any\ value\ from\ the\ \(\(interval[\(-0.5\), 0.5]\)\(.\)\)\)

Pauli-Matrices

Spin-1/2 particles are described by the Pauli matrices. One representation of 
them is:

(Subscript[σ, 1]) = (({{0, 1}, {1, 0}}));  (Subscript[σ, 2]) = (({{0, -I}, {I, 0}}));  (Subscript[σ, 3]) = (({{1, 0}, {0, -1}}));

Commutator relations

Levi-Civita tensor, epsilon tensor

For the commutator relations, we need the epsilon or Levi-Civita-tensor:

(Subscript[ϵ, i_ , j_, k_])  := Signature[{i, j, k}];

Commutator

The Pauli-matrices are defined by their commutator relations:

Com[i_, 
          j_]  :=  (Subscript[σ, i])  .  (Subscript[σ, j])  -  (Subscript[σ, j])  .  (Subscript[σ, i]);
V[i_, j_]  :=  
        Com[i, j]  ==  I 2 (∑_(k = 1)^3)  (Subscript[ϵ, i, j, k])  (Subscript[σ, k]);

Verify Commutator Relations

We can now use these relations to check whether the matrices defined above 
are a correct description of a Spin-1/2 particle:

Do[Print["[σ", i, ",σ", j, "]:   ", 
        V[i, j]], {i, 1, 3}, {j, 1, 3}]

Time Evolution

Description

Time evolution

Time evolution operator

U[t_]  =  
        MatrixExp[- 2 I π t (Subscript[σ, 1])] // 
          FullSimplify;
MatrixForm[U[t]]

Initial state vector

We choose an initial state with  <(Subscript[S, y])>  =  0. This choice merely defines the orientation of the y-axis, the x-axis has  already 	
	been defined by the direction of the external B-field. <(Subscript[S, y])>  =  0 is fulfilled by the normalized vector

Ψ[α_]  =  {Cos[α], 
          Sin[α]};

As we will see later, α is determined by <(Subscript[S, x])>.

Time dependent state vector

w[t_, α_] = 
        U[t] . Ψ[α] // Simplify;

Expectation values <(Subscript[S, α])>

x[
          t_, α_] = Conjugate[w[t, α]] . (Subscript[σ, 1]) . 
                w[t, α]/2 // ComplexExpand // 
          Simplify;
y[
          t_, α_] = Conjugate[w[t, α]] . (Subscript[σ, 2]) . 
                w[t, α]/2 // ComplexExpand // 
          Simplify;
z[
          t_, α_] = Conjugate[w[t, α]] . (Subscript[σ, 3]) . 
                w[t, α]/2 // ComplexExpand // 
          Simplify;
\(Print[\*""<\!\(S\_x\)> ="", 
      x[t, α], "
", \*""<\!\(S\_y\)> ="", 
      y[t, α], "
", \*""<\!\(S\_z\)> ="", 
      z[t, α]]\)

We see that the expectation value of(Subscript[S, x]), i . e .  the component parallel to the B - field, is time independent .   As required,the  y - component vanishes for t=0.  The angle α is fixed by the choice of <(Subscript[S, x])> =: κ.

α[κ_]  =  ArcSin[ 2  κ]/2;

Expectation values <(((Δ(Subscript[S, α])))^2)>

For the calculation of \(\(\(<\)\(\((ΔS\_α)\)\^2\)\(>=\)\(<\)\(S\_α\^2\)\(>\)\(-\)\)\)\(\(\(<\)\(S\_α\)\( > \^2\)\)\), we make use of (Subscript[σ, α])\(TraditionalForm\`\^2\)= 1.

dx[t_, α_] = 
        1/4 - x[t, α]^2;
dy[t_, α_] = 
        1/4 - y[t, α]^2;
dz[t_, α_] = 
        1/4 - z[t, α]^2;
\(Print[\*""<(Δ(Subscript[S, x]))^2> ="", 
      dx[t, α], "
", \*""<(Δ(Subscript[S, y]))^2> ="", 
      dy[t, α], "
", \*""<(Δ(Subscript[S, z]))^2> ="", dz[t, α]]\)

Simulation

Rotation of(Overscript[σ, →])  :

start = 
        ButtonBox[
          StyleBox["Run Animation", "Title", 
            FontFamily -> "Helvetica"], 
          ButtonFunction :> 
            FrontEndExecute[{SelectionMove[EvaluationNotebook[], Before, 
                    ButtonCell]
SelectionMove[
                    EvaluationNotebook[], Previous, 
                    CellGroup]
SelectionAnimate[
                    SelectedNotebook[], 
                    AnimationDisplayTime -> simtime]}], 
          ButtonEvaluator -> Automatic, 
          Active -> True];
α1 = α[x0[[2]]];
tr = 
        ParametricPlot3D[{x[t, α1], y[t, α1], 
            z[t, α1]}, {t, 0, 1}, 
          PlotRange -> {{-0.5, 0.5}, {-0.5, 0.5}, {-0.5, 
                0.5}}, ViewPoint -> {1.473,  -2.440,  1.823}, 
          Boxed  ->  False, 
          DisplayFunction -> Identity];
\(\(tr = 
        Show[tr, 
	
