Stark and Zeeman splitting: static perturbation theory

Two static, first-order perturbative effects on the hydrogen atom. In a magnetic field \(B\), the weak-field Zeeman effect shifts each \(\lvert l, m_l; s, m_s\rangle\) sublevel by

\[\Delta E = \mu_B B\,(m_l + g_s m_s);\]

in a uniform electric field \(F\), the linear Stark effect splits hydrogen’s 4-fold degenerate \(n=2\) shell (using parabolic quantum numbers \(n_1, n_2, m\) with \(n_1+n_2+\lvert m\rvert+1=n\)) by

\[\Delta E = \tfrac{3}{2}\, n\, (n_1 - n_2)\, a_0 F.\]

The Zeeman effect is the textbook example of non-degenerate perturbation theory; the linear Stark effect of hydrogen specifically is the degenerate variety, made possible only by hydrogen’s accidental level degeneracy in \(l\).

import matplotlib.pyplot as plt
import numpy as np

from physicskit.quantum.chapters.perturbation import stark_n2_quartet, zeeman_spectrum

The Stark quartet and the Zeeman splitting

Left: the three Stark-shifted energies of the \(n=2\) quartet vs. field \(F\) (the \(m=\pm1\) sublevels stay unshifted since only the \(m=0\) pair couples to the field). Right: the Zeeman sublevel shifts vs. \(B\) for \(l=0,1,2\).

fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))

# Linear Stark effect: hydrogen n=2 quartet
F_values = np.linspace(-0.02, 0.02, 50)
shifts = np.array([[lv.shift for lv in stark_n2_quartet(F)] for F in F_values])
for i in range(shifts.shape[1]):
    axes[0].plot(F_values, shifts[:, i])
axes[0].set_xlabel("Field F")
axes[0].set_ylabel("Energy shift")
axes[0].set_title("Linear Stark effect: hydrogen n=2 quartet")

# Zeeman splitting
B_values = np.linspace(0, 2, 40)
for l in [0, 1, 2]:
    levels = np.array([zeeman_spectrum(l, B) for B in B_values])
    for i in range(levels.shape[1]):
        axes[1].plot(B_values, levels[:, i], color=f"C{l}", alpha=0.6, label=(f"l={l}" if i == 0 else None))
axes[1].set_xlabel("Magnetic field B")
axes[1].set_ylabel("Energy shift")
axes[1].set_title("Zeeman sublevel splitting")
axes[1].legend(fontsize=8)

fig.tight_layout()

print(f"Stark n=2 shifts at F={F_values[-1]:.3f}: {shifts[-1]}")
print(f"Zeeman l=1 shifts at B={B_values[-1]:.3f}: {zeeman_spectrum(1, B_values[-1])}")

Stark n=2 shifts at F=0.020: [-0.06  0.    0.    0.06]
Zeeman l=1 shifts at B=2.000: [-4. -2.  0.  0.  2.  4.]