### A Pluto.jl notebook ###
# v0.19.4

using Markdown
using InteractiveUtils

# ╔═╡ a1ebb7c2-c46a-11ec-3b27-e355bd83eec7
begin
	using Markdown
	using QuantumOptics
	using Plots
	using LsqFit
	using QuadGK
	using FFTW
	using PlutoUI
	using Statistics
	using PlutoUI
	plotly()
end
	

# ╔═╡ b5c5be4f-82e7-4d17-9519-d246a202063d
md"""
## Viļņu funkcijas un Heizenberga nenoteiktība
### Ievads
Šodien apskatīsimies viļņu funkcijas un Heizenberga nenoteiktībsa princīpu. Atklāsim fundamentālus jēdzienus kvantu mehānikā, tai skaitā
* Brīva daļiņas viļņu funkcija
* Grupas ātrums un fāzes ātrums
* Viļņu funkcijas representācija pozicījas vai kustības daudzuma bāze
* Furjē (_Fourier_) transformācijas
* Viļņu paketes
* Heizenberga nenoteiktība
* Dispersija

"""


# ╔═╡ eea47c51-5e7e-467b-a9bf-109f1d2d7fbc
md"""
### Mācību mērķi
1. Studenti iepazīsies brīvas daļiņas viļņu funkciju. 
2. Studenti sapratīs brīvas daļiņas viļņu funkcijas izplatīšanas telpā un laikā. 
3. Studenti spēs atšķirt fāzes ātrumu un grupas ātrumu un tos novērtēt. 
4. Studenti sapratīs, kā Furjē transformācija dod viļņu funkcijas ``k``-komponentes un kā tas ir saistītas ar kustības daudzumu.
5. Studenti sapratīs, kā veidojās viļņu paketes. 
6. Studenti saistīts viļņu paketes platumu ar platumu ``k``-telpā un Heizenberga nenoteiktības principu. 
6. Studenti sapratīs, kāpēc notiek viļņu pakešu dispersija.
7. Studenti spējs risināt uzdevumus ar Heizenberga nenoteiktības principu. 
8. Studenti saistīs Heizenberga nenoteiktības princpipu ar Debroljī viļņiem. 
"""

# ╔═╡ 328d888c-5b1c-464c-8aa4-1fb73157b3b2
md"""
### Šrēdingera vienādojuma risinājums brīvā telpā
Šrēdingera vienādojums postūla tādu funkciju, ko sauc par viļņu funkciju, kas apraksta kād fizikālu sistēmu, un saista viļņu funkcijas izmaiņa laikā ar Hamiltoniāna operatoru, kas apraksta sistēmas enerģiju:

`` i \hbar \frac{\partial}{\partial t} \Psi = \hat{H} \Psi  ``.

Mēs gribam aprakstīt kādas daļiņas kustību, tātad

``\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(x), `` 

kur ``V(x)`` ir potenciālā enerģija. Brīvās daļiņas gadījumā, ``V(x)=0``. Turklāt, ja strādājām tikai vienā dimensija tad ``\nabla^2 = \frac{\partial^2}{\partial x^2}``.

Viļņu funkciju varam rakstīt kā no laika atkārīgā un no laika neatkarīgas daļas reizinājumu ``\Psi(x,t) = \psi(x)\phi(t)``. No laika neatkārīgā Šrēdingera vienādojums ir

``\hat{H}\psi(x)=E\psi(x)``.

Brīvajai daļiņai šis vienādojums kļūst par

``\frac{\partial^2}{\partial x^2}\psi(x)+\frac{2mE}{\hbar^2}\psi(x)=0. ``

Mēs atpazīstam viļņu vienādojumu, kuru risinājumus mēs zinām:

``\psi(x)=A e^{ikx} + Be^{-ikx}``,
kur 
``k^2=\frac{2mE}{\hbar^2} ``,
``e^{ikx}=\cos(kx) + i\sin(kx)`` 
un ``A`` un ``B`` ir konstantes.

