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

using Markdown
using InteractiveUtils

# ╔═╡ 83d05132-b257-4c45-ad70-a24c92c8f8ee
begin
	using QuantumOptics
	using LinearAlgebra
	using SparseArrays
	using PlutoUI
	using Plots
	plotly()
end

# ╔═╡ cd1bc8b0-d5c0-11ec-2a3b-831b0b8d8b3a
md"""
## Šrēdingera vienādojuma risinājumi

Šodien mēs apskatīsimies Šrēdingera vienādojuma risinājumus daļiņai dažādos potenciālos. Strādāsim vienā dimensijā. 

Parasti rīkosimies šādi: vispirms risināsim no laika neatkārīgo uzdevumu

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

kur 

``\hat{H} = \frac{\hat{p}^2}{2m} + V(x)``.

``V(x)`` ir potenciālā enerģija, bet ``\hat{p}`` ir kustības daudzuma operators ``\hat{p}=-i\hbar \frac{\partial}{\partial x}``.

Pēc tam pievienosim atkarību no laika:

``\Psi(x,t)=\sum_{n} \psi_n(x)e^{i E_n t/\hbar} ``,

kur ``psi_n(x)`` ir īpašfunkcijas un ``E_n`` ir īpašenerģijas, vai

``\hat{H}\psi_n(x) = E_n \psi_n(x)``.

"""

# ╔═╡ be74a383-df10-463c-92a2-a093b8a2d784
md"""
### Daļiņa bedrē
Lecijas laika tika risināts Šrēdingera vienādojums daļiņai bedrē ar bezgalīgi augšām sieniņām. To var aprakstīt ar potenciālu ``V(x)=0``, kad ``0<=x<=L``, bet ``V(x)=\infty`` citur. 

Savukārt ``\psi_n(x)=\sqrt(\frac{2}{L}) sin(n \pi x/L)`` bedrē, bet ``\psi(x)=0`` ārpus bedres.

Mēs izmantosim paketi [QuantumOptics.jl](https://qojulia.org/). Potenciālās funkcijas ar bezgalībām mēs nevaram izmantot, bet varam pētīt, kā risināt bedri ar galīgi augstām sieniņām. Tāds uzdevums nav tik risināms analitiski, tomēr varēsim risināt skaitliski bez lielām problēmām. 
"""

# ╔═╡ 5afc4a98-b3cb-49c0-a5a3-acbcd6509ed2
begin
	xmin=-5    # minimalā x-ass vērtība (Nevar būt bezgalīgi garša ass)
	xmax=5     # maximālā x-ass vērtība
 	xstep=0.1  # soļu lielums starp x-vērtībām. 
	xrange=xmin:xstep:xmax # iterators ar x-vērtībām.  
	
	basis = PositionBasis(xmin, xmax, length(xrange)) # bāze
	
	x = position(basis)   # posicijas operators
	p = momentum(basis)   # kustības daudzuma operators
	
	V0 = 20. # Bedres dziļums
	d = 2.0    # Bedres platums

	# Tagad rakstam funkiju, kas definē poteniālu
	function V_box(x)
	    if x < -d/2 || x > d/2
	        return 0.
	    else
	        return -V0
	    end
	end
	
	V = potentialoperator(basis, V_box)   # no funkcijas veidojām operatoru
	
	H = p^2/2 + V   # Hamiltona operators

	Nsols=5   # risinājumu skaits, ko meklējām no Šrēdingera vienādojuma. 
	# Ar vienu rindu risinām Šrēdingera vienāojumu:
	energies, states = eigenstates((H+dagger(H))/2, Nsols)    
end

# ╔═╡ 9a8b8479-f5a2-46a5-94b7-e66d2d2a3a02
typeof(p)

# ╔═╡ a64c019b-4c80-4f48-8cfa-4646c26c27e6
md"""
Risinājumā ir gan enerģijas ```energies``` un īpašfunkcijas ```states```.
Energijas vērtības ir vektorā. Īpašfunkcijas ir ```Ket``` objektā. Piemēram, enerģija ```energies[1]``` ir saistīts ar īpašfunkciju ```states[1]```.
"""

