Hydrogen Atom

Energy levels, orbitals and spectral transitions

Parameters
Energy Diagram
Bohr Model

Energy levels of hydrogen-like atoms follow from Bohr quantization. Z is the nuclear charge (H=1, He⁺=2, Li²⁺=3...).

$$ E_n = -\frac{Z^2 \cdot 13.6\,\text{eV}}{n^2}, \quad n = 1, 2, 3, \ldots $$
Schrödinger Equation

The full wave function separates into radial and angular parts:

$$ \psi_{nlm}(r,\theta,\phi) = R_{nl}(r)\cdot Y_l^m(\theta,\phi) $$

Quantum numbers: n=1,2,3... (principal); l=0..n-1 (orbital: s,p,d,f); m=-l..+l (magnetic).

Radial Function
$$ R_{nl}(r) = \sqrt{\left(\frac{2Z}{na_0}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}}\, e^{-Zr/na_0} \left(\frac{2Zr}{na_0}\right)^l L_{n-l-1}^{2l+1}\!\!\left(\frac{2Zr}{na_0}\right) $$

L — associated Laguerre polynomial. a₀=0.0529 nm — Bohr radius.

Selection Rules

Electromagnetic transitions are only allowed for specific changes in quantum numbers:

$$ \Delta n \text{ — ნებისმიერი}, \quad \Delta l = \pm 1, \quad \Delta m = 0, \pm 1 $$
Spectral Series
SeriesLower levelλ (nm)Region
Lyman191–122UV
Balmer2365–656VIS/UV
Paschen3820–1875IR
Brackett41458–4051IR
Step-by-Step Guide
1
Energy Levels
Choose atom (Z) and number of levels. For H: E₁=-13.6 eV, E₂=-3.4 eV, ...
2
Orbital Wave Functions
In the Wave Functions tab — choose n and l. Available: 1s, 2s, 2p, 3s, 3p, 3d orbitals.
3
Radial Probability
The r²|R|² plot shows where the electron is most likely to be found.
4
Spectrum
In the Spectrum tab — Lyman, Balmer, Paschen series lines. Balmer series is visible light.
5
Compare He⁺, Li²⁺
Increasing Z → energies scale as Z², orbitals become smaller.
Interpreting Results

n=1 (1s) — ground state, most stable. n→∞ — ionization (E=0). Number of radial nodes: n-l-1. 1s orbital — Gaussian shape, 2p — node at center.

Self-Assessment

Q1. What is the ground state energy of the hydrogen atom?

A 0 eV
B -13.6 eV
C -3.4 eV
D +13.6 eV

Q2. How many orbitals exist for n=2 (counting l and m)?

A 2
B 3
C 4
D 6

Q3. The Balmer series corresponds to transitions to which level?

A n→1
B n→3
C n→4
D n→5

Q4. He⁺ (Z=2) ground state energy compared to H?

A 2× larger
B 2× smaller
C same
D 4× larger