Hyperfine Splitting of the Dressed Hydrogen Atom Ground State in Non-Relativistic Qed

Abstract

Abstract. We consider a spin-1 2 electron and a spin-1 2 nucleus interacting with the quantized electromagnetic field in the standard model of non-relativistic QED. For a fixed total momentum sufficiently small, we study the multiplicity of the ground state of the reduced Hamiltonian. We prove that the coupling between the spins of the charged particles and the electromagnetic field splits the degeneracy of the ground state.

Hyperfine Splitting in Non-relativistic QED: Uniqueness of the Dressed Hydrogen Atom Ground State

“…In [1], using a contradiction argument, we proved under this assumption that the multiplicity of the ground state is strictly less than 4. This shows that a hyperfine splitting does occur in non-relativistic QED (we refer the reader to the introduction of [1] for a more detailed discussion on the hyperfine structure of Hydrogen). Refining our previous analysis, we shall prove in the present paper that the ground state of the dressed Hydrogen atom is unique.…”

“…It suffices to introduce the expression (1.8) of W g,≥σ (P ) into the operator Π 0,≥σ W g,≥σ (P ) K 0,≥σ (P ) − E g,≥σ (P ) −1 Π0,≥σ W g,≥σ (P )Π 0,≥σ , and next to estimate each term separately. An explicit computation then leads directly to the statement of the lemma (see the proof of Lemma A.9 in [1] for more details).…”

“…We recall the definition of the Hamiltonian associated with a freely moving hydrogen atom at a fixed total momentum P in non-relativistic QED. For more details, we refer to [1,Section 2].…”

“…If the electron spin is taken into account and the nucleus spin is neglected, then the ground state is twice degenerate [20]. Now, following the physical prescription, we assume that both the electron and the nucleus have a spin equal to 1 2 . In [1], using a contradiction argument, we proved under this assumption that the multiplicity of the ground state is strictly less than 4.…”

“…Now, following the physical prescription, we assume that both the electron and the nucleus have a spin equal to 1 2 . In [1], using a contradiction argument, we proved under this assumption that the multiplicity of the ground state is strictly less than 4. This shows that a hyperfine splitting does occur in non-relativistic QED (we refer the reader to the introduction of [1] for a more detailed discussion on the hyperfine structure of Hydrogen).…”