Symmetry in Physics

Symmetry is the deepest organizing principle in theoretical physics. Noether's theorem establishes the profound connection between continuous symmetries and conservation laws. Discrete symmetries (parity, charge conjugation, time reversal) impose selection rules and constrain interactions. Gauge symmetry underlies all fundamental forces, and its spontaneous breaking generates particle masses through the Higgs mechanism. This chapter explores these ideas systematically.

1. Noether's Theorem: Symmetries and Conservation Laws

Noether's theorem (1918) is one of the most beautiful results in theoretical physics. It states that every continuous symmetry of the action implies a conserved quantity. We derive it from the Lagrangian formalism.

Consider a system with generalized coordinates $q_i(t)$ and Lagrangian $L(q_i, \dot{q}_i, t)$. The action is:

$$S[q] = \int_{t_1}^{t_2} L(q_i, \dot{q}_i, t) \, dt$$

Derivation. Consider an infinitesimal transformation parameterized by $\epsilon$:

$$q_i \to q_i + \epsilon \, \delta q_i, \quad t \to t + \epsilon \, \delta t$$

This is a symmetry if the action changes by at most a boundary term, i.e., there exists a function $\Lambda$ such that:

$$\delta L = \epsilon \frac{d\Lambda}{dt}$$

Expanding the variation of the Lagrangian explicitly:

$$\delta L = \sum_i \left(\frac{\partial L}{\partial q_i} \delta q_i + \frac{\partial L}{\partial \dot{q}_i} \delta \dot{q}_i\right) + \frac{\partial L}{\partial t} \delta t$$

Using the Euler-Lagrange equations $\frac{\partial L}{\partial q_i} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}$and the identity $\delta \dot{q}_i = \frac{d}{dt}(\delta q_i) - \dot{q}_i \frac{d}{dt}(\delta t)$, we can write:

$$\delta L = \frac{d}{dt}\left(\sum_i \frac{\partial L}{\partial \dot{q}_i} \delta q_i\right) + \frac{d}{dt}(L \, \delta t) - \frac{dL}{dt} \delta t + \frac{\partial L}{\partial t}\delta t$$

Setting $\delta L = \epsilon \frac{d\Lambda}{dt}$ and rearranging, we obtain the Noether charge:

Noether's Theorem

If the action is invariant under a continuous transformation, the following quantity is conserved:

$$Q = \sum_i \frac{\partial L}{\partial \dot{q}_i} \delta q_i + \left(L - \sum_i \frac{\partial L}{\partial \dot{q}_i} \dot{q}_i\right) \delta t - \Lambda$$

That is, $\frac{dQ}{dt} = 0$ along solutions of the equations of motion.

Key examples of Noether's theorem:

Time translation invariance ($\delta t = \epsilon$, $\delta q_i = 0$): conserved quantity is the energy$H = \sum_i p_i \dot{q}_i - L$
Spatial translation invariance ($\delta q_i = \epsilon \hat{n}_i$): conserved quantity is linear momentum$P = \sum_i p_i \hat{n}_i$
Rotational invariance ($\delta \mathbf{r} = \epsilon \hat{n} \times \mathbf{r}$): conserved quantity is angular momentum$\mathbf{L} = \mathbf{r} \times \mathbf{p}$

Field theory version. For a field $\phi(x^\mu)$ with Lagrangian density$\mathcal{L}(\phi, \partial_\mu \phi)$, Noether's theorem gives a conserved current:

$$j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \delta\phi - T^{\mu\nu} \delta x_\nu, \quad \partial_\mu j^\mu = 0$$

where $T^{\mu\nu}$ is the energy-momentum tensor. The conserved charge is$Q = \int j^0 \, d^3x$.

2. Discrete Symmetries: P, C, T

Unlike continuous symmetries, discrete symmetries do not have associated Noether charges. However, they impose powerful constraints on physical processes through selection rules.

Parity (P)

Parity is spatial inversion: $\mathbf{x} \to -\mathbf{x}$. Under parity:

$$P: \quad \mathbf{x} \to -\mathbf{x}, \quad \mathbf{p} \to -\mathbf{p}, \quad \mathbf{L} \to +\mathbf{L}, \quad \mathbf{S} \to +\mathbf{S}$$

Vectors that flip sign under P are polar vectors; those that do not are axial vectors (pseudovectors). The parity operator satisfies $P^2 = 1$, so eigenvalues are $\pm 1$.

Charge Conjugation (C)

Charge conjugation replaces particles with antiparticles:

$$C: \quad e^- \to e^+, \quad \mathbf{E} \to -\mathbf{E}, \quad \mathbf{B} \to -\mathbf{B}$$

Like parity, $C^2 = 1$ and eigenvalues are $\pm 1$. Only particles that are their own antiparticles (like $\pi^0$ or the photon) can be C eigenstates.

Time Reversal (T)

Time reversal is the transformation $t \to -t$:

$$T: \quad t \to -t, \quad \mathbf{p} \to -\mathbf{p}, \quad \mathbf{L} \to -\mathbf{L}, \quad \mathbf{S} \to -\mathbf{S}$$

CPT Theorem. The combination CPT is an exact symmetry of any local Lorentz-invariant quantum field theory. Individual symmetries can be violated:

P violation: the weak interaction maximally violates parity (Wu experiment, 1957)
C violation: neutrinos are left-handed, antineutrinos right-handed
CP violation: observed in the neutral kaon system (Cronin-Fitch, 1964) and B mesons
T violation: follows from CP violation via the CPT theorem

3. Gauge Symmetry

A global symmetry is one where the transformation parameter is the same at every spacetime point. A gauge (local) symmetry allows the parameter to vary:$\alpha \to \alpha(x)$. Promoting a global symmetry to a local one requires introducing gauge fields.

U(1) gauge theory (electrodynamics). Start with a free Dirac field:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi$$

This is invariant under the global U(1) transformation $\psi \to e^{i\alpha}\psi$. But if we make $\alpha = \alpha(x)$ local, the derivative term fails:

$$\partial_\mu \psi \to e^{i\alpha(x)}(\partial_\mu + i\partial_\mu\alpha)\psi$$

To restore invariance, we introduce the covariant derivative:

$$D_\mu = \partial_\mu + ieA_\mu, \quad A_\mu \to A_\mu - \frac{1}{e}\partial_\mu \alpha$$

The gauge field $A_\mu$ is the photon! Adding its kinetic term gives QED:

$$\mathcal{L}_{\mathrm{QED}} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

Non-abelian gauge theory (Yang-Mills). For a non-abelian group like SU(N), the gauge field becomes matrix-valued: $A_\mu = A_\mu^a T_a$. The field strength acquires a non-linear term:

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig[A_\mu, A_\nu]$$

The commutator term means gauge bosons carry charge and interact with each other (gluon self-interaction in QCD, W boson self-interaction in the weak force). The Yang-Mills Lagrangian is:

$$\mathcal{L}_{\mathrm{YM}} = -\frac{1}{2}\mathrm{tr}(F_{\mu\nu}F^{\mu\nu}) = -\frac{1}{4}F_{\mu\nu}^a F^{a\mu\nu}$$

4. Spontaneous Symmetry Breaking

A symmetry is spontaneously broken when the Lagrangian is symmetric but the ground state is not. The classic example is the "Mexican hat" potential.

Consider a complex scalar field $\phi$ with Lagrangian:

$$\mathcal{L} = |\partial_\mu \phi|^2 - V(\phi), \quad V(\phi) = \mu^2 |\phi|^2 + \lambda |\phi|^4$$

This has a global U(1) symmetry: $\phi \to e^{i\alpha}\phi$.

Case 1: $\mu^2 > 0$. The minimum is at $\phi = 0$, which respects the U(1) symmetry. No symmetry breaking.

Case 2: $\mu^2 < 0$. The potential has a circle of minima at$|\phi| = v/\sqrt{2}$ where $v = \sqrt{-\mu^2/\lambda}$. The vacuum chooses one point on this circle, breaking the U(1) symmetry.

Expanding around the chosen vacuum $\phi = \frac{1}{\sqrt{2}}(v + h(x))e^{i\xi(x)/v}$:

$$V = \lambda v^2 h^2 + \lambda v h^3 + \frac{\lambda}{4}h^4 + \text{const}$$

The field $h$ has mass $m_h = \sqrt{2\lambda} v$ (the Higgs-like radial mode). The field $\xi$ is massless — this is the Goldstone boson, corresponding to oscillations along the flat direction of the potential.

Goldstone's Theorem

For every spontaneously broken continuous symmetry generator, there exists a massless scalar particle (a Goldstone boson or Nambu-Goldstone boson).

If $G$ is broken to a subgroup $H$, the number of Goldstone bosons equals$\dim G - \dim H$.

Proof sketch. Let $V(\phi)$ be the potential with minimum at $\phi = \phi_0$. Symmetry under generator $T_a$ implies $V(\phi_0 + i\epsilon T_a \phi_0) = V(\phi_0)$to all orders. Expanding to second order:

$$\frac{\partial^2 V}{\partial \phi_i \partial \phi_j}\bigg|_{\phi_0} (T_a \phi_0)_j = 0$$

The mass matrix $M^2_{ij} = \partial^2 V/\partial\phi_i\partial\phi_j$ has the vector$T_a \phi_0$ as a zero eigenvector whenever $T_a \phi_0 \neq 0$ (i.e., when generator $T_a$ does not leave the vacuum invariant). Each such zero eigenvalue corresponds to a massless Goldstone boson. $\square$

The Higgs mechanism. When the broken symmetry is a gauge symmetry, the Goldstone bosons are "eaten" by the gauge fields, which acquire mass:

$$m_A^2 = g^2 v^2$$

The would-be Goldstone boson becomes the longitudinal polarization of the now-massive gauge boson. In the Standard Model, the SU(2)$_L$ $\times$ U(1)$_Y$ gauge symmetry is broken to U(1)$_{\mathrm{EM}}$, giving masses to the $W^\pm$ and $Z^0$bosons while leaving the photon massless. Three of the four Higgs field components become the longitudinal polarizations of $W^\pm$ and $Z^0$; the remaining component is the physical Higgs boson, discovered at the LHC in 2012.

5. Physical Examples of Symmetry Breaking

Spontaneous symmetry breaking appears throughout physics:

Ferromagnetism: The Heisenberg Hamiltonian $H = -J\sum_{\langle ij\rangle} \mathbf{S}_i \cdot \mathbf{S}_j$has SO(3) rotational symmetry, but below the Curie temperature the spins align, breaking SO(3) $\to$ SO(2). The 2 Goldstone bosons are magnons (spin waves).
Chiral symmetry breaking in QCD: The approximate SU(2)$_L$ $\times$ SU(2)$_R$ chiral symmetry of light quarks is spontaneously broken to SU(2)$_V$ (isospin) by the quark condensate$\langle\bar{q}q\rangle \neq 0$. The 3 pseudo-Goldstone bosons are the pions ($\pi^+, \pi^0, \pi^-$).
Superconductivity: The U(1) electromagnetic gauge symmetry is broken by Cooper pair condensation. The "eaten" Goldstone boson gives the photon an effective mass inside the superconductor (Meissner effect).
Superfluidity: Global U(1) symmetry (particle number conservation) is spontaneously broken. The Goldstone boson is the phonon mode of the superfluid.

The pattern of symmetry breaking $G \to H$ determines the low-energy physics. The Goldstone bosons parameterize the coset space $G/H$, and their interactions are dictated by the broken symmetry through effective Lagrangians (chiral perturbation theory).

Explicit vs. spontaneous breaking. When the Lagrangian itself lacks the symmetry (e.g., quark masses break chiral symmetry), the would-be Goldstone bosons acquire small masses proportional to the symmetry-breaking parameter. These are pseudo-Goldstone bosons. The pion mass satisfies the Gell-Mann-Oakes-Renner relation:

$$m_\pi^2 f_\pi^2 = -(m_u + m_d)\langle\bar{q}q\rangle$$

relating the pion mass to quark masses and the chiral condensate.

5.5. The Standard Model Higgs Mechanism in Detail

The Standard Model Higgs field is an SU(2)$_L$ doublet with hypercharge $Y = 1/2$:

$$\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \quad \langle\Phi\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v \end{pmatrix}$$

The four generators of SU(2)$_L \times$ U(1)$_Y$ are $T_1, T_2, T_3$ and $Y$. Acting on the vacuum: $T_1\langle\Phi\rangle \neq 0$, $T_2\langle\Phi\rangle \neq 0$,$(T_3 - Y)\langle\Phi\rangle \neq 0$, but $(T_3 + Y)\langle\Phi\rangle = 0$. So 3 generators are broken (giving masses to $W^\pm$ and $Z^0$) while the combination $Q = T_3 + Y$ (electric charge) remains unbroken, keeping the photon massless.

The gauge boson masses are:

$$m_W = \frac{gv}{2}, \quad m_Z = \frac{v}{2}\sqrt{g^2 + g'^2}, \quad m_\gamma = 0$$

where $g$ and $g'$ are the SU(2) and U(1) coupling constants. The ratio$m_W / m_Z = \cos\theta_W$ (the Weinberg angle) is a testable prediction confirmed by experiment.

6. Energy-Momentum Tensor from Noether's Theorem

Spacetime translation invariance $x^\mu \to x^\mu + a^\mu$ gives the canonical energy-momentum tensor via Noether's theorem:

$$T^{\mu\nu}_{\mathrm{can}} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial^\nu \phi - \eta^{\mu\nu} \mathcal{L}$$

Conservation $\partial_\mu T^{\mu\nu} = 0$ gives four conserved charges:

$$P^\nu = \int d^3x \, T^{0\nu}$$

where $P^0 = H$ is the Hamiltonian (energy) and $P^i$ is the momentum.

Similarly, Lorentz invariance ($x^\mu \to \Lambda^\mu_{\ \nu} x^\nu$) yields the angular momentum tensor:

$$M^{\mu\nu\rho} = x^\nu T^{\mu\rho} - x^\rho T^{\mu\nu} + S^{\mu\nu\rho}$$

where $S^{\mu\nu\rho}$ is the spin contribution. The conserved charges$M^{\nu\rho} = \int d^3x \, M^{0\nu\rho}$ generate rotations ($M^{ij}$) and boosts ($M^{0i}$).

Internal symmetries also produce conserved charges. For a complex scalar field with U(1) symmetry $\phi \to e^{i\alpha}\phi$:

$$j^\mu = i(\phi^*\partial^\mu\phi - \phi\partial^\mu\phi^*), \quad Q = \int d^3x \, j^0$$

The conserved charge $Q$ is interpreted as electric charge (or particle number in non-relativistic contexts).

7. Ward Identities and Quantum Symmetries

At the quantum level, symmetries manifest as Ward identities — constraints on correlation functions. For a conserved current $j^\mu$:

$$\partial_\mu \langle j^\mu(x) \mathcal{O}_1(x_1) \cdots \mathcal{O}_n(x_n) \rangle = -i \sum_{k=1}^n \delta(x - x_k) \langle \mathcal{O}_1 \cdots \delta\mathcal{O}_k \cdots \mathcal{O}_n \rangle$$

In QED, the Ward identity takes the form:

$$q_\mu \Gamma^\mu(p, p') = S^{-1}(p') - S^{-1}(p)$$

relating the vertex function $\Gamma^\mu$ to the electron propagator $S$. This ensures that the photon remains massless to all orders in perturbation theory and that charge is conserved at each vertex.

Anomalies occur when a classical symmetry cannot be maintained at the quantum level. The chiral anomaly modifies the conservation of the axial current:

$$\partial_\mu j^{\mu 5} = \frac{e^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}$$

This anomaly explains $\pi^0 \to \gamma\gamma$ decay and constrains the particle content of consistent gauge theories (anomaly cancellation in the Standard Model).

Anomaly cancellation is a non-trivial consistency requirement. In the Standard Model, the contributions from quarks and leptons cancel precisely:

$$\sum_{\text{fermions}} Q_f^3 = 3\left(\frac{2}{3}\right)^3 + 3\left(-\frac{1}{3}\right)^3 + (-1)^3 + 0^3 = 0$$

The factor of 3 comes from color. This cancellation requires quarks to come in three colors and relates the electric charges of quarks and leptons — a hint toward grand unification. The anomaly cancellation condition was historically used to predict the existence of the top quark before its discovery.

Gravitational anomalies impose even stronger constraints: the total number of chiral fermion species must satisfy $\sum_f (n_L - n_R) = 0$ for consistency of quantum gravity. In string theory, anomaly cancellation in 10 dimensions restricts the gauge group to $\mathrm{SO}(32)$ or $E_8 \times E_8$.

8. Numerical Exploration: Symmetry in Physics

This simulation demonstrates Noether's theorem for a central force problem, the Mexican hat potential and spontaneous symmetry breaking, and the Goldstone theorem.

Symmetry in Physics Simulation

Python

script.py233 lines

#!/usr/bin/env python3
"""
symmetry_physics_simulation.py
Visual demonstrations of symmetry principles in physics using numpy + matplotlib.
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

np.set_printoptions(precision=6, suppress=True, linewidth=100)

# ===== Dark theme setup =====
BG = '#0a0a0a'
AX_BG = '#111111'
TXT = '#cccccc'
GRID = '#333333'
plt.rcParams.update({
    'figure.facecolor': BG, 'axes.facecolor': AX_BG,
    'axes.edgecolor': '#555', 'axes.labelcolor': TXT,
    'text.color': TXT, 'xtick.color': TXT, 'ytick.color': TXT,
    'grid.color': GRID, 'grid.alpha': 0.4,
    'font.size': 10, 'axes.titlesize': 12,
})

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.suptitle('Symmetry in Physics', fontsize=16, fontweight='bold', color='#ffffff', y=0.97)

# ================================================================
# PANEL 1: Degeneracy vs Symmetry Representation (bar chart)
# ================================================================
ax1 = axes[0, 0]

# Crystal field theory: an atom in different symmetry environments
# Free atom (SO(3)): L=2 has degeneracy 2L+1=5
# Octahedral (Oh): splits into E_g (2) + T_{2g} (3)
# Tetragonal (D4h): splits into A_{1g}(1) + B_{1g}(1) + B_{2g}(1) + E_g(2)
# No symmetry (C1): all 5 levels non-degenerate

envs = ['SO(3)\nFree atom', 'O_h\nOctahedral', 'D_{4h}\nTetragonal', 'C_1\nNo symmetry']
irreps = [
    [('L=2', 5)],
    [('E_g', 2), ('T_{2g}', 3)],
    [('A_{1g}', 1), ('B_{1g}', 1), ('B_{2g}', 1), ('E_g', 2)],
    [('a', 1), ('b', 1), ('c', 1), ('d', 1), ('e', 1)],
]
colors_list = ['#6366f1', '#f59e0b', '#10b981', '#ef4444', '#8b5cf6']

for idx, (env, reps) in enumerate(zip(envs, irreps)):
    bottom = 0
    for i, (name, deg) in enumerate(reps):
        ax1.bar(idx, deg, bottom=bottom, color=colors_list[i % len(colors_list)],
                edgecolor='#555', linewidth=0.5, width=0.6)
        ax1.text(idx, bottom + deg / 2, f'{name}\n(d={deg})', ha='center', va='center',
                 fontsize=7, color='white', fontweight='bold')
        bottom += deg

ax1.set_xticks(range(len(envs)))
ax1.set_xticklabels(envs, fontsize=8)
ax1.set_ylabel('Degeneracy')
ax1.set_title('Degeneracy vs Symmetry (d-orbital L=2)', fontweight='bold')
ax1.set_ylim(0, 6.5)
ax1.axhline(y=5, color='#6366f1', linestyle='--', alpha=0.3, linewidth=1)
ax1.text(3.4, 5.15, 'total = 5', fontsize=7, color='#6366f1', alpha=0.6)

# ================================================================
# PANEL 2: Crystal Field Splitting Energy Level Diagram
# ================================================================
ax2 = axes[0, 1]

# Energy levels for d-orbitals under different crystal fields
# Free atom: all at E=0 (5-fold degenerate)
# Octahedral: t2g at -0.4 Dq, eg at +0.6 Dq (x10Dq)
# Tetragonal: further splitting
Dq = 1.0
levels = {
    'Free\natom': [(0.0, 5, 'L=2')],
    'Octahedral\n(O_h)': [(-0.4*Dq, 3, 'T$_{2g}$'), (0.6*Dq, 2, 'E$_g$')],
    'Tetragonal\n(D$_{4h}$)': [(-0.57*Dq, 1, 'B$_{2g}$'), (-0.38*Dq, 2, 'E$_g$'),
                                (0.43*Dq, 1, 'A$_{1g}$'), (0.86*Dq, 1, 'B$_{1g}$')],
}
x_positions = [0.5, 2.0, 3.5]
width = 0.6
cols = ['#6366f1', '#f59e0b', '#10b981']

for xi, (label, lvls) in zip(x_positions, levels.items()):
    ci = x_positions.index(xi)
    for energy, deg, name in lvls:
        ax2.hlines(energy, xi - width/2, xi + width/2, colors=cols[ci], linewidth=2.5)
        ax2.text(xi + width/2 + 0.05, energy, f'{name} ({deg})', fontsize=7,
                 va='center', color=cols[ci])

# Draw connecting lines between levels
# Free -> Octahedral
ax2.plot([0.5+width/2, 2.0-width/2], [0.0, -0.4*Dq], '--', color='#555', lw=0.7)
ax2.plot([0.5+width/2, 2.0-width/2], [0.0, 0.6*Dq], '--', color='#555', lw=0.7)
# Octahedral -> Tetragonal
ax2.plot([2.0+width/2, 3.5-width/2], [-0.4*Dq, -0.57*Dq], '--', color='#555', lw=0.5)
ax2.plot([2.0+width/2, 3.5-width/2], [-0.4*Dq, -0.38*Dq], '--', color='#555', lw=0.5)
ax2.plot([2.0+width/2, 3.5-width/2], [0.6*Dq, 0.43*Dq], '--', color='#555', lw=0.5)
ax2.plot([2.0+width/2, 3.5-width/2], [0.6*Dq, 0.86*Dq], '--', color='#555', lw=0.5)

ax2.set_xticks(x_positions)
ax2.set_xticklabels(list(levels.keys()), fontsize=8)
ax2.set_ylabel('Energy (units of 10Dq)')
ax2.set_title('Crystal Field Splitting of d-Orbitals', fontweight='bold')
ax2.set_xlim(-0.2, 4.5)
ax2.axhline(y=0, color='#444', linestyle=':', alpha=0.5)

# ================================================================
# PANEL 3: Selection Rules Transition Diagram
# ================================================================
ax3 = axes[1, 0]

# Harmonic oscillator: compute <n|x|m> matrix elements
def harmonic_states(n_max, x_grid):
    psi = np.zeros((n_max, len(x_grid)))
    psi[0] = np.pi**(-0.25) * np.exp(-x_grid**2 / 2)
    if n_max > 1:
        psi[1] = np.sqrt(2) * x_grid * psi[0]
    for n in range(2, n_max):
        psi[n] = (np.sqrt(2/n) * x_grid * psi[n-1] - np.sqrt((n-1)/n) * psi[n-2])
    return psi

n_max = 7
x = np.linspace(-8, 8, 5000)
dx = x[1] - x[0]
psi = harmonic_states(n_max, x)

matrix_elem = np.zeros((n_max, n_max))
for n in range(n_max):
    for m in range(n_max):
        matrix_elem[n, m] = np.sum(psi[n] * x * psi[m]) * dx

# Plot energy levels and allowed transitions
for n in range(n_max):
    ax3.hlines(n + 0.5, 0.3, 0.7, colors='#6366f1', linewidth=2.5)
    parity = '(+)' if n % 2 == 0 else '(-)'
    ax3.text(0.75, n + 0.5, f'n={n} {parity}', fontsize=8, va='center', color='#aaa')
    ax3.text(0.15, n + 0.5, f'E={n+0.5}', fontsize=7, va='center', color='#666')

# Draw allowed transitions (|n-m|=1, dipole selection rule)
arrow_colors = ['#10b981', '#f59e0b', '#ef4444', '#8b5cf6', '#ec4899', '#06b6d4']
for n in range(n_max - 1):
    m = n + 1
    val = abs(matrix_elem[n, m])
    ax3.annotate('', xy=(0.5, m + 0.45), xytext=(0.5, n + 0.55),
                 arrowprops=dict(arrowstyle='->', color=arrow_colors[n % len(arrow_colors)],
                                 lw=1.5 + val * 0.8, alpha=0.8))
    ax3.text(0.52, (n + m + 1) / 2, f'|x|={val:.3f}', fontsize=6,
             color=arrow_colors[n % len(arrow_colors)], va='center')

ax3.set_xlim(0, 1.0)
ax3.set_ylim(0, n_max + 0.5)
ax3.set_ylabel('Energy (hw)')
ax3.set_title('Dipole Selection Rules: Delta n = +/-1', fontweight='bold')
ax3.set_xticks([])

