Looking at a plot of the electron probability density for the hydrogen state

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Radiative transitions, external fields and collisions

Reading the Spectrum, Ch Atoms in an electric field: the Stark effect and polarizability

Problem

Looking at a plot of the electron probability density ∣ψ∣2|\psi|^2 for the hydrogen 2p02p_0 state, someone argues from its shape. It is shaped like a dumbbell, with two lobes stretched out along the zz-axis, one above the nucleus and one below. The argument runs: "the cloud reaches a long way out along zz, so its charge must sit mostly away from the nucleus, and the atom must have a permanent electric dipole moment along zz, even with no field applied." ((a)) Is this reasoning correct? Justify your answer using the symmetry of ∣ψ∣2|\psi|^2 under reflection through the nucleus, r⃗→−r⃗\vec r \to -\vec r. ((b)) Show that the first-order shift eE⟨z⟩eE\langle z\rangle of the 2p02p_0 state taken on its own is zero in a weak uniform field EE along zz. Then explain why hydrogen's n=2n = 2 level nevertheless shifts linearly in that field, naming the state that 2p02p_0 is mixed with and the shifts that result.

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