# ╔═╡ c4b5d3df-2ea1-48c8-9598-48fc1caaba16
energies

# ╔═╡ cef656c7-2cda-41f8-bca1-c17138f35e73
states[1]   # states[1] ir īpašfunkcija, kas atbilst energies[1].

# ╔═╡ 277d6d2c-5ce2-479a-b9b1-006afbd7529b
md"""
Tagad zīmēsim viļņu funkcijas. Zīmējām tikai viļņa funkcijas reālo daļu.(Atcerēsimies, ka varētu būt arī imaginārā daļa - to varētu zīmēt aizvietojot ```real``` ar ```imag```) koda bloka 6. rindā. 
"""

# ╔═╡ 8dde5436-b5cc-4ac3-b217-c374d888a345
begin
	xpoints = samplepoints(basis)
	plot(xpoints, V_box.(xpoints),label="V(x)")
	for i=1:length(states)
		label_string="ψ"*string(i)
    	plot!(xpoints, real.(states[i].data)*5 .+ energies[i],label=label_string) # viļņu funkcija ψ reizināta ar 5, lai būtu vieglāk redzama.
	end
	plot!(title="Finite Potential Well",xlabel="Position", ylabel="Energy")
end

# ╔═╡ a4bc121f-9002-46d5-a688-e2c865726af3
md"""
#### Uzdevums 1. Daļiņa bedrē ar galīgi augstām sienām. 

a. Kad bedres dzilms ```V0=20``` un bedres platums ```d=2```.

i)  cik daudz saistīto viļņu funkcijas varam dabūt?

ii) Salīdziniet iegūtos saistītā stāvokļa enerģijas vērtībām ar attiecīgām enerģijas vērtībām bedrē ar bezgalīgi augstām sieniņām. 

b. Raksturojiet viļņu funkcijas.

i) Raksturojiet funkcijas, kas apraksta viļņu funkciju bedrē, t.i.,  kad ``-d<x<d``

ii) Raksturojiet funkcijas, kas apraksta viļņu funkciju āprus bedres, t.i., kad ``x>d`` vai ``x<-d``.

iii) Kāda ir attiecība starp mezgļu punktu skaitu un viļņu funkcijas kvantu skaitli ``n``? (Mezglu punkts ir punkts, kur ``\psi_n(x)=0``.)


c. Paspēlējiet ar parametriem ```V0```, ```d``` un ```Nsols```. 

i) Kā saistīto viļņu funkciju skaits ir atkarīgi no parametra ```V0```. (Varbūt noderēs zīmēt kādu grafiku. Pietiek ar kvalitatīvo aprakstu. )

ii) Kā saistīto viļņu funkciju skaits ir atkarīgi no parametra ```d```.

iii) Vai ir kāds platums vai dziļums, kuram neatbilst saistītā stāvokļa?
"""

# ╔═╡ 612f9b30-f156-40cd-8835-f5b0ce7865fa
md"""
### Varbūtība atrast daļiņu

Viļņu funkcija apraksta daļiņas dinamiku, bet to nevaram novērot tieši. Varam novērot daļiņas varbūtības sadaļijumu ``P(x)=|\psi(x)|^2=\psi^{\star}(x)\psi(x)``, kur ``\psi^{\star}(x)`` ir viļņu funkcijas kompleksais saistītais. (Piemēram, ja ``\psi(x)=e^{ix}``, tad ``\psi^{\star}(x)=e^{-ix}``.)

Tagad zīmēsim varbūtības sadalījumus. 

"""

# ╔═╡ 8338eac7-9dab-4343-9b71-b4b71fb9848e
begin
	plot(xpoints, V_box.(xpoints),label="V(x)")
	for i=1:length(states)
		label_string2="|ψ"*string(i)*"|²"
    	plot!(xpoints, abs2.(states[i].data)*30 .+ energies[i],label=label_string2) # |ψ|² reizināts ar 30, lai būtu vieglāk redzams. 
	end
	plot!(title="Finite Potential Well",xlabel="Position", ylabel="Energy")
end

# ╔═╡ 1e6a2be5-8f21-4d89-9050-c64bc7509be4
md"""
## Viļņu funkcijas evolūcija laikā
Tagad varam skatīties, kā viļņu funkcija attīstās laikā.  
"""

# ╔═╡ 775f9389-8040-42c4-9b3f-cd55cd7ba013
begin
	gr()
	x0box = 0
	p0box = 1.5
	sigma0box = 0.25
	Ψ0box = gaussianstate(basis, x0box, p0box, sigma0box);
	boxAnimation=Animation()
	#Ψ0box=states[1]+states[2] # Sāksim ar stāvokli, kuru veido divi no īpašstāvokļiem.
	
