⚡ Pluto.jl ⚡

Problem 2. Let $S_{1}$ , $S_{2}$ , and $S_{3}$ be the spin matrices for spin 1/2 particles (i.e. fermion Pauli spin matrices) given by:

$S_{1} = \frac{1}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) S_{2} = \frac{1}{2} (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) S_{2} = \frac{1}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) .$

If we consider the following Hamiltonians expressed in terms of these matrices:

$H = \frac{1}{ℏ ω} \hat{H} = S_{1} \otimes S_{1} + S_{2} \otimes S_{2} + S_{3} \otimes S_{3}$

$K = \frac{1}{ℏ ω} \hat{K} = S_{1} \otimes S_{2} + S_{2} \otimes S_{3} + S_{3} \otimes S_{1} .$

What are the eigenvalues and eigenvectors.

14.0 μs

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using SymPy

11.1 s

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begin
    𝑖,(ħ,ω) = sympy.I,sympy.symbols("ħ,ω")
    half = sympy.Rational(1,2)
    ⊗(x,y) = kron(x,y) #This is the same as tensor product.
    ⋅ = *
end;

174 ms

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begin
    σ₁ = half⋅[0 1;
               1 0]
    σ₂ = half⋅[0 -𝑖;
               𝑖 0]
    σ₃ = half⋅[1 0;
               0 -1]
end;

588 ms

$[\begin{array}{rrrr} \frac{1}{4} & 0 & 0 & 0 \\ 0 & - \frac{1}{4} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & - \frac{1}{4} & 0 \\ 0 & 0 & 0 & \frac{1}{4} \end{array}]$

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 Ĥ = (σ₁ ⊗ σ₁) + (σ₂ ⊗ σ₂) + (σ₃ ⊗ σ₃)

74.8 ms

K̂

$[\begin{array}{rrrr} 0 & \frac{1}{4} & - \frac{i}{4} & - \frac{i}{4} \\ \frac{1}{4} & 0 & \frac{i}{4} & \frac{i}{4} \\ \frac{i}{4} & - \frac{i}{4} & 0 & - \frac{1}{4} \\ \frac{i}{4} & - \frac{i}{4} & - \frac{1}{4} & 0 \end{array}]$

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K̂ = (σ₁ ⊗ σ₂) + (σ₂ ⊗ σ₃) + (σ₃ ⊗ σ₁)

3.0 ms

The eigenvectors and eigenvalues for these operators are then:

5.0 μs

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begin
    ℍ,𝕂 = Ĥ.eigenvects(chop=true),K̂.eigenvects(chop=true)
    #unpack :e - eigenvalue, :m - multiplicty (degeneracy)
    eₕ = [Dict(:e=>e[1],:m=>e[2]) for e in ℍ]
    eₖ = [Dict(:e=>e[1],:m=>e[2]) for e in 𝕂]
    vₕ = [v[end][1:end] for v in ℍ]
    vₖ = [v[end][1:end] for v in 𝕂]
end;

2.5 s

16.4 ms

indices 1 to 2

11.0 μs

$\hat{H} : [λ_{1} = - 3 / 4, λ_{2} = 1 / 4]$

$\hat{K} : [λ_{1} = - 3 / 4, λ_{2} = 1 / 4]$

353 ms

As you can see the eigensepectrum[1] is the same for both operators. In addition there is a 3-fold degeneracy for the second eigenstate. Lets take a look at the eigenvectors $\hat{H}$ :

5.3 μs

51.8 μs

eigenvector index 2

10.0 μs

Array{SymPy.Sym,2}1

$[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}]$

$[\begin{array}{r} 0 \\ 1 \\ 1 \\ 0 \end{array}]$

$[\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \end{array}]$

615 ns

and $\hat{K}$ :

34.3 μs

Array{SymPy.Sym,2}1

$[\begin{array}{r} 1 \\ 1 \\ 0 \\ 0 \end{array}]$

$[\begin{array}{r} - i \\ 0 \\ 1 \\ 0 \end{array}]$

$[\begin{array}{r} - i \\ 0 \\ 0 \\ 1 \end{array}]$

335 ns

Note that the $\frac{1}{ℏ ω}$ isn't included here.

9.5 μs