Stark Effect Energy Shift Calculator

The Stark effect is the shifting and splitting of an atom's energy levels in the presence of an external electric field, the electrical analogue of the magnetic Zeeman effect. In hydrogen, the effect is unusually large and appears already at first order in the field strength, called the linear Stark effect.

Hydrogen's energy levels depend only on the principal quantum number n, so states with the same n but different orbital angular momentum l are exactly degenerate. An electric field mixes these degenerate states freely, producing a shift proportional to the field strength F. In other atoms, different l states have different energies, so the field can only mix them weakly, giving a much smaller shift proportional to F squared (the quadratic Stark effect).

n1 and n2 are quantum numbers that arise when the hydrogen Schrödinger equation is solved in parabolic coordinates instead of the usual spherical coordinates, the natural coordinate system once a uniform electric field picks out a preferred direction. They replace the orbital and magnetic quantum numbers l and m, and satisfy n1 + n2 + |m| + 1 = n for a state with principal quantum number n.

The first-order energy shift is ΔE = (3/2) n (n1 − n2) e a0 F, where e is the elementary charge, a0 is the Bohr radius, and F is the applied electric field strength. The shift is exactly zero when n1 = n2, and reaches its largest magnitude when the parabolic quantum numbers are as unequal as the n1 + n2 + |m| + 1 = n constraint allows.

The sign of n1 − n2 sets the sign of the shift: states with n1 > n2 shift up in energy, while states with n1 < n2 shift down by the same amount. This is what splits a single hydrogen energy level into a symmetric fan of evenly-spaced sublevels once the field is applied.

It is the exact result of first-order (linear) degenerate perturbation theory, valid whenever the Stark shift stays small compared to the energy spacing to neighboring principal quantum numbers. At very strong fields, higher-order corrections and eventual field ionization become important and this linear formula breaks down.

For a valid physical state, the three parabolic-style quantum numbers must satisfy n1 + n2 + |m| + 1 = n, where n1 and n2 are non-negative integers and m is any integer. This calculator checks that constraint and reports an error with the correct n for your chosen n1, n2, and m if it is not satisfied.

For a hydrogen n=2 state in a field of 10 million volts per metre (100 kV/cm), the maximum shift is about 1.6 x 10⁻³ eV, roughly a thousandth of the unperturbed level spacing. Laboratory Stark effect experiments typically use fields from a few kV/cm up to several hundred kV/cm.

Yes, in principle it applies to any hydrogen-like one-electron system, such as ionized helium (He+) or muonic hydrogen, with the appropriate scaled Bohr radius and energy levels. This calculator is scoped to ordinary hydrogen, where the Bohr radius and Rydberg energy are the standard tabulated values.

The Stark effect is used in Stark spectroscopy to probe atomic and molecular structure, in Rydberg-atom experiments where the huge n values make the shift enormous, and historically it provided early evidence for the quantization of atomic energy levels alongside the Zeeman effect.

The Zeeman effect splits energy levels using a magnetic field coupling to a state's magnetic moment, while the Stark effect splits them using an electric field coupling to the atom's electric dipole. Hydrogen's Stark effect is linear in the field because of its l-degeneracy, whereas the Zeeman effect is linear in the field for essentially all atoms.