	# Time evolution
	Trange = [0:0.05:50;]     # iterators ar laika vērtībām
	toutBox, ΨtBox = timeevolution.schroedinger(Trange, Ψ0box, H);

	# Animācijas veidošana
	for i=1:length(Trange)
	    Ψbox = ΨtBox[i]
	    n2 = abs2.(Ψbox.data)
		plot(xpoints, V_box.(xpoints),xlabel="x",ylabel="|Ψ(t)|^2",ylim=(-50,50))
	    plt=plot!(xpoints, 100*n2)# , "C0", alpha=0.9*(float(i)/length(T))^8+0.1)
		frame(boxAnimation,plt)
	end

	gif(boxAnimation)
end

# ╔═╡ 5165e0bd-8196-4a58-97b6-2bff9d920185
md"""
## Daļiņa harmoniskā oscilatora potenciālā

Tagad apskatīsimies daļiņas viļņu funkcijas harmoniskajā oscilātora potenciālā. 

Hamiltona operators ir

``\hat{H}=\frac{\hat{p}^2}{2 m} + \frac{1}{2}k \hat{x}^2 = \frac{\hat{p}^2}{2 m} + \frac{1}{2}m \omega^2 \hat{x}^2,`` 

kur ``k`` ir atsperes konstante, ``m`` ir daļiņas masa, un ``\omega = \sqrt{\frac{k}{m}}``.

No lekcijām zinām, ka enerģijas ``E_n=(n+\frac{1}{2})\hbar \omega``.

Attiecīgās īpašfunkcijas ir 

``\psi_n(x)=\frac{1}{\sqrt{2^n n!} \left(\frac{m \omega}{\pi \hbar} \right)} e^{\frac{m \omega x^2}{2 \hbar}}H_n \left(\sqrt{\frac{m \omega}{\hbar}} \right)``, 

kur ``H_n`` ir [Hermīta polinomiāli](https://en.wikipedia.org/wiki/Hermite_polynomials). 
"""

# ╔═╡ c710d71c-62f9-4a3e-abf0-c1338f3d0c5f
begin
	basis_position_HO = PositionBasis(-5, 5, 100)
	basis_momentum_HO = MomentumBasis(basis_position_HO)
	xHO = position(basis_position_HO)
	pHO = momentum(basis_position_HO)
	
	m=1.0 # daļiņas masa
	ω=0.5 # oscilātora potenciāla stiprums

	HHO = pHO^2/(2*m) + 0.5*m*ω^2*xHO^2
	energiesHO, statesHO = eigenstates((HHO+dagger(HHO))/2, 5);
end

# ╔═╡ 5ce6769e-2eea-49c7-99cf-01fff2a8a193
begin
	X = samplepoints(basis_position_HO)
	plot(X, 2*X.^2)
	for i=1:length(statesHO)
		labelHO="|ψ"*string(i)*"|²"
	    plot!(X, abs2.(statesHO[i].data).*200 .+ 20*energiesHO[i],label=labelHO)
	end
	plot!(title="Harmonic Oscillator",xlabel="Position", ylabel="Energy")
end

# ╔═╡ 21bc8a72-5790-4e81-b420-d6d47047a24a
md"""
### Uzdevums 2. Harmoniskais oscilātors. 

a. Kāda ir attiecībā starp kvantu skaitli ``n`` un mezgļu punktu skaitu?

b. Kāpēc viļņu funkciju telpiskā perioda pie augstākām enerģijām ir mazākā? Ar kādu fizikālo lielumu tas ir saistīts?

c. Kāpēc viļņu funkciju telpiskā perioda pie oscilātora potenciāla malām ir lielākā? Kāpēc tur amplitūda ir lielākā? Kā šis novērojums saistās ar klasisko oscilātoru?