Viļņu funkcijas
``\psi(x)=e^{ikx}`` reālā daļa ar ``k=1.0`` izskatās šādi:
"""


# ╔═╡ a434d498-0fb8-4e4d-9f6e-84814580966a
begin
	Xmax=50*pi
	Xmin=-50*pi
	Npts=1000
	ΔX=(Xmax-Xmin)/Npts
	X=Xmin:ΔX:Xmax
	freePartY=real(exp.(im*1.0 .* X))
	plot(X[450:550],freePartY[450:550],label="real(ψ(x))",xlabel="Position",ylabel="real(ψ(x))")
end

# ╔═╡ 80aee5cd-a27d-4bfd-a73b-bc7d1abdace9
md"""
Redzam, ka brīvai daļiņai visas pozitīvas enerģijas vērtības ir atļautas. 
Atgriežoties pie no laika atkārīga uzdevuma redzam, ka 

``\phi(t)=e^{\pm i E/\hbar t} = e^{\pm i \omega t}``, 

kur ``E=\hbar \omega`` un ``\omega=2\pi \nu``. Līdz ar to, ``E=h\nu`` un ``ω=\hbar k^2/2m``.

Tātad pilnā viļņu funkcija ir

``\Psi(x,t)=\psi(x)\phi(t)=e^{i(kx-\omega t)}``

vai

``\Psi(x,t)=\psi(x)\phi(t)=e^{i(kx+\omega t)}``

Pirmais vilnis izplatās uz labo (pozitīvo) pusi, bet otrais - uz kreiso (negatīvo) pusi. 

Tas ar ``k=1`` izskatās šādi:
"""

# ╔═╡ 5409746f-27fc-405b-8d01-c95a924542e6
begin
	gr()
	myWaveAnimation=Animation()
	m=1
	ħ=1
	k=1.0
	ω=ħ*k^2/(2*m)
	smallX=X[450:550]
	for t in 0:0.1:10
		plt=plot(smallX,[real.(exp.(im*(k.*smallX .- ω*t))), real.(exp.(im*(k.*smallX .+ ω*t)))],label=["exp(i(kx-ωt))" "exp(i(kx+ωt))"],xlabel="Position",ylabel="Ψ(x)")
		frame(myWaveAnimation,plt)
	end
	gif(myWaveAnimation)

end


# ╔═╡ c42d5a5d-500c-4011-a08d-9f2c3bad1ff5
md"""
### Uzdevums 1. Fāzes ātrums un grūpas ātrums
a. Cik ātri izplatās viļņu funkcija  ``\Psi(x,t)=e^{i(kx-\omega t)}``? Tas ir fāzes ātrums. 

b. Rakstiet fāzes ātruma izteiksmi un pārbaudiet animācijā.  

c. Kāpēc saucās "fāzes ātrums"?

d. Kāds ir kustības daudzums ``p`` daļiņai, kura kustību apraksta viļņu funkcija ``\Psi(x,t)=e^{i(kx-\omega t)}``? 

e. Kādam ātrumam atbilst šis kustības daudzums? Šis ir grupas ātrums. Vai atšķiras no fāzes ātruma?

f. Kā frekvence ``\omega``, fāzes ātrums ``v_g``, kustības daudzums ``p`` un enerģija ``E`` ir saistītas savā starpā?
"""

# ╔═╡ 7156f3fe-3de5-43e6-9992-c99096336b7d
md"""
### Furjē transformācija
Apskatīsimies periodisku funkciju ar periodu ``2L``:

``f(x)=f(x+2L)``.
Furjē serija pa intervālu (-L,L) ir

``f(x)=\sum_{n=-\inf}^{\infty}a_n e^{in\pi x/L} \Delta n,

kur ``a_n`` apraksta viļņa ``exp(in\pi x/L)`` ar ``k=n\pi x/L`` stiprumu amplitūda, un ``\Delta n=1` ir starpība starp veseļiem skaitļiem. 

Tā kā Furjē modi ir ortonormāli,

``\frac{1}{2L}\int_{-L}^{L}dx(e^{in\pi x/L})^{\star}e^{im\pi x/L}=\frac{1}{2L}\int_{-L}^{L}dxe^{-in\pi x/L}e^{im\pi x/L}=\delta_{mn}``,

kur ``\delta_{mn}=1`` kad ``m=n`` un ``\delta_{mn}=0`` citādi.

