0.000318
eV
5.086670e-23
J
-3.4
eV
0.000318
eV
5.086670e-23
J
-3.4
eV
The Stark Effect Calculator computes the energy level shift of a hydrogen atom in an external electric field, known as the Stark effect. Discovered by Johannes Stark in 1913 (Nobel Prize 1919), the Stark effect is the electric analogue of the Zeeman effect: an external electric field lifts the degeneracy of atomic energy levels, splitting spectral lines into multiple components.
For hydrogen, the Stark effect in the n=2 level provides the most famous example of a linear Stark shift, where the energy change is proportional to the electric field strength. This linear effect (first-order perturbation theory) exists because the n=2 states 2s and 2p are degenerate in hydrogen (same energy), allowing mixing and a permanent electric dipole moment. The ground state (n=1) shows only a quadratic Stark shift because it is non-degenerate.
The linear Stark shift in parabolic coordinates is delta_E = (3/2) * n * (n1 - n2_parabolic) * e * a0 * E, where n1 and n2 are parabolic quantum numbers, a0 is the Bohr radius, and E is the electric field. The splitting grows as n^2 for higher shells because the electron is further from the nucleus and more easily polarized.
The Stark effect is the basis of several important modern techniques: Stark spectroscopy determines molecular properties (dipole moments, polarizabilities), Stark decelerators slow polar molecules for trapping experiments, and the AC Stark effect (light shift) is fundamental to laser trapping and optical lattices used in atomic clocks and quantum computing.
In astrophysics, the Stark effect broadens spectral lines in dense stellar atmospheres and white dwarf stars, where electric fields from neighboring charged particles shift each atom's energy levels by a slightly different amount, producing pressure broadening of lines.
The linear Stark shift in hydrogen is delta_E = (3/2) * e * a0 * n * q * E, where q = n1 - n2 = n1 - (n - |m| - 1 - n1) is the difference of parabolic quantum numbers, e = 1.602e-19 C is the electron charge, a0 = 5.292e-11 m is the Bohr radius, n is the principal quantum number, and E is the electric field in V/m. The unperturbed hydrogen energy is E0 = -13.6/n^2 eV.
For n=2 states in a 10^6 V/m field, the Stark shift is on the order of 10^-6 eV. Fields of 10^8 V/m are needed to shift energy levels by ~0.1 eV. For Rydberg atoms with large n, the Stark shift grows rapidly as n^4. The quadratic (second-order) Stark effect applies to the hydrogen ground state and non-degenerate levels.
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For the n=2 level with parabolic n1=1, m=0 in a 1 MV/m field, the linear Stark shift is about 9.6 microelectronvolts, shifting from the -3.4 eV unperturbed energy.
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High Rydberg states are extremely sensitive to electric fields because the electron's orbital radius scales as n^2. A modest 10^5 V/m field already shifts the n=10 level by 2 meV.
The splitting and shifting of atomic spectral lines in an external electric field. Discovered by Johannes Stark in 1913, it is the electric analogue of the Zeeman effect (magnetic field splitting).
The linear Stark effect (shift proportional to E) occurs for degenerate states like hydrogen n=2. The quadratic Stark effect (shift proportional to E^2) occurs for non-degenerate states and all states of most atoms at low fields.
The 2s and 2p states of hydrogen are accidentally degenerate in the Coulomb approximation. This degeneracