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Hydrogen-Like Bound-State Energy for an Ideal Nucleus

Calculate negative bound-state energy for a one-electron, infinite-mass point nucleus, with editable integer n and nuclear charge Z.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For Z = 1 and n = 1, the infinite-mass nonrelativistic bound energy is approximately −13.605693123 eV. At n = 2 it is approximately −3.401423281 eV.

En = −hcR∞ Z²/n². Energy zero is the free electron at infinite separation; R∞ = 10,973,731.568157 m⁻¹ (2022 CODATA central value).

Worked example

Enter these known values and leave the other values blank.

Principal quantum number n
1
Nuclear charge number Z
1
Ideal bound-state energy
-13.606 eV

Assumptions and limitations

  • The model has one electron and an ideal point nucleus of infinite mass. It is the nonrelativistic Coulomb model; it omits reduced mass, isotope differences, relativistic effects, finite nuclear size, fine structure and external fields.
  • The negative result is binding-state energy relative to a separated electron. Its magnitude is the ionization energy within this ideal model; it is not a positive emitted photon energy.
  • Actual measured hydrogen levels differ from the infinite-nucleus prediction. For large Z the nonrelativistic approximation becomes inaccurate; use this as an ideal model rather than precision spectroscopy.
  • The supported numerical range for n and Z is positive exact integers from 1 to 1,000,000. Z denotes nuclear charge; a one-electron ion is assumed, not a neutral many-electron atom.
  • All three quantities can be edited to solve an inverse. Clear a default n or Z when it is the unknown. An arbitrary energy need not correspond to a whole allowed state.
  • The six energy choices are neV, μeV, meV, eV, keV and MeV. The electronvolt uses the exact SI elementary charge; unit changes preserve the negative energy.
  • Only computed integer inverses are recovered within an absolute 10⁻⁵⁰ arithmetic remainder. This is not a measurement tolerance and does not round manually entered fractional quantum numbers. R∞ has measured uncertainty, which is not propagated.

Common questions

Why is bound-state energy negative?

The zero is a free electron infinitely far from the nucleus. A bound electron has less energy than that reference; its negative energy represents the binding.

Does this give the exact measured energy of hydrogen?

No. It assumes infinite nuclear mass and a nonrelativistic point nucleus. Actual isotope masses and further corrections shift the levels.

How does energy change when n doubles?

Its magnitude falls by a factor of four. At fixed Z, En is proportional to −1/n².

Can I infer n from a measured energy?

You can solve the ideal inverse, but it must correspond to a positive whole n in the supported range. Ordinary rounded or measured energies may not exactly match this ideal model.

References

Use 2022 CODATA Rydberg and exact SI photon constants for the ideal infinite-nucleus model. An inferred integer state or charge is recovered only for a numerical remainder within 10⁻⁵⁰; ordinary fractional inverses and manual fractions remain invalid. Bound-state energy is strictly negative relative to ionization. The state and nuclear charge are exact positive integers from 1 to 1,000,000; inverse square roots require negative bound energy and positive known state or charge. The ideal model omits isotope and relativistic corrections.

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