Izmantojot faktu, kā

``\frac{1}{2L}\int_{-L}^{L}dxe^{-in\pi x/L}f(x)=\frac{1}{2L}\int_{-L}^{L}dxe^{-in\pi x/L}\sum_{m=-\infty}^{\infty}a_m e^{im\pi /L}``,

var iegūt Furjē amplitūdas

``a_n=\frac{1}{2L}\int_{-L}^Ldx f(x)e^{-in \pi x/L}``.

Tagad var definēt 

``\Delta k= \frac{\pi \Delta n}{L}``

un

`` A(k)=\frac{\sqrt{2\pi}La_n}{\pi}``.

Tātad var rakstīt funkciju kā

`` f(x)=\sum_k \frac{A(k)}{\sqrt{2\pi}} e^{ikx} \Delta k``.

Ņemot limitu ``L \rightarrow \inf``, ``\Delta k`` kļūs bezgalīgi mazs un summa kļūst par integrālu:

``f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} A(k)e^{ikx}dk``

un līdzigi,

``A(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-ikx}dx``.

Sīs transformācijas sauc par Furjē transformācijām, kur pirmā ir apgrieztā transformācija. 

"""

# ╔═╡ 0ce471fc-e45d-46c0-a58f-ec432ab54df5
md"""
#### Brīvas daļiņas viļņu funkcija k-telpā
Kas notiek, kad veiksim brīvas dalīņas viļņu funkcijas Furjē transformāciju?
Definēsim brīvas daļiņas viļņu funkciju ar viļņu vektoru ``k_0`` kā:

``f(x)=e^{ik_0(x-x_0)}``.

Šeit ``x_0`` ir kāds atsauces punkts kuram būs nozīmē tikai vēlāk kad runāsim par viļņu paketēm. 

``A(k)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} e^{ik_0 (x-x_0)} e^{-ikx}dk``.

Rezultāts ir 

``A(k)=\delta(k-k_0)``.

Lai to redzētu, mums ir jāveic [diskrētā Furjē transformācija](https://www.matecdev.com/posts/fft-tutorial.html). Definēsim mūsu funkciju uz ``N`` punktiem no ``X_{min}`` līdz ``X_{max}`` un starpību ``\Delta x=(X_{max}-X_{min})/N``. 

Tālāk, k-telpā definēsim
``\Delta K=\left( \frac{\pi}{ΔX}+\frac{pi}{ΔX} \right)/N``.

Rezultāts izskatās šādi:
"""

# ╔═╡ cfa6936c-22fd-4063-b585-c547ea02c711
begin
	#display(ui)
	plotly()
	ΔK=(pi/ΔX+pi/ΔX)/Npts
	K=-pi/ΔX:ΔK:pi/ΔX
	k0=1.0
	ψ0=exp.(im*k0 .* X)	
	A0=fftshift(fft(ψ0))/Npts  # Jādalās ar N normēšanai
	plotX=plot(X,real.(ψ0),label="ψ(x)",title="Posicīja")
	plotProb=plot(X,abs.(ψ0),ylim=(0,1.1),label="|ψ(x)|²",title="Varbūtība",color=:blue)
	plotK=plot(K,real.(A0),label="A(k)",title="K-telpā",color=:red)
	plot(plotX,plotProb,plotK,layout=(3,1))
end

# ╔═╡ ca9762ae-ed4a-4b13-9a60-2789e96bb452
md"""
Mēs redzam ``\delta``-funkcijas pie ``k=+1``, kas atbilst viļņu funkcijai ``\psi(x)=e^{ikx}``. Viļņu skaitlis ar ``k=+1`` atbilst viļņim, kas kustās uz labo pusi. 

