A hydrogen atom in the level sits in a uniform electric field

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

A hydrogen atom in the n=2n=2 level sits in a uniform electric field E=5.0×106 V m−1E = 5.0\times10^{6}\,\mathrm{V\,m^{-1}}. The field mixes the degenerate 2s2s and 2p02p_0 states into the combinations (2s±2p0)/2(2s\pm2p_0)/\sqrt2, which carry dipole moments ±3ea0\pm3ea_0 once the field has selected them, so their energies shift by ∓3ea0E\mp3ea_0E relative to the field-free level. ((a)) Calculate the magnitude of this shift in eV, in cm−1\mathrm{cm^{-1}} and in GHz. ((b)) The hydrogen 1s1s ground state has polarizability α=7.42×10−41 C m2 V−1\alpha = 7.42\times10^{-41}\,\mathrm{C\,m^2\,V^{-1}}. Calculate its quadratic shift ΔE=−12αE2\Delta E = -\tfrac12\alpha E^2 at the same field, in eV. ((c)) Find the ratio of the magnitude of the n=2n=2 linear shift to the magnitude of the 1s1s quadratic shift, and comment on why the two levels respond so differently to the same field.

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