"""

# ╔═╡ 42b3c114-ece5-47bf-8ae5-498e3d517ec4
md"""
Arī harmonisko oscilātoru viļņu funkcijām varam pievienot laika atkarību. Vispirms definēsim Furjē transformāciju no pozīcijas bāzes uz kustības daudzuma bāzi.
"""

# ╔═╡ 20f36616-c2fd-4ba0-93a5-a8fdf95d0619
begin
	# Transforms a state multiplied from the right side from real space
	# to momentum space.
	T_px = transform(basis_momentum_HO, basis_position_HO)
end

# ╔═╡ 204c3f78-deca-4547-a2f5-ed8d3d69b7e3
md"""
Varam arī definēt transforāciju otrā virzienā. 
md"""

# ╔═╡ 74bed55e-3150-4b75-a087-11ba127143fb
begin
	T_xp = dagger(T_px)   # Transformācija no p-bāzes uz x-bāzi.
	
	x_HO2 = position(basis_position_HO)   # x-bāzes definīcja
	p_HO2 = momentum(basis_momentum_HO)   # b-bāzes definīcija
	
	H_kin_HO2 = LazyProduct(T_xp, p_HO2^2/2m, T_px)  # Hamiltona operatora kinētiskā daļa
	V_HO2 = ω*x_HO2^2  # Hamiltona operatora potenciālā daļa
	H_HO2 = LazySum(H_kin_HO2, V_HO2)  # Hamiltona operators
end

# ╔═╡ fa269ca9-01fc-4a6d-bf35-4e1ba561a7ba
md"""
### Gausa viļņa pakete Harmoniskā oscilatora potenciālā

Tagad definēsim Gausa viļņu pakete Harmoniskā oscilatora potenciālā un novērosim laika evolūciju. 

Interesanti ir tas, ka viļņu pakete patur savu formu laikā: nenotiek dispersija. Tādu stāvokli sauc par koherentu stāvokli. 
"""

# ╔═╡ 1341e7a1-ee6e-4e60-b46b-8b392f1989b5
begin
	# Initial state
	gr()
	HOanimation=Animation()
	
	x0 = 1.5
	p0 = 0
	sigma0 = 0.6
	Ψ0 = gaussianstate(basis_position_HO, x0, p0, sigma0);
	
	# Time evolution
	T = [0:0.1:6;]
	tout, Ψt = timeevolution.schroedinger(T, Ψ0, H_HO2);
	
	# Plot dynamics of particle density
	x_points = samplepoints(basis_position_HO)
	
	n = abs.(Ψ0.data).^2
	##V = ω*x_points.^2
	VV=diag(V_HO2.data)    # get diagonal elements of sparse operator as matrix
	C = maximum(abs.(VV))/maximum(n)
	

	plt_bg=plot(x_points, (abs.(VV).-3)./C,xlabel="x",ylabel="|Ψ(t)|^2") #, "k--")

	
	for i=1:length(T)
	    Ψ = Ψt[i]
	    n = abs.(Ψ.data).^2
		plot(x_points, (abs.(VV).-3)./C,xlabel="x",ylabel="|Ψ(t)|^2",ylim=(-0.03,0.1),label="V(x)")
	    plt=plot!(x_points, n,label="|Ψ(x,t)|²")
		frame(HOanimation,plt)
	end

	gif(HOanimation,fps=10)
end

# ╔═╡ 8609b62a-66a7-4186-bd3e-0db3794fd8d5
md"""
#### Uzdevums 3. Elektrons bedre ar bezgalīgi augstām sieniņām
Elektrons atrodas bezgalīgi dziļa bedrē (``0<x<L``). Pieņemiet, ka elektrons atrodas pamatstāvoklī. Aprēķiniet varbūtību, ka elektrons atrodas starp ``x=0`` un ``x=L/4``.
"""

# ╔═╡ feb57b20-7c4e-4d60-9a63-35b76c341a01
md"""
#### Uzdevums 4. Spiediens bedrē.
Daļiņa atrodas viendimensiju potenciālā bedrē ar bezgalīgi augstām sienām. Bedres platums ``L``. Nosakiet daļiņas spiediena spēku uz bedres sieniņām, ja daļiņai ir minimālā enerģija. [Saspiežot kasti nepieciešams padarīt darbu, kas pāriet daļiņas iekšējā enerģijā.]
"""

# ╔═╡ 46310bf4-b49e-42de-a18a-d2e34093c6cb
md"""
#### Uzdevums 5. ``\delta-``funkcijas potenciāls.

Izskatiet potenciālu ``V(x) = −aV_0 \delta(x)``. 