"""

# ╔═╡ 3a8d16de-d777-44ca-9503-7628fb3f9a63
md"""
#### Viļņu funkcija ar divām kustības daudzuma komponentēm
Kas notiek, kad mums ir divi viļņi? Pieņemsim, ka mums ir ``\psi(x)=\psi_1(x)+\psi_2(x)``, kur ``\psi_1(x)=e^{ik_1x}`` un ``\psi_2=e^{ik_2x}``. Apskatīsim nākamo attēlu:
"""

# ╔═╡ a5e783cb-4a1b-454b-9601-e77c3255bb7b
begin
	plotly()
	k1a=1.0
	k1b=1.1
	Y1a=exp.(im*k1a.*X)
	Y1b=exp.(im*k1b.*X)
	Y1sum=Y1a+Y1b
	P1=real(fftshift(fft(Y1sum)))/Npts 
	label1="ψ₁=exp(i($k1a) x)"
	label2="ψ₂ exp(i($k1b) x)"
	p1a=plot(X,[real.(Y1a),real(Y1b)],label=[label1 label2],title="Position x")
	p1b=plot(X,real.(Y1sum),label="ψ₁+ψ₂",title="Position x",color=:green)
	p1c=plot(X,abs.(Y1sum),label="|ψ₁+ψ₂|²",title="Position x",color=:cyan)
	p1d=plot(K,P1,label="k",title="Wave vector k",color=:blue)
	plot(p1a,p1b,p1c,p1d,layout=(4,1))
end


# ╔═╡ 4b46bdd1-ecaa-4b7f-9bb6-ba951050b5df
md"""
Šajā gadījumā, mēs redzam divas ``k``-komponentes, vienu pie ``k_1=1.0``, bet otro, pie ``k_2=1.1``. Ļoti interesanti ir rezultējošā viļņu funkcijas summa ``x``-telpā. 

[Šeit](https://phet.colorado.edu/en/simulations/fourier-making-waves/about) var atrast noderīgu simulāciju par Furjē transforācijām. 
"""

# ╔═╡ b694faef-1b52-476d-98e1-b0de28c714aa
md"""
#### Uzdevums 2. Viļņu sitienu frekvence. 

a. Kad divi viļņi at ašķirīgām frekvencēm sumējās, rezultāts sauc pa sitieniem. Kāpēc?

b. Kā sitiena frekvence ir saistīta ar divu komponenšu frekvenci? (Varat spēlēt ar kodu, kas veido grafiku.)

c. Ar kādu trigonometrisko identitāti tas ir saistīts?
"""

# ╔═╡ 9919160a-3bde-458d-a82a-c4e9e8f12a44
md"""
### Viļņu paketes
Tagad apskatīsim viļņu paketes. 
Pieņemsim funkciju

``\psi_x(x)=\frac{1}{\sqrt{\sqrt{2\pi}*σ}}*e^{-(x- x0)^2 /(4*σ^2)}``.

Tā mums dos varbūtības sadalījuma funkciju 

``P(x)=\psi^{\star}(x)\psi(x)=\frac{1}{\sqrt{2\pi}*σ}*e^{-(x- x0)^2 /(2*σ^2)}``.

Mēs varam runāt par lokalizēto funkciju un raksturot iespējamību atrast daļiņu ar parametru ``\Delta x``, kur

``(\Delta x)^2=\left< (x-<x>)^2\right>``, 

kur iekavas ``\left< \right>`` nozīmē iekļautā lieluma vidējo vērtību. 

To var vienkāršot šādi:

``\Delta x = \sqrt{\left<x^2 \right> - <x>^2}``, kur

``\left< x \right> = \frac{\int_{-infty}^{infty}\psi^{\star}(x)x\psi(x)dx}{\int_{-infty}^{infty}\psi^{\star}(x)\psi(x)dx}``,

kur ``\psi^{\star}`` ir ``\psi`` kompleksais saistītais.

Līdzīgi 

``\left< x^2 \right> = \frac{\int_{-inf}^{inf}\psi^{\star}(x)x^2\psi(x)dx}{\int_{-inf}^{inf}\psi^{\star}(x)\psi(x)dx}``.

