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4.1.3. One-Dimensional Hydrogen Atom, as a Limiting Situation

The behavior of the ground state energy of the confined two-dimensional hydrogen atom, when one dimension becomes infinitely small, will be analyzed in this section.

If the nucleus of the atom is located at the origin and the confining region is a symmetrical narrow strip of width 2y0, the Hamiltonian for the system is (in atomic units)

where Vb(y) is the confining potential, defined as

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