          Graphics3D[{Thickness[0.015], 
              Hue[ .6], {Line[{{0.5, 0, 0}, {0, 0, 0}}], 
                Line[{{0.5, 0, 0}, {0.45, \(-0.05\), 0}}], 
                Line[{{0.5, 0, 0}, {0.45, 0.05, 
                      0}}]}}, 
		\ \ \ PlotRange -> {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 
                  0.5}}, 
		\ \ ViewPoint -> {1.473, \ \(-2.440\), \ 
                1.823}
		\ \ ]
	];\)\)
\(\(planim = 
        Table[Show[tr, 
            Graphics3D[{Thickness[0.015], Hue[1], 
                Line[{{0, 0, 0}, {x[t, α1], y[t, α1], 
                      z[t, α1]}}]}, 
		\ \ \ PlotRange -> {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 
                    0.5}}, 
		\ \ ViewPoint -> {1.473, \ \(-2.440\), \ 1.823}
	], 
            DisplayFunction -> $DisplayFunction], {t, 0, 0.5, 
            0.05}];\)\)
SelectionMove[EvaluationNotebook[], After, 
        EvaluationCell];
SelectionMove[EvaluationNotebook[], Next, CellGroup];
FrontEndTokenExecute["OpenCloseGroup"];
start /. {simtime -> 0.1} // DisplayForm

The red line represents the expectation value, the blue arrow gives the  direction of the magnetic field. 
We see that<(Subscript[S, z])> and <(Subscript[S, y])>are time-dependent and that the spin rotates around the direction of the  magnetic field.

If you want to save this animation as an animated GIF file, you have to 
uncomment the following cell. Just delete the (* and the *), saving may take 
fairly long.

Variance of (Subscript[σ, α]):

\(tr2 = 
      Plot[{\@dx[t, α1], \@dy[t, α1], \@dz[t, α1]}, {t, 
          0, 1}, PlotRange -> {{0, 1}, {\(- .1\), 0.5}}, 
        PlotStyle -> {{Thickness[0.015], Hue[0.6]}, {Thickness[0.01], 
              Hue[1]}, {Thickness[0.01], Hue[0.3]}}, 
        AxesLabel -> \ {"t", \*""(Subscript[v, α])=(Sqrt[<(Δ(Subscript[S, α]))^2>])""}, DisplayFunction -> Identity]; 
    Show[{tr2, 
        Graphics[{Text[\*""(Subscript[v, x])"", {1.1, \@dx[1, α1]}, 
              TextStyle -> {FontColor -> Hue[0.6]}], 
            Text[\*""(Subscript[v, y])"", {1.1, \@dy[1, α1]}, 
              TextStyle -> {FontColor -> Hue[1]}], 
            Text[\*""(Subscript[v, z])"", {1.1, \@dz[1, α1]}, 
              TextStyle -> {FontColor -> Hue[0.3]}]}]}, 
      TextStyle -> {FontSize -> 10}, 
      PlotRange -> {{0, 1.1}, {\(- .1\), 0.5}}, 
      DisplayFunction -> $DisplayFunction];\)
\(\(Print[\*""<\!\(S\_x\)>(0) ="", 
        x[0, α1], \*""\t<\!\(S\_x\)>(\!\(1\/8\)) ="", 
        x[0.125, α1], "
", \*""<\!\(S\_y\)>(0) ="", 
        y[0, α1], \*""\t<\!\(S\_y\)>(\!\(1\/8\)) ="", 
        y[0.125, α1], "
", \*""<\!\(S\_z\)>(0) ="", 
        z[0, α1], \*""\t<\!\(S\_z\)>(\!\(1\/8\)) ="", 
        z[0.125, α1]];\)\)
\(\(Print[\*""<(Δ(Subscript[S, x]))^2>(0) ="", 
        dx[0, α1], \*""\t<(Δ(Subscript[S, x]))^2>((1/8)) ="", 
        dx[0.125, α1], "
", \*""<(Δ(Subscript[S, y]))^2>(0) ="", 
        dy[0, α1], \*""\t<(Δ(Subscript[S, y]))^2>((1/8)) ="", 
        dy[0.125, α1], "
", \*""<(Δ(Subscript[S, z]))^2>(0) ="", 
        dz[0, α1], \*""\t<(Δ(Subscript[S, z]))^2>((1/8)) ="", 
        dz[0.125, α1]];\)\)

We see that (Subscript[v, x]) = const., that (∑_α) (Subscript[v, α]) = 1at all times and that (Subscript[v, y])and (Subscript[v, z])have a period of π/4, i.e. they rotate twice as fast as (σ^⇀) .