latex_schrodinger

Solution of Schrödinger’s equation for the motion of a particle in one dimension in a parabolic potential well. This example demonstrates the use of mathtext on Label and Title annotations.

Details

Bokeh APIs:

figure.line, figure.varea, bokeh.models.Label, bokeh.models.Title

More info:

Mathematical notation

Keywords:

mathtext, latex

import numpy as np
from scipy.special import factorial, hermite

from bokeh.models import Label, Range1d, Title
from bokeh.plotting import figure, show

p = figure(
    width=800, height=600, x_range=Range1d(-6, 6), y_range=Range1d(0, 8), toolbar_location=None,
    title=r"$$\text{Wavefunction } \psi_v(q) \text{ of first 8 mode solutions of Schrodinger's equation }"
          r" -\frac{1}{2}\frac{d^2\psi}{dq^2} + \frac{1}{2}q^2\psi = \frac{E}{\hbar\omega}\psi$$")
p.xaxis.axis_label = r"$$q$$"
p.yaxis.visible = False
p.xgrid.visible = False
p.ygrid.visible = False
p.add_layout(Title(text=r"$$\text{Each wavefunction is labelled with its quantum number } v \text{ and energy } E_v$$"), "above")
p.add_layout(Title(text=r"$$\text{in a potential } V(q) = \frac{q^2}{2} \text{ shown by the dashed line.}$$"), "above")

q = np.linspace(-6, 6, 100)
yscale = 0.75
number_of_modes = 8

for v in range(number_of_modes):
    H_v = hermite(v)
    N_v = (np.pi**0.5 * 2**v * factorial(v))**(-0.5)
    psi = N_v*H_v(q)*np.exp(-q**2/2)
    E_v = v + 0.5  # Use energy level as y-offset.

    y = yscale*psi + E_v
    yupper = np.where(y >= E_v, y, E_v)
    ylower = np.where(y <= E_v, y, E_v)

    p.varea(q, yupper, E_v, fill_color="coral")
    p.varea(q, ylower, E_v, fill_color="orange")
    p.line(q, y, color="red", line_width=2)

    p.add_layout(Label(x=-5.8, y=E_v, y_offset=-21, text=r"$$v = " + str(v) + r"$$"))
    p.add_layout(Label(x=3.9, y=E_v, y_offset=-25, text=r"$$E_" + str(v) + r" = (" + str(2*v+1) + r"/2) \hbar\omega$$"))

V = q**2 / 2
p.line(q, V, line_color="black", line_width=2, line_dash="dashed")

show(p)