Integrālus mēs aproximēsim ar summam. Tātad ``X`` būs ``N \times 1`` vektors ar ``x-``vertībām un ``Y`` būs ``N \times 1`` vektors ar ``\psi(x)``-vērtībām, savukārt ``Y'`` būs ``1 \times N`` (horizontālais vektors) kurā vērtības ir ``Y`` vektora kompleksi saistīti. Tātad vektoriālais reizinājums ``Y'*Y=\sum_{i=1}^{N}\psi^{\star}(x_i)\psi(x_i)`` aproximē integrālu ``\int_{-\infty}^{\infty}psi^{\star}(x)\psi(x)``. 

Šis lielums nebūs atkarīgs no funkcijas formās. Tāpēc līdzīgo lielumu varēsim aprēķināt arī priekš ``\Delta k``. 

Kas mums tad sanāks?

"""

# ╔═╡ ac9c4e60-d7fd-408f-bd43-18ead5a9cff7
begin
	σ=5.0
	x0=0
	Y2=(1/sqrt((sqrt(2*pi)*σ)))*exp.(-(X .- x0).^2 ./(4*σ^2))  # viļņu funkcija
	p2a=plot(X,Y2,label="ψ(x)", xlabel="Position x", ylabel="ψ(x)")
	P2=(fftshift(fft(Y2)))/Npts
	p2b=plot(K,abs.(P2), label="A(k)", xlabel="Wavenumber k", ylabel="A(k)")
	plot(p2a,p2b,layout=(2,1))
end

# ╔═╡ 41fbf8f0-0d7f-45a0-b13b-ba075bf4b1ba
begin
	with_terminal() do
		println("Δx = ", sqrt(Y2'*((X.^2).*Y2)/(Y2'*Y2)) )
#20/sqrt(2)
		println("Δk = ", sqrt(P2'*((K.^2).*P2)/(P2'*P2)) )
		println("ΔxΔk = ", sqrt(Y2'*((X.^2).*Y2)/(Y2'*Y2))*sqrt(P2'*((K.^2).*P2)/(P2'*P2)) )

	end
end


# ╔═╡ 7c203b16-bef9-4996-97ab-a0f470658054
md"""
#### Uzdevums 3.
a. Atrodiet ``\Delta x``, ``\Delta k`` un ``\Delta x \Delta k`` vairākām viļņu paketes platuma vērtībām (mainiet mainīgā ``\sigma`` vērtību iepriekšējā kodā). 

b. Kāda ir Jūsu rezultāta interpretācija?
"""

# ╔═╡ 06be7f7e-2769-4276-bb92-765213e16e35
md"""
### Dispersija

Apskatīsimies viļņu paketi, ko apraksta ar funkciju

``\psi(x)=\frac{1}{\sqrt(2\pi)σ}e^{-(x - x_0)^2 /(2σ^2)}``.

