Hydrogen and hydrogen-like atoms

Hydrogen.py module

Bohr model / analytical solution of the Schrödinger equation for a one-electron system (H, He⁺, Li²⁺, …), with an optional finite nuclear-mass (reduced-mass) correction. Covers energy levels, ionization energies, spectral series, and their visualization as energy-level diagrams and line spectra.

Note: this module covers energy levels, ionization energies, and spectra (lines and diagrams). The atomic orbitals themselves — radial parts, angular parts, and their 2D/3D visualization — are in the separate AO.py module.

Quick Start

from pyphyschemtools import (
    rydberg_Z, En_hydrogenic, Ei_hydrogenic,
    spectral_series, plot_energy_levels, plot_line_spectrum
)

# Ground-state energy of hydrogen, in eV
En_hydrogenic(1)

# Ionization energy of He+ (Z=2)
Ei_hydrogenic(Z=2)

# Print the Lyman to Humphreys series, transitions up to n2=12
spectral_series(n2_max=12)

# Energy-level diagram, absorption transitions
plot_energy_levels(n2_max=6, transition_type="absorption")

# Line spectrum on a log wavelength axis
plot_line_spectrum(n2_max=12)

ou bien

import pyphyschemtools.hydrogen as H
H.Ei_hydrogenic(Z=1)
H.plot_line_spectrum(n2_max=12)

Reduced-mass correction

All energy calculations go through a Rydberg constant that can be corrected for the finite mass of the nucleus:

\[R_M = R_\infty \frac{\mu}{m_e}, \quad \mu = \frac{m_e M_{\text{nucleus}}}{m_e + M_{\text{nucleus}}}\]

rydberg_Z(Z=1, A=None, M_atom=None) returns \(R_M\) in m⁻¹. The nuclear mass is looked up via mendeleev for isotope A of element Z (the most abundant isotope by default), or can be given directly as M_atom (kg), bypassing the lookup entirely.
Every other function accepts reduced_mass=True (default, uses \(R_M\)) or reduced_mass=False (uses \(R_\infty\), the infinite-nucleus limit — A and M_atom are then ignored).
The correction is small (~10⁻⁴ relative) and shrinks as the nucleus gets heavier; it is included mainly for pedagogical completeness rather than to close the gap with experiment, which — for Z > 1 — is dominated by relativistic and QED effects that this module does not include.

Energy levels

\[E_n = -h c R \frac{Z^2}{n^2}\]

En_hydrogenic(n, Z=1, A=None, M_atom=None, reduced_mass=True, unit='eV') returns the (negative, bound-state) energy of level \(n\) of a hydrogenic atom/ion of nuclear charge \(Z\), in joules or eV.

Ionization energy

\[E_i = h c R Z^2\]

Ei_hydrogenic(Z, A=None, M_atom=None, reduced_mass=True) returns the energy (eV) needed to remove the single electron from the ground state to the continuum. The \(Z\)-th ionization energy of an element matches the hydrogen-like ion of nuclear charge \(Z\), which makes this a convenient reference point against tabulated ionization energies.

Spectral series

Named series are keyed by their lower level \(n_1\):

\(n_1\)	Name	Short
1	Lyman	Ly
2	Balmer	Ba
3	Paschen	Pa
4	Brackett	Br
5	Pfund	Pf
6	Humphreys	Hu

For \(Z > 1\) these names are historically defined for hydrogen only and are used “by analogy” — printed and labeled with a “-like” suffix (e.g. the Lyman-like series of He⁺).

spectral_series(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=12, series_names=None) prints, for each series in series_names (default: SERIES_NAMES, Lyman to Humphreys), its series limit (\(n_2 \to \infty\)) and every transition \(n_2 \to n_1\) with its energy (eV), wavelength (nm), and frequency (Hz).

Energy-level diagrams

plot_energy_levels(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=7, series_names=None, to_scale=True, transition_type="absorption", size=(9, 11), show_level_labels=True, show_transition_labels=True, save_as=None) draws:

a horizontal line for each level \(n = 1 \ldots n_2^{\max}\), labeled with its energy;
the ionization limit at 0 eV, with the continuum of free-electron states shaded above it;
the transitions of each series as colored arrows labeled with Greek letters (\(\alpha, \beta, \ldots\), up to \(\epsilon\) — there is no standard name for higher members).

Two display modes, controlled by to_scale:

to_scale=True (default): levels sit at their true eV energy. Since \(E_n \propto 1/n^2\), levels \(n \geq 2\) crowd together near the ionization limit while \(n=1\) sits far below — itself a useful illustration of why the spectrum bunches up with increasing \(n\).
to_scale=False: levels are evenly spaced by rank — the usual simplified textbook diagram, more readable for \(n \geq 2\) but no longer a true energy axis.

transition_type controls both arrow direction and layout:

"absorption" (default): arrows point up (\(n_1 \to n_2\)), ordered left to right by increasing transition size (α first).
"emission": arrows point down (\(n_2 \to n_1\), the electron falling back to \(n_1\)), ordered left to right by decreasing transition size, all converging on the same \(n_1\).

