bessel_roots.py

"""
The energy eigenvalues for solutions of the Schrodinger equation in an infinite circular
well are proportional to the squares of the roots of ordinary Bessel functions of the 1st
kind.

The quantum numbers (l, n) imply the `n`th root of the order `l` Bessel function.

I don't know of a clever way to the order the `n`th roots across function orders, so I've
calculated by brute for and hard coded the first 50 roots `alphas` with the corresponding
quantum numbers (`lns`) below.
"""

alphas = [
    2.404825557695773,
    3.8317059702075125,
    5.135622301840683,
    5.520078110286311,
    6.380161895923984,
    7.015586669815619,
    7.5883424345038035,
    8.417244140399864,
    8.653727912911013,
    8.771483815959954,
    9.76102312998167,
    9.936109524217684,
    10.173468135062722,
    11.064709488501185,
    11.086370019245084,
    11.61984117214906,
    11.791534439014281,
    12.225092264004656,
    12.338604197466944,
    13.015200721698434,
    13.323691936314223,
    13.35430047743533,
    13.589290170541217,
    14.37253667161759,
    14.475500686554541,
    14.795951782351262,
    14.821268727013171,
    14.930917708487787,
    15.589847884455486,
    15.70017407971167,
    16.03777419088771,
    16.223466160318768,
    16.470630050877634,
    16.698249933848246,
    17.003819667816014,
    17.24122038248913,
    17.615966049804832,
    17.80143515328244,
    17.959819494987826,
    18.071063967910924,
    18.28758283248173,
    18.43346366696658,
    18.899997953174022,
    18.98013387517992,
    19.409415226435012,
    19.554536430997054,
    19.615858510468243,
    19.61596690396692,
    19.994430629816385,
    20.320789213566506,
    20.789906360078444,
]

lns = [
    (0, 1),
    (1, 1),
    (2, 1),
    (0, 2),
    (3, 1),
    (1, 2),
    (4, 1),
    (2, 2),
    (0, 3),
    (5, 1),
    (3, 2),
    (6, 1),
    (1, 3),
    (4, 2),
    (7, 1),
    (2, 3),
    (0, 4),
    (8, 1),
    (5, 2),
    (3, 3),
    (1, 4),
    (9, 1),
    (6, 2),
    (4, 3),
    (10, 1),
    (2, 4),
    (7, 2),
    (0, 5),
    (11, 1),
    (5, 3),
    (8, 2),
    (3, 4),
    (1, 5),
    (12, 1),
    (6, 3),
    (9, 2),
    (4, 4),
    (13, 1),
    (2, 5),
    (0, 6),
    (7, 3),
    (10, 2),
    (14, 1),
    (5, 4),
    (3, 5),
    (8, 3),
    (1, 6),
    (11, 2),
    (15, 1),
    (6, 4),
    (12, 2),
]