(Kodā implementācija tas izksatās šādi:

```Y2=(1/(sqrt(2*pi)*σ))*exp.(-(X .- x0).^2 ./(2*σ^2))```.

Konstrukcija ```exp.(X)``` nozīmē, ka funkcija ```exp``` izpildās uz katra vektora ```X``` elementa, un rezultāts būs vektorā ar to pašu dimensiju kā ```X```: ```[exp(X[1]), exp(X[2]), ... , exp(X[N])]```. Līdzīgi, konkstrukcija ```X.^2``` nozīmē, ka katrs vektora ```X``` elements tiks reizināts ar sevi, un rezultāts būs vektorā ar to pašu dimensiju. Līdzīgi konstrukcija ```X .- x0``` nozīmē, ka skalārs ```x0``` tiks atņemts no katra vektora ```X``` elementa, utt.) 

Tomēr, attēlu mēs neiegūsim no šī eksponentiālās funkcijas, bet rekonstruēsim no tās Furjē komponentēm:

``\psi(x)=\sum_{l=1}^{l=Npts} A_l e^{i k_l x}``.

Mēs tāda veidā iegūsim to pašu funkciju ``\psi(x)``.

"""

# ╔═╡ bb89c480-a252-49f1-ab3f-b5994176ac7b
begin
	plotly()
	A2=fftshift(fft(Y2))              # Furjē transformācija 
	ψWPcomps=[]
	ψWPsum=zeros(Complex,length(A2))
	for i in 1:length(A2)
		if abs(A2[i]) > 0.001
			ψWPcomp=fftshift(A2[i]*(exp.(-im*(K[i]*X ))))
			ψWPsum +=ψWPcomp 
			push!(ψWPcomps,ψWPcomp)  # Veido vektoru, kura komponentes ir vektori ar iļņu funkcijas komponentēm. 
		end
	end

	plot(X,abs.(ψWPsum),label="ψ(x)")
	#plot(X,real.(ψWPcomps),label=false)
end

# ╔═╡ 730fe489-a10a-4cd1-97e3-f56b4384ebdd
md"""
#### Viļņu paketes evolūcija laikā

Tagad apskatīsimies, kā viļņu pakete evolucionē laikā. 

Reizinām ``\psi(x)`` ar ``\phi(t)=e(-i\omega t)``, bet atkaļ, pa Furjē komponentēm, jo katrai komponentei būs sava frekvence ``\omega``:

``\Psi(x,t)=\sum_{l=1}^{l=Npts} A_l e^{i k_l x} e^{-i \omega_i t}=\sum_{l=1}^{l=Npts} A_l e^{i( k_l x- \omega_l t)}``.
"""

# ╔═╡ fb434a2b-a455-4936-bac3-5f64e3bcc209
begin
	gr()   # izmantot GR grafia "backend" animācijai. 
	myDispersionAnimation=Animation()    #"Īnicializē animācijas objektu"
	ΨWPcomps = []     # tukšs vektors
	Kmoving = K .+ 1.0   # pievieno k_0, t.i., kustības virzienu un ātrumu

	for t in 0:1:100.0
		WPsum = zeros(Complex,length(A2))
		for i in 1:length(A2)
			if abs(A2[i]) > 0.001
				ωWP=ħ*Kmoving[i]^2/(2*m)
				ΨWPcomp=fftshift(A2[i]*(exp.(-im*(Kmoving[i]*X ))))*exp(im*ωWP*t)
				WPsum = WPsum + ΨWPcomp 
				#push!(ΨWPcomps,ΨWPcomp)
							
			end
		end
		plt=plot(X,abs.(WPsum),ylim=(-5,90))
		frame(myDispersionAnimation,plt)
	end
	gif(myDispersionAnimation)
end

# ╔═╡ 1010ad5b-1411-4448-adee-5a9776853149
md"""
#### Uzdevums 4. Dispersija.
Paspēlējiet ar parametriem animācija: ```Kmoving```, ```σ``` (vairāki paneļi augstāk)

a. Paskaidrojiet, kāpēc viļņu pakete maina savu formu ar laiku. 

b. Kā viļņu paketes izplatīšanas ātrums ietekmē dispersiju?

c. Kā viļņu paketes sākotnējais platums ietekmē dispersiju?

"""

# ╔═╡ a0eb5dac-f1a8-4786-a98d-eafbeec0572c
md"""
#### Uzdevums 5. Heizenberga nenoteiktības sakarības bedrē.

Pieņemiet, ka daliņa ar masu ``m`` atrodas bedrē ar bezgalīgi augstām sieniņām, kura platums ir L. 

a. Izvertējiet daliņas ``\Delta x`` un pamatojiet savu pieeju. 

b. Izsakiet daliņas enerģiju izmantojot ``\Delta k``.

c. Izmantojiet Heizenberga nenoteiktības sakarības, lai novērtētu minimālu daliņas enerģiju. (A)

"""

# ╔═╡ bd3ae1b9-34ff-4f3b-a19f-d68939c17cb4
md"""
#### Uzdevums 6. Difrakcija.

a. Paralēls ūdeņraža atomu kūlis ar ātrumu ``v`` = 1,2 km/s krīt perpendikulāri šauri spraugai, veidojot interferences ainu uz ``l``  = 1m attālumā novietota ekrāna. Izejot no nenoteiktības sakarībām, novērtēt spraugas platumu, pie kuras difrakcijas centrālais maksimums būs vismazākais.
