Energy levels calculator
See quantization in nature: pick two levels of a hydrogen-like atom to get the photon's energy, wavelength and colour, or confine a particle to a box and watch its allowed energies.
Atomic transition
Famous lines
Bohr-model values with an infinitely heavy nucleus. Including the reduced mass of the electron shifts hydrogen lines by about 0.05%, so Hα comes out at 656.1 nm here versus about 656.5 nm measured in vacuum.
Particle in a box
Constants: h = 6.62607015 × 10⁻³⁴ J·s (exact in the SI since 2019), mₑ and mₚ from CODATA 2022. An electron in a 1 nm box has a ground-state energy of about 0.376 eV.
Why the answers come in steps
An electron bound to a nucleus, or a particle trapped between walls, behaves as a wave that must fit its boundaries. Only certain standing-wave patterns fit, and each has a definite energy. Change level and the energy difference leaves or arrives as a single photon with E = hf = hc/λ. That is why hydrogen glows at a few sharp wavelengths instead of a smooth rainbow. Read the full story in quantization in physics.
Frequently asked questions
Why is the ground-state energy of hydrogen −13.6 eV?
The negative sign means the electron is bound: it takes 13.6 eV to pull it free from the ground state. Higher levels are less tightly bound, approaching 0 eV as n grows.
Which hydrogen lines are visible?
The Balmer series, which ends on level 2. Its first four lines are near 656, 486, 434 and 410 nm: red, blue-green and two violets.
How is this related to quantization in AI or audio?
Only by analogy. In computing we choose to round values onto a grid; in physics the discreteness comes from nature. Both produce a ladder of allowed values with gaps in between.