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schrodinger_eq_scipy.jl
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schrodinger_eq_scipy.jl
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### A Pluto.jl notebook ###
# v0.19.43
using Markdown
using InteractiveUtils
# ╔═╡ 17750a7c-4273-11ef-2bea-9b56de603b65
begin
using Printf
using Markdown
using CondaPkg
using Plots
using PythonCall
CondaPkg.add_pip("numpy")
CondaPkg.add_pip("scipy")
end
# ╔═╡ f3ec2781-9ba9-4acf-863e-2f2c693951da
md"""
# SciPy を利用したシュレディンガー方程式の数値計算結果を Pluto ノートブックから利用する
これは [大槻純也, Pythonによる計算物理, 森北出版](https://j-otsuki.github.io/comput-phys-book/) の題材にある [時間発展シュレディンガー方程式](https://www.physics.okayama-u.ac.jp/~otsuki/lecture/CompPhys2/pde/schro.html)
のトピックと対応する実装 [schr_wall.py](https://github.com/j-otsuki/comput-phys-book/blob/main/python/schr_wall.py) を学ぶために用いたノートである．
理論的な背景は[大槻氏のウェブサイト](https://www.physics.okayama-u.ac.jp/~otsuki/lecture/CompPhys2/index.html) を，具体的な実装は [j-otsuki/comput-phys-book](https://github.com/j-otsuki/comput-phys-book) を参考にすると良い．
このノートブックはパッケージ化されていない１画面に収まる程度の小規模な Python の実装を Pluto ノートブックから利用するサンプルになっている．
"""
# ╔═╡ be3a7a86-efda-4bdd-9876-653bb504d340
md"""
Pluto のセルに下記のような変数を定義する
```julia
my_python_script = raw"# write Python script here>"
```
次に上記で定義した `my_python_script` を一度 mylib.py のようにしてファイルに落とし込む． (もしかしたら Python の `eval` を使えば楽にできるかもしれないけれど)
```julia
mylib = mktempdir() do d
rm(joinpath(d, "mylib.py"), force=true)
open(joinpath(d, "mylib.py"), "w") do io
print(io, my_python_script)
end
pyimport("sys").path.append(d)
mylib = pyimport("mylib")
importlib = pyimport("importlib")
importlib.reload(mylib)
end
```
`mylib.py` の格納場所を Python の `sys.path` に `append` によって追加する．これにより
```python
# Python
import mylib
```
相当の実装ができる．
試行錯誤する(=`my_python_script` を編集する) 際に `mylib.py` を更新する必要がある． どうやら mylib.py を格納する一時ディレクトリ `d` は使いまわされるようなので `rm` 関数で一度消すことにしている． また更新した mylib モジュールを再ロードするために Python 標準の `importlib` パッケージを利用して `importlib.reload(mylib)` を実行している．
"""
# ╔═╡ c2e4b721-10fb-4e2d-8300-58808d179d26
begin
# Edit Python script
my_python_script = raw"""
import numpy as np
from scipy.sparse import csr_matrix as sparse_matrix
from scipy.integrate import solve_ivp
def make_differential_op(nx, dx):
f0 = np.identity(nx)
f1 = np.roll(f0, 1, axis=1)
f2 = np.roll(f0, 2, axis=1)
f_1 = f1.transpose()
f_2 = f2.transpose()
D1 = sparse_matrix(f1 - f_1) / (2 * dx)
D2 = sparse_matrix(f1 - 2 * f0 + f_1) / (dx * dx)
D3 = sparse_matrix(f2 - 2 * f1 + 2 * f_1 - f_2) / (2 * dx * dx * dx)
return (D1, D2, D3)
def schrodinger(t, u, D2, v):
return -1.0j * (-0.5 * (D2 @ u) + v * u)
def solve(x, t, u0, v):
dx = x[1] - x[0]
_, op_df2, _ = make_differential_op(x.size, dx)
tspan = (t[0], t[-1])
args = (op_df2, v)
sol = solve_ivp(
schrodinger, tspan, u0, dense_output=True, args=args, rtol=1e-8,
)
return sol
def main():
xmin, xmax = -50, 50
nx = 1000
x = np.linspace(xmin, xmax, nx, endpoint=False)
k = 4
x0 = -20
delta = 4
u0 = np.exp(1.0j * k * x - 0.5 * ((x - x0) / delta) ** 2)
dx = x[1] - x[0]
integ = np.sum(np.absolute(u0) ** 2) * dx
u0 /= np.sqrt(integ)
v_height = 10.0
width = 0.2
v = np.where(np.abs(x) < width, v_height, 0)
tmax = 16
nt = 81
t = np.linspace(0, tmax, nt)
sol = solve(x, t, u0, v)
return x, t, sol
"""
end;
# ╔═╡ 3e692a8b-0aa4-4941-bad4-7cef12fc4032
Markdown.parse(raw"```python\n" * my_python_script * raw"```")
# ╔═╡ 21e28ce0-6a8b-438e-a89e-c13cc081dd5f
begin
mylib = mktempdir() do d
rm(joinpath(d, "mylib.py"), force=true)
open(joinpath(d, "mylib.py"), "w") do io
print(io, my_python_script)
end
pyimport("sys").path.append(d)
mylib = pyimport("mylib")
importlib = pyimport("importlib")
importlib.reload(mylib)
end
np = pyimport("numpy")
x, t, sol = mylib.main()
println(sol.t.size)
println(np.diff(sol.t).min())
println(np.diff(sol.t).max())
end
# ╔═╡ fff48d97-69b1-42b6-8222-a3679520ed95
md"""
トンネル効果の様子を可視化できる．
"""
# ╔═╡ 46f0f644-bdc0-441a-b76b-8df0fb816995
begin
u = np.absolute(sol.sol(t)) ^ 2
u = u.T
end
# ╔═╡ 65d2c521-8b84-46db-8646-fc0ccebceca9
begin
anim = @animate for i in pyrange(u.shape[0])
f = pyconvert(Float64, t[i])
s = @sprintf "%.2f" f
plot(PyVector(x), PyVector(u[i]), xlim=[-30, 30], ylim=[0, 0.4], title="t=$(s)")
end
gif(anim, "wall.gif")
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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PythonCall = "6099a3de-0909-46bc-b1f4-468b9a2dfc0d"
[compat]
CondaPkg = "~0.2.22"
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PythonCall = "~0.9.20"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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