Attēla platums ``\Delta = b + 2H``, kur ``H`` ir papildus platums, kuru izsauc atomu impulsa nenoteiktība, izejot cauri spraugai.

b. Kāds ir katra ūdenraza atoma Debroljī viļņa garums?

c. Salīdziniet rezultātu no (a) ar rezultātu, ko iegustu, rēķinot Debroljī viļnas difrakciju pa spraugu. 

d. Paskaidrojiet kādas atšķirības, ja ir. 


"""

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uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[deps.TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "5ce79ce186cc678bbb5c5681ca3379d1ddae11a1"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.7.0"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.ThreadingUtilities]]
deps = ["ManualMemory"]
git-tree-sha1 = "f8629df51cab659d70d2e5618a430b4d3f37f2c3"
uuid = "8290d209-cae3-49c0-8002-c8c24d57dab5"
version = "0.5.0"

[[deps.TreeViews]]
deps = ["Test"]
git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
version = "0.3.0"

[[deps.TriangularSolve]]
deps = ["CloseOpenIntervals", "IfElse", "LayoutPointers", "LinearAlgebra", "LoopVectorization", "Polyester", "Static", "VectorizationBase"]
git-tree-sha1 = "b8d08f55b02625770c09615d96927b3a8396925e"
uuid = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf"
version = "0.1.11"

[[deps.URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.VectorizationBase]]
deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "Hwloc", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static"]
git-tree-sha1 = "9d1b533f597d87ce9b4abd36a2ce4664f08e08ed"
uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f"
version = "0.21.29"

[[deps.VertexSafeGraphs]]
deps = ["Graphs"]
git-tree-sha1 = "8351f8d73d7e880bfc042a8b6922684ebeafb35c"
uuid = "19fa3120-7c27-5ec5-8db8-b0b0aa330d6f"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.WignerSymbols]]
deps = ["HalfIntegers", "LRUCache", "Primes", "RationalRoots"]
git-tree-sha1 = "960e5f708871c1d9a28a7f1dbcaf4e0ee34ee960"
uuid = "9f57e263-0b3d-5e2e-b1be-24f2bb48858b"
version = "2.0.0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╠═a1ebb7c2-c46a-11ec-3b27-e355bd83eec7
# ╟─b5c5be4f-82e7-4d17-9519-d246a202063d
# ╟─eea47c51-5e7e-467b-a9bf-109f1d2d7fbc
# ╟─328d888c-5b1c-464c-8aa4-1fb73157b3b2
# ╠═a434d498-0fb8-4e4d-9f6e-84814580966a
# ╟─80aee5cd-a27d-4bfd-a73b-bc7d1abdace9
# ╠═5409746f-27fc-405b-8d01-c95a924542e6
# ╟─c42d5a5d-500c-4011-a08d-9f2c3bad1ff5
# ╟─7156f3fe-3de5-43e6-9992-c99096336b7d
# ╟─0ce471fc-e45d-46c0-a58f-ec432ab54df5
# ╠═cfa6936c-22fd-4063-b585-c547ea02c711
# ╟─ca9762ae-ed4a-4b13-9a60-2789e96bb452
# ╟─3a8d16de-d777-44ca-9503-7628fb3f9a63
# ╠═a5e783cb-4a1b-454b-9601-e77c3255bb7b
# ╟─4b46bdd1-ecaa-4b7f-9bb6-ba951050b5df
# ╟─b694faef-1b52-476d-98e1-b0de28c714aa
# ╠═9919160a-3bde-458d-a82a-c4e9e8f12a44
# ╠═ac9c4e60-d7fd-408f-bd43-18ead5a9cff7
# ╠═41fbf8f0-0d7f-45a0-b13b-ba075bf4b1ba
# ╟─7c203b16-bef9-4996-97ab-a0f470658054
# ╟─06be7f7e-2769-4276-bb92-765213e16e35
# ╠═bb89c480-a252-49f1-ab3f-b5994176ac7b
# ╟─730fe489-a10a-4cd1-97e3-f56b4384ebdd
# ╠═fb434a2b-a455-4936-bac3-5f64e3bcc209
# ╟─1010ad5b-1411-4448-adee-5a9776853149
# ╟─a0eb5dac-f1a8-4786-a98d-eafbeec0572c
# ╟─bd3ae1b9-34ff-4f3b-a19f-d68939c17cb4
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