(Atcerēsimies, ka ``\delta(x-x_0)=\infty``, kad ``x=x_0``, bet 0 citur. 

``\int_{-\infty}^{\infty} \delta(x-x_0) \mathrm{dx} = 1``.)

a. Pierādiet, ka potenciālam ir saistīts stāvoklis ar enerģiju ``E=ma^2 V_0^2/2\hbar^2``. 

b. Atrodiet normēto viļņu funkciju.

Varbūt noder atcerēties: 
1. ``\lim_{\epsilon \rightarrow 0}[\psi'(\epsilon)-\psi'(\epsilon)] = \lim_{\epsilon \rightarrow 0} \int_{-\epsilon}^{\epsilon} \psi''(x) \mathrm{dx}``.

2. ``\int_{-\infty}^{\infty} f(x)\delta(x-x_0) \mathrm{dx} = f(x_0)``.
"""

# ╔═╡ d0d9ee0b-b1db-4e47-8788-37bf879167b9
md"""
#### Uzdevums 6. Bedre, kas pekšņi palielināsies
Daļiņa atrodas bezgalīgi dziļas bedres pamatstāvoklī. Bedres platums ir ``L``.  Pēkšņi bedres platums palielinās divas reizes uz ``2L``, nemainot daļiņas viļņu funkciju. Pierādiet, ka varbūtība, ka daļiņa palīks jaunās bedres pamatstāvoklī ir ``(8/3\pi)^2``.

Varbūt noder atcerēties:

1. ``\psi(x)=\sum_{n=1}^{n=\infty} a_n \phi_n(x)``, 

kur ``\phi_n(x)`` ir konkrētā Hamiltoniāna ``\hat{H}`` īpašfunkcijas.

2. ``\int_{-\infty}^{\infty} \phi_n(x) \phi_m(x) \mathrm{dx} = \delta_{mn}``

3. ``a_n = \int_{-\infty}^{\infty} \psi(x) \phi_n(x) \mathrm{dx} = \int_{-infty}^{\infty} \left(\sum_{m=1}^{m=\infty}\phi_m(x) \right)  \phi_n(x) \mathrm{dx} `` 
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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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:
# ╟─cd1bc8b0-d5c0-11ec-2a3b-831b0b8d8b3a
# ╠═83d05132-b257-4c45-ad70-a24c92c8f8ee
# ╟─be74a383-df10-463c-92a2-a093b8a2d784
# ╠═5afc4a98-b3cb-49c0-a5a3-acbcd6509ed2
# ╠═9a8b8479-f5a2-46a5-94b7-e66d2d2a3a02
# ╟─a64c019b-4c80-4f48-8cfa-4646c26c27e6
# ╠═c4b5d3df-2ea1-48c8-9598-48fc1caaba16
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# ╟─8dde5436-b5cc-4ac3-b217-c374d888a345
# ╟─a4bc121f-9002-46d5-a688-e2c865726af3
# ╟─612f9b30-f156-40cd-8835-f5b0ce7865fa
# ╠═8338eac7-9dab-4343-9b71-b4b71fb9848e
# ╟─1e6a2be5-8f21-4d89-9050-c64bc7509be4
# ╠═775f9389-8040-42c4-9b3f-cd55cd7ba013
# ╟─5165e0bd-8196-4a58-97b6-2bff9d920185
# ╠═c710d71c-62f9-4a3e-abf0-c1338f3d0c5f
# ╠═5ce6769e-2eea-49c7-99cf-01fff2a8a193
# ╟─21bc8a72-5790-4e81-b420-d6d47047a24a
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# ╠═20f36616-c2fd-4ba0-93a5-a8fdf95d0619
# ╟─204c3f78-deca-4547-a2f5-ed8d3d69b7e3
# ╠═74bed55e-3150-4b75-a087-11ba127143fb
# ╟─fa269ca9-01fc-4a6d-bf35-4e1ba561a7ba
# ╠═1341e7a1-ee6e-4e60-b46b-8b392f1989b5
# ╟─8609b62a-66a7-4186-bd3e-0db3794fd8d5
# ╟─feb57b20-7c4e-4d60-9a63-35b76c341a01
# ╟─46310bf4-b49e-42de-a18a-d2e34093c6cb
# ╟─d0d9ee0b-b1db-4e47-8788-37bf879167b9
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
