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Plot the eigenvalues of a spin-1 system in a magnetic field?

Posted 1 month ago
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Hi,
so my general goal is to calculate the eigenstates of a spin-1 system in a magnetic field which is not parallel to the spin axis. Therefore the system has B and α as variables.
This is what I got so far:

Manipulate[
 Plot[{\[Lambda]1, \[Lambda]2, \[Lambda]3} = 
   Eigenvalues[
    h*D1*(Sz)^2 + g*\[Mu]B*B*Cos [\[Alpha]]*Sz/h + 
     g*\[Mu]B*(Sqrt[B^2 - (B*Cos [\[Alpha]])^2]*Sx + By*Sy)/h + 
     h*E1*(Sx.Sx - Sy.Sy)], {B, 0, 0.040}, 
  GridLines -> {{{2.87*10^9, Red}}}, 
  PlotRange -> {{0, 0.04}, {-0.5*10^9, 4*10^9}}, 
  PlotRangeClipping -> True, Frame -> True], {\[Alpha], 0, Pi/2}]

with the following constants used:

D1 = 2.87*10^9 1/h;
E1 = 0.005*10^9 1/h; 
g = 2.00231930436182; 
\[Mu]B = 9.274009994*10^(\[Minus]24);
h = 6.626070040*10^(\[Minus]34);
By = 0

and S are the spin-1 Pauli matrizes

Sx = 1/Sqrt[2] * ( {
    {0, 1, 0},
    {1, 0, 1},
    {0, 1, 0}
   } ); Sy = 1/Sqrt[2] * ( {
    {0, -I, 0},
    {I, 0, -I},
    {0, I, 0}
   } ); Sz = ( {
   {1, 0, 0},
   {0, 0, 0},
   {0, 0, -1}
  } )

Is their a way to do a similar plot but with λ1-λ3 and λ2-λ3?
Thanks for your help,
Stefan

3 Replies

define a function which gives the eigenvalues as a function of alpha and use that to plot the differences

Posted 1 month ago

Do you mean like this:

f[B_] := {\[Lambda]1, \[Lambda]2, \[Lambda]3} = 
  Eigenvalues[
   h*D1*(Sz)^2 + g*\[Mu]B*B*Cos [\[Alpha]]*Sz/h + 
    g*\[Mu]B*(Sqrt[B^2 - (B*Cos [\[Alpha]])^2]*Sx + By*Sy)/h + 
    h*E1*(Sx.Sx - Sy.Sy)]

But how do I address the eigenvalues individually now to get the differences?

not a good idea to use capital letters for variables. f[b_]:= Eigenvalues[...]

Plot[ f[b][[1]] - f[b][[2]] ,..]

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