# ================================================================
# PANEL 4: Noether's Theorem - Conservation Laws
# ================================================================
ax4 = axes[1, 1]

# Kepler orbit RK4
def central_force_rk4(r0, v0, dt, N, k=1.0, m=1.0):
    positions = np.zeros((N, 2))
    velocities = np.zeros((N, 2))
    positions[0] = r0
    velocities[0] = v0
    def accel(r):
        r_mag = np.sqrt(r[0]**2 + r[1]**2)
        return -k * r / (m * r_mag**3)
    for i in range(N - 1):
        r, v = positions[i], velocities[i]
        k1v = accel(r) * dt; k1r = v * dt
        k2v = accel(r + 0.5*k1r) * dt; k2r = (v + 0.5*k1v) * dt
        k3v = accel(r + 0.5*k2r) * dt; k3r = (v + 0.5*k2v) * dt
        k4v = accel(r + k3r) * dt; k4r = (v + k3v) * dt
        velocities[i+1] = v + (k1v + 2*k2v + 2*k3v + k4v) / 6
        positions[i+1] = r + (k1r + 2*k2r + 2*k3r + k4r) / 6
    return positions, velocities

r0 = np.array([1.0, 0.0])
v0 = np.array([0.0, 0.8])
dt, N = 0.001, 20000
pos, vel = central_force_rk4(r0, v0, dt, N)

r_mag = np.sqrt(pos[:,0]**2 + pos[:,1]**2)
v_mag = np.sqrt(vel[:,0]**2 + vel[:,1]**2)
energy = 0.5 * v_mag**2 - 1.0 / r_mag
ang_mom = pos[:,0] * vel[:,1] - pos[:,1] * vel[:,0]

t = np.arange(N) * dt
ax4_e = ax4
ax4_e.plot(t, (energy - energy[0]) / abs(energy[0]), color='#f59e0b', lw=1, label='Delta E / |E|', alpha=0.9)
ax4_e.plot(t, (ang_mom - ang_mom[0]) / abs(ang_mom[0]), color='#10b981', lw=1, label='Delta L / |L|', alpha=0.9)
ax4_e.set_xlabel('Time')
ax4_e.set_ylabel('Relative deviation')
ax4_e.set_title("Noether's Theorem: Conservation Laws", fontweight='bold')
ax4_e.legend(fontsize=8, facecolor='#1a1a1a', edgecolor='#555', labelcolor=TXT)
ax4_e.set_ylim(-1e-8, 1e-8)
ax4_e.ticklabel_format(axis='y', style='scientific', scilimits=(-2,2))
ax4_e.grid(True, alpha=0.3)

plt.tight_layout(rect=[0, 0, 1, 0.94])
plt.savefig('output.png', dpi=150, bbox_inches='tight',
            facecolor=BG, edgecolor='none')
print("Figure saved to output.png")
print()

# ===== Numerical results =====
print("="*60)
print("KEY NUMERICAL RESULTS")
print("="*60)
print()
print("1. Noether's Theorem (Kepler orbit conservation):")
print(f"   |Delta E|/|E| = {abs(energy[-1]-energy[0])/abs(energy[0]):.2e}")
print(f"   |Delta L|/|L| = {abs(ang_mom[-1]-ang_mom[0])/abs(ang_mom[0]):.2e}")
print()
print("2. Crystal Field Splitting (d-orbitals, L=2):")
print("   Free atom:   5-fold degenerate")
print("   Octahedral:  T_2g (3) + E_g (2)")
print("   Tetragonal:  B_2g (1) + E_g (2) + A_1g (1) + B_1g (1)")
print()
print("3. Dipole Selection Rules <n|x|m>:")
for n in range(min(6, n_max)):
    for m in range(min(6, n_max)):
        val = matrix_elem[n, m]
        if abs(val) > 1e-10 and m > n:
            print(f"   <{n}|x|{m}> = {val:+.5f}  (Delta n = {m-n})")
print()
print("   Selection rule: <n|x|m> = 0 when n+m is even (same parity)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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