Complex Harmonic Oscillator¶

In this tutorial, we show a few interesting examples of the complex simple harmonic oscillator.

In [1]:

import math

import numpy as np
from plotly import express as px

from hamilflow.models.harmonic_oscillator import ComplexSimpleHarmonicOscillator

import math import numpy as np from plotly import express as px from hamilflow.models.harmonic_oscillator import ComplexSimpleHarmonicOscillator

Basic setups¶

In [2]:

t = np.linspace(0, 3, 257)
system_specs = {"omega": 2 * math.pi}

t = np.linspace(0, 3, 257) system_specs = {"omega": 2 * math.pi}

Positive-frequency, circular-polarised mode¶

Also known as the left-rotating mode.

In [3]:

csho = ComplexSimpleHarmonicOscillator(system_specs, initial_condition={"x0": (1, 0)})

df = csho(t)

arr_z = df["z"].to_numpy(copy=False)

px.line_3d(
    x=arr_z.real,
    y=arr_z.imag,
    z=t,
    labels={"x": "real", "y": "imag", "z": "time"},
)

csho = ComplexSimpleHarmonicOscillator(system_specs, initial_condition={"x0": (1, 0)}) df = csho(t) arr_z = df["z"].to_numpy(copy=False) px.line_3d( x=arr_z.real, y=arr_z.imag, z=t, labels={"x": "real", "y": "imag", "z": "time"}, )

Negative-frequency, circular-polarised mode¶

Also known as the right-rotating mode.

In [4]:

csho = ComplexSimpleHarmonicOscillator(system_specs, initial_condition={"x0": (0, 1)})

df = csho(t)

arr_z = df["z"].to_numpy(copy=False)

px.line_3d(
    x=arr_z.real,
    y=arr_z.imag,
    z=t,
    labels={"x": "real", "y": "imag", "z": "time"},
)

csho = ComplexSimpleHarmonicOscillator(system_specs, initial_condition={"x0": (0, 1)}) df = csho(t) arr_z = df["z"].to_numpy(copy=False) px.line_3d( x=arr_z.real, y=arr_z.imag, z=t, labels={"x": "real", "y": "imag", "z": "time"}, )

Positive-frequency, elliptic-polarised mode¶

In [5]:

csho = ComplexSimpleHarmonicOscillator(
    system_specs,
    initial_condition={"x0": (math.cos(math.pi / 12), math.sin(math.pi / 12))},
)

df = csho(t)

arr_z = df["z"].to_numpy(copy=False)

px.line_3d(
    x=arr_z.real,
    y=arr_z.imag,
    z=t,
    labels={"x": "real", "y": "imag", "z": "time"},
)

csho = ComplexSimpleHarmonicOscillator( system_specs, initial_condition={"x0": (math.cos(math.pi / 12), math.sin(math.pi / 12))}, ) df = csho(t) arr_z = df["z"].to_numpy(copy=False) px.line_3d( x=arr_z.real, y=arr_z.imag, z=t, labels={"x": "real", "y": "imag", "z": "time"}, )

End of Notebook¶