Hydrogen Energy Levels Calculator
Enter a principal quantum number (n) to get the electron energy in electronvolts and joules using the Bohr model. Choose an upper and lower level to compute the transition energy, photon wavelength, frequency, and spectral series. Works for hydrogen and one-electron ions such as He+, Li2+, and Be3+. Results update as you type.
The Bohr model and hydrogen energy levels
In 1913, Niels Bohr proposed that the electron in a hydrogen atom occupies discrete circular orbits, each with a fixed energy. The energy of the nth level is E_n = -(13.606 eV) / n^2 for hydrogen. Negative energy means the electron is bound: it takes work to pull it free. At n=1 (the ground state) the energy is about -13.6 eV, the lowest (most stable) level. As n increases the energy rises toward zero; at n = infinity the electron is no longer bound and the atom is ionised. Each level is called a shell, labelled K (n=1), L (n=2), M (n=3) and so on. Although the full quantum-mechanical picture (Schrodinger equation, quantum numbers l, m, s) reveals structure the Bohr model misses, the energy formula remains exact for hydrogen and any species with exactly one electron.
Hydrogen-like ions and the role of atomic number Z
Any atom or ion with a single electron (He+, Li2+, Be3+, B4+, C5+ ...) obeys the same formula with Z inserted: E_n = -(Z^2 * 13.606 eV) / n^2. Because Z appears squared, helium-ion (Z=2) levels are four times deeper than hydrogen levels at the same n; lithium (Z=3) levels are nine times deeper. This means hydrogen-like ions require much more energy to ionise from their ground state, and their spectral lines shift toward shorter (more energetic) wavelengths. Astrophysicists routinely observe highly charged hydrogen-like species in stellar atmospheres and plasmas.
Spectral series and photon emission
When an electron drops from a higher level n2 to a lower level n1, it releases a photon whose energy equals the difference: delta-E = |E_n2 - E_n1|. The Rydberg formula gives the photon wavelength directly: 1/lambda = R_H * Z^2 * (1/n1^2 - 1/n2^2), where R_H is the Rydberg constant (1.097e7 m^-1). Transitions ending at n=1 form the Lyman series (ultraviolet); those ending at n=2 form the Balmer series (mostly visible light, the familiar red H-alpha at 656 nm, green H-beta at 486 nm); those ending at n=3 form the Paschen series (infrared). Absorption works in reverse: a photon of exactly the right wavelength can kick the electron up to a higher level.
Practical uses: from neon signs to astrophysics
Hydrogen spectral lines appear in an enormous range of applications. The Balmer series gives hydrogen discharge tubes their characteristic pink glow and gives emission nebulae their red hue (H-alpha at 656 nm). Astronomers use the 21 cm radio line (hyperfine, not electronic) to map galactic hydrogen, and use UV Lyman-alpha to probe the intergalactic medium. In the laboratory, Rydberg atoms (n > 20) are used in quantum computing experiments because their loosely bound electrons are extremely sensitive to electric fields. Laser cooling of hydrogen also relies on precision knowledge of these transitions. For students, the hydrogen spectrum was the first quantitative evidence that atoms have discrete energy levels, launching the era of quantum mechanics.
Hydrogen spectral series (Z=1)
| Series | Lower level n1 | EM region | Wavelength range |
|---|---|---|---|
| Lyman | 1 | Ultraviolet | 91 - 122 nm |
| Balmer | 2 | Visible / near-UV | 365 - 656 nm |
| Paschen | 3 | Near-infrared | 820 - 1875 nm |
| Brackett | 4 | Infrared | 1460 - 4051 nm |
| Pfund | 5 | Mid-infrared | 2279 - 7460 nm |
| Humphreys | 6 | Far-infrared | 3282 - 12372 nm |
The six named series of hydrogen spectral lines, their lower-level quantum number, electromagnetic region, and the wavelength range.
Frequently asked questions
What does a negative energy level mean?
A negative energy means the electron is bound to the nucleus: you must supply energy to free it. The zero of energy is defined as the electron infinitely far from the nucleus (unbound). At n=1, the electron is at -13.6 eV, so you need at least 13.6 eV to ionise a ground-state hydrogen atom. As n grows, the electron is farther out and less tightly held, so the energy is less negative, approaching zero.
How does the calculator handle He+, Li2+ and other hydrogen-like ions?
The Bohr energy formula scales as Z^2, where Z is the number of protons. Selecting He+ (Z=2) multiplies every energy level by 4; Li2+ (Z=3) multiplies by 9. The photon energies scale the same way, so all spectral lines shift toward shorter wavelengths compared with neutral hydrogen.
What is the Rydberg constant and how is it related to the 13.6 eV ground state?
The Rydberg energy R_inf = m_e * e^4 / (8 * epsilon_0^2 * h^2) equals approximately 13.6058 eV. When n=1 and Z=1 (hydrogen ground state), E_1 = -R_inf = -13.6058 eV. The Rydberg constant quoted as a wavenumber (R_H = 1.0974e7 m^-1) is just R_inf divided by h*c.
How do I identify the spectral series from the quantum numbers?
The series is determined by the lower quantum number n1: n1=1 is Lyman (UV), n1=2 is Balmer (visible/near-UV), n1=3 is Paschen (near-IR), n1=4 is Brackett (IR), n1=5 is Pfund (mid-IR), n1=6 is Humphreys (far-IR). The upper number n2 can be any integer greater than n1; larger n2 means a shorter wavelength within the same series.
What is the H-alpha line and why is it famous?
H-alpha is the n=3 to n=2 transition in neutral hydrogen, emitting a photon at 656 nm (red). It is the brightest visible hydrogen line, responsible for the red colour of emission nebulae such as the Orion Nebula. In stellar spectroscopy, H-alpha absorption appears as a dark line in stellar atmospheres and is used to measure stellar radial velocities via Doppler shift.
Can I use this calculator for non-hydrogen elements?
Only for one-electron species (hydrogen-like ions). Once an atom has two or more electrons, electron-electron repulsion breaks the simple Z^2/n^2 pattern and you need quantum-mechanical perturbation theory or computation. This calculator covers H, He+, Li2+, Be3+, B4+, and C5+ only.