Figures are saved through pyphyschemtools.core.save_fig (missing directories created automatically).

Line spectra

plot_line_spectrum(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=12, series_names=None, show_visible=True, log_scale=True, ax=None) draws a wavelength axis (nm) with one colored vertical line per transition \(n_2 \to n_1\), colored by series, the α-line of each series labeled (e.g. “Ly-α”), and — by default — the visible range (380–780 nm) shaded and a log wavelength scale (the series span several decades, from ~100 nm to several microns).

API reference

hydrogen.py — hydrogen atom and hydrogenic ions

Bohr model / analytical solution of the Schrodinger equation for a one-electron system (H, He+, Li2+, …), with an optional finite nuclear-mass (reduced-mass) correction.

Public functions: - rydberg_Z : Rydberg constant corrected for the reduced mass - En_hydrogenic : energy of level n (in joules) - Ei_hydrogenic : ionization energy (in eV)

pyphyschemtools.hydrogen.Ei_hydrogenic(Z, A=None, M_atom=None, reduced_mass=True)[source]

Ionization energy (in eV) of a hydrogenic (hydrogen-like) ion of nuclear charge Z, i.e. the energy to remove its single electron from the ground state (n=1 to infinity). Ei = h*c*R*Z^2, the opposite of the ground-state energy.

input:

Z = nuclear charge / atomic number (1 = H, 2 = He+, 3 = Li2+, …)
A = mass number of the isotope (for the mendeleev mass lookup)
M_atom = atomic mass in kg (overrides the lookup)
reduced_mass = if True (default), R = R_M is corrected for the finite nuclear mass of the isotope (see rydberg_Z); if False, R = Rinf is the infinite-nucleus limit (A and M_atom are then ignored)

output:

ionization energy of the hydrogen-like ion, in eV

note: this is a non-relativistic Bohr/Schrodinger result. The reduced-mass correction is tiny (~1e-4 relative) and shrinks as Z grows, whereas the gap to experiment grows with Z because relativistic and QED effects (absent here) scale steeply with Z. To compare with mendeleev, the Z-th ionization energy of the element matches the hydrogen-like ion of charge Z.

pyphyschemtools.hydrogen.En_hydrogenic(n, Z=1, A=None, M_atom=None, reduced_mass=True, unit='eV')[source]

Energy of level n of a hydrogenic (hydrogen-like) atom or ion (Bohr model / analytical solution of the Schrodinger equation), in joules or eV.

E_n = -h*c*R * Z^2 / n^2, with R the Rydberg constant. With reduced_mass=True (default) R = R_M is corrected for the finite nuclear mass of the isotope (see rydberg_Z); with reduced_mass=False R = Rinf is the infinite-nucleus limit. Equivalently E_n = -Ei * Z^2 / n^2 with Ei = h*c*R.

input:

n = principal quantum number (integer > 0)
Z = atomic number (default: 1 for hydrogen)
A = mass number of the isotope (for the mendeleev mass lookup)
M_atom = atomic mass in kg (overrides the lookup)
reduced_mass = if True (default), use the reduced-mass-corrected R_M; if False, use the infinite-nucleus constant Rinf
unit = ‘J’ or ‘eV’ (default)

output:

energy of level n, in joules or eV (negative value, bound state)

pyphyschemtools.hydrogen.plot_energy_levels(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=7, series_names=None, to_scale=True, transition_type='absorption', size=(9, 11), show_level_labels=True, show_transition_labels=True, save_as=None)[source]

Draw an energy-level diagram of a hydrogenic atom/ion of nuclear charge Z: a horizontal line for each level n = 1 .. n2_max (labeled “n = …” on the right, against a true energy axis in eV on the left), a dashed line for the ionization limit at 0 eV with the continuum of free-electron states shaded above it, and the transitions n2 <-> n1 of each series in series_names, drawn as colored arrows and labeled with Greek letters (alpha, beta, …) in the series’ color.

