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stern-gerlach.jl
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stern-gerlach.jl
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### A Pluto.jl notebook ###
# v0.19.17
using Markdown
using InteractiveUtils
# ╔═╡ 7428b4e8-6b45-4e3d-b7fa-cb4df9f4e159
using CairoMakie
# ╔═╡ ba4d71e1-ac3b-4387-b55c-6d26452c098a
using Statistics
# ╔═╡ 693a8baa-7fd0-11ed-0f8b-29728c9a8460
using StaticArrays
# ╔═╡ 493830a2-a7ec-4c6f-9dcd-a01e40d88377
using LinearAlgebra
# ╔═╡ 2e342331-b75e-4276-975b-1b5a51506bc5
using Random: AbstractRNG, default_rng
# ╔═╡ f5f2f83c-ee8a-4440-8b29-404167e41619
md"Links
- [Julia manual section on types](https://docs.julialang.org/en/v1/manual/types/#man-types)
- [Julia manual section on methods](https://docs.julialang.org/en/v1/manual/methods/#Methods)
- [Docs for Linear Algebra package](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#man-linalg)
- [Julia manual section on complex numbers](https://docs.julialang.org/en/v1/manual/complex-and-rational-numbers/#Complex-Numbers)
- [StaticArrays.jl readme page](https://github.com/JuliaArrays/StaticArrays.jl)
- [Plotting example catalog](https://beautiful.makie.org/dev/)
"
# ╔═╡ c104ea4a-45e5-42ce-bea3-e97192b9873d
const σz = SMatrix{2, 2, Complex{Float64}}([1.0 0.0; 0.0 -1.0])
# ╔═╡ 35c30329-4dc4-47f3-913e-f1175545ddfd
eigen(σz)
# ╔═╡ e8a044f4-1787-4325-890d-157a0be6a9b8
md"## Measurement"
# ╔═╡ cb412b72-525f-4ad3-a16f-4b2da0fafd39
function zmeasure(rng::AbstractRNG, state::V) where V <: AbstractVector{<:Real}
pp, pm = abs2.(state)
if rand(rng) <= pp / (pp + pm) # measured +1
# The V(...) and eltype bits make sure the output types match the input types
return V(SVector{2, ComplexF64}(1, 0)), eltype(V)(1)
else # measured -1
return V(SVector{2, ComplexF64}(0, 1)), eltype(V)(-1)
end
end
# ╔═╡ def29bec-c12a-4d60-82d8-4aef3e18e25c
zmeasure(state::V) where V <: AbstractVector{<:Real} = zmeasure(default_rng(), state)
# ╔═╡ 4f2c11df-ccd2-4720-9822-59fca4ec078d
md"## Time Evolution"
# ╔═╡ 84f69043-b1b6-4eb9-97c1-96c4655c93ba
μ = 1
# ╔═╡ c1409361-87ad-4bba-bf6b-242861e50c46
H = -μ * σz
# ╔═╡ fc195f6f-dcdc-4a4c-a288-1a55b6c9505a
U(t) = exp(-im*H*t)
# ╔═╡ 8b30811b-4c1b-4ee1-adf2-6a399aec31c9
U(10.0) * [1, 0]
# ╔═╡ 8a155d09-820c-4eed-911f-65455004a264
function meanzspin(rng::AbstractRNG, t::Real, ψ0::AbstractVector, n::Integer)
ψ = U(t) * ψ0
return mean(1:n) do _
pp, pm = abs2.(ψ)
rand(rng) < pp / (pp + pm) ? 1 : -1
end
end
# ╔═╡ 76b361dc-3a01-4065-a506-2b46ed200d8c
meanzspin(t::Real, ψ0::AbstractVector, n::Integer) = meanzspin(default_rng(), t, ψ0, n)
# ╔═╡ 214e63a5-0131-44ec-b5d1-2ad35f95cf0e
let fig = Figure(), ψ0 = [0.5, 0.5], n = 1000
ax = Axis(fig[1, 1])
ts = 0:0.1:10
zs = [meanzspin(t, ψ0, n) for t in ts]
lines!(ax, ts, zs)
fig
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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[compat]
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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