Hydrogen Energy Levels Calculator

For single-electron systems like hydrogen, singly ionized helium, or doubly ionized lithium, the electron's energy levels depend primarily on the nuclear charge and the principal quantum number. The underlying equation used in this calculator is derived from relativistic quantum mechanics and fine-structure approximations: Energy = - (1/137.036)^2 * 510,998 * (atomic_number^2 / (2 * n^2)). Here, the atomic number represents the number of protons in the nucleus, while n denotes the principal quantum number corresponding to the electron shell distance from the nucleus.

Let us calculate the energy level for a singly ionized helium atom (Helium-4). For helium, the atomic number is 2, and let us evaluate it at the first excited state where the principal quantum number n equals 2. Substituting these values into our formula: Energy = - (1/137.036)^2 * 510,998 * (2^2 / (2 * 2^2)). Simplifying the nuclear charge term (4 / 8), we get 0.5. Multiplying this by the rest mass energy constant and the fine structure constant square yields an energy level of approximately -13.6 electron-volts, perfectly matching theoretical quantum expectations.

When working with single-electron atomic systems, ensure you accurately identify the principal quantum number n as an integer greater than or equal to 1. Remember that higher values of n place electrons further from the nucleus, resulting in energy levels that approach zero. Always verify your input atomic number against the specific ion you are modeling to avoid calculation errors.