By default (to_scale=True) the levels are placed at their true energy (eV) on the left-hand axis: since E_n scales as 1/n^2, n=2..n2_max then crowd together near the ionization limit while n=1 sits far below – itself a good illustration of why the levels bunch up as n increases. With to_scale=False the levels are evenly spaced by rank instead (the usual simplified “schematic” textbook diagram), which keeps the n>=2 levels readable but no longer reflects the true energy axis.

transition_type controls both the arrow direction and, within a series, the left-to-right layout: “absorption” (default) draws arrows pointing up (n1 -> n2), ordered left to right by increasing transition size (alpha first); “emission” draws arrows pointing down (n2 -> n1, the electron falling back to n1), ordered left to right by decreasing transition size, all arriving at the same n1.

input:

Z = nuclear charge / atomic number (default: 1 for H)
A = mass number of the isotope (for the mass lookup)
M_atom = atomic mass in kg (overrides the lookup)
reduced_mass = if True (default), use the reduced-mass-corrected R_M; if False, use the infinite-nucleus constant Rinf
n2_max = highest principal quantum number shown
series_names = {n1: name} mapping of lower levels to draw; defaults to SERIES_NAMES (Lyman to Humphreys, n1 = 1 to 6)
to_scale = if True (default), levels are placed at their true energy on the left-hand eV axis; if False, they are evenly spaced by rank instead (schematic)
transition_type = “absorption” (default) or “emission”; see above
show_level_labels = if True (default), label every level “n = … (… eV)” on the right; if False, only n=1 to 5 are labeled (the others still get their line, just no text). The ionization limit label is kept either way.
show_transition_labels = if True (default), label each transition with its Greek letter, alpha to epsilon only (the first five transitions of a series) – there is no standard name for higher members, so none is invented and those stay unlabeled regardless of this setting. If False, no per-transition label is drawn at all (the arrows and series names are unaffected).
size = (width, height) in inches for a new figure
save_as = file path to save the figure to (e.g. “levels.png” or “levels.svg”), passed to pyphyschemtools.core’s save_fig (format taken from the extension, missing directories created automatically). Not saved if None (default)

pyphyschemtools.hydrogen.plot_line_spectrum(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=12, series_names=None, show_visible=True, log_scale=True, ax=None)[source]

Draw the line spectrum of a hydrogenic atom/ion of nuclear charge Z: a wavelength axis (nm) with one colored vertical line per transition n2 –> n1, colored by series, and the alpha line of each series labeled with an arrow (e.g. “Ly-alpha”).

input:

Z = nuclear charge / atomic number (default: 1 for hydrogen)
A = mass number of the isotope (for the mendeleev mass lookup)
M_atom = atomic mass in kg (overrides the lookup)
reduced_mass = if True (default), use the reduced-mass-corrected R_M; if False, use the infinite-nucleus constant Rinf
n2_max = highest principal quantum number n2 included
series_names = {n1: name} mapping of lower levels to draw; defaults to SERIES_NAMES (Lyman to Humphreys, n1 = 1 to 6)
show_visible = if True (default), shade the visible range (380-780 nm)
log_scale = if True (default), use a log wavelength axis (the series span several decades, from ~100 nm to several microns)
ax = matplotlib Axes to draw on; a new figure is created if None (default)

output:

the matplotlib Axes used

pyphyschemtools.hydrogen.rydberg_Z(Z=1, A=None, M_atom=None)[source]

Rydberg constant (in m^-1) corrected for the finite nuclear mass of the given isotope: R = Rinf * M_nucleus / (me + M_nucleus).

Note: the value depends on the nuclear mass only, not on Z. The Z^2 factor belongs to the level/transition formula, not to the Rydberg constant itself.

input:

Z = atomic number (default: 1 for hydrogen); used to look up the element and, together with M_atom, to get the nuclear mass
A = mass number of the isotope to use. If None (default), the most abundant isotope of element Z (highest abundance among its isotopes with known abundance) is used
M_atom = atomic mass in kg; overrides the mendeleev lookup entirely (A is then ignored)

output:

Rydberg constant corrected for the reduced mass, in m^-1

pyphyschemtools.hydrogen.spectral_series(Z=1, A=None, M_atom=None, reduced_mass=True, n2_max=12, series_names=None)[source]

Print the spectral series of a hydrogenic atom/ion of nuclear charge Z: for each n1 in series_names, the series limit (as n2 tends to infinity) and every transition n2 –> n1 for n1 < n2 <= n2_max, …, with the transition energy (eV), wavelength (nm) and frequency (Hz).

The series names (Lyman, Balmer, …) were historically defined for the hydrogen atom (Z=1). For Z>1 they are only used “by analogy” (the term structure is the same, just scaled by Z^2; e.g. the “Lyman-like” series of He+ is sometimes called its Lyman series in the literature) – printed here with a “-like” suffix to make that explicit.

input:

Z = nuclear charge / atomic number (default: 1 for hydrogen)
A = mass number of the isotope (for the mendeleev mass lookup)
M_atom = atomic mass in kg (overrides the lookup)
reduced_mass = if True (default), use the reduced-mass-corrected R_M; if False, use the infinite-nucleus constant Rinf
n2_max = highest principal quantum number n2 to print
series_names = {n1: name} mapping of lower levels to print; defaults to SERIES_NAMES (Lyman to Humphreys, n1 = 1 to 6)

output:

None (prints to stdout)