Worked example: First-order Stark shift — a hydrogen atom in state |100⟩ placed in a uniform electric field E along z. Estimate the energy shift. Step 1: Perturbation V̂ = eEz (electron charge −e times potential −Ez gives +eEz; sign conventions vary). Step 2: First-order shift E¹ = ⟨100|eEz|100⟩ = eE⟨100|z|100⟩. Step 3: The 1s state is spherically symmetric: ⟨z⟩ = 0, so E¹ = 0 — no linear Stark shift for the ground state. Step 4: Second order is non-zero: the field mixes |100⟩ with |n10⟩ states, giving E² = −½αE² (induced dipole energy, α = polarizability). Conclusion: symmetry can kill first order entirely — the atom has no permanent dipole, only an induced one; excited states (n = 2) do show linear Stark splitting, a classic perturbation result.