Stark effect in neutral hydrogen by direct integration of the Hamiltonian in parabolic coordinates

We present a theoretical study to determine the energies, widths, and wave functions of the neutral hydrogen atom under a constant electric field by the direct integration of the Hamiltonian in parabolic coordinates. We work in terms of the complex coordinate rotation to distinguish the resonances from the continuum sea, and the wave functions are expanded in a basis set of Laguerre-mesh polynomials. We obtain ab initio results for the first five atomic shells of neutral hydrogen for a field intensity up to ${10}^9$ V/m.

Figure 1

Diagram showing the behavior of a root in the hydrogen atom under a constant electric field.

Figure 2

Obtained eigenvalues for a basis set $N=30$ for $m=0$ and a field intensity $F=0.0020\mathrm{a}.\mathrm{u}.$ Some found resonances are marked.

Figure 3

Calculated energies for the $n=2$–5 shells of the hydrogen atom vs the field intensity.

Figure 4

Calculated widths for the $n=2$–5 shells of the hydrogen atom vs the field intensity.

Figure 5

Wave function of the Stark states $|100\rangle$ and $|201\rangle$ for a field intensity of $F=0.0020\mathrm{a}.\mathrm{u}.$

Figure 6

Wave function of the Stark states $|2-10\rangle$ and $|210\rangle$ for a field intensity of $F=0.0020\mathrm{a}.\mathrm{u}.$

Figure 7

Wave function of the Stark states $|3-20\rangle$ and $|320\rangle$ for a field intensity of $F=0.0020\mathrm{a}.\mathrm{u}.$

Figure 8

Wave function of the Stark states $|3-11\rangle$ and $|311\rangle$ for a field intensity of $F=0.0020\mathrm{a}.\mathrm{u}.$

Figure 9

Wave function of the Stark states $|300\rangle$ and $|302\rangle$ for a field intensity of $F=0.0020\mathrm{a}.\mathrm{u}.$

Figure 10

Stark splitting of the ${\mathrm{D}}_{\alpha }$ line of deuterium vs the electric field intensity.

Figure 11

Einstein spontaneous emission coefficients of neutral hydrogen vs the electric field intensity. The values for the Rydberg hydrogen atom for zero field intensity are marked.

Figure 12

Emission profile of the Balmer ${\mathrm{D}}_{\alpha }$ and ${\mathrm{H}}_{\alpha }$ lines for two different field intensities in corona equilibrium.