⚡ Pluto.jl ⚡

Problem 3. Find the eigenvalues and normalized eigenvectors of the Hamiltonian operator:

$\hat{K} = \frac{1}{ℏ ω} \hat{H} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix})$

19.2 μs

Solution 3.

10.9 μs

xxxxxxxxxx
 
begin
    using SymPy
    using Plots
    pyplot()
end;

779 μs

Definitions:

5.3 μs

xxxxxxxxxx
 
begin
    π,𝑖 = sympy.symbols("π 𝑖")
    sqrthalf = 1/sympy.sqrt(2)
    ⋅(x::SymMatrix,y::SymMatrix) = sympy.Matrix.dot(x,y).simplify()
end;

360 ms

This operator is the Hadamard gate.

4.7 μs

xxxxxxxxxx
 
K̂ = sqrthalf * sympy.Matrix([1 1;
                             1 -1]);

1.4 s

Now calculating the eigenvalues and eigenvectors using SymPy.

6.7 μs

xxxxxxxxxx
 
begin
    eᵢ = K̂.eigenvals(multiple=true,rational=true)
    vᵢ = [sympy.simplify(v[end][1]) for v in K̂.eigenvects(chop=true)]
end;

2.6 s

The eigenvectors, which are not normalized, and eigenvalues are:

$v_{1} = [\begin{matrix} 1 - \sqrt{2} \\ 1 \end{matrix}], e_{1} = - 1$

$v_{2} = [\begin{matrix} 1 + \sqrt{2} \\ 1 \end{matrix}], e_{2} = 1$

63.2 ms

The characteristics that make these eigenvectors is that the inner product is a scalar value equal to 0, as seen below,

6.3 μs

$0$

xxxxxxxxxx
 
begin
    v₁, v₂ = vᵢ[1], vᵢ[2];
    v₁'⋅ v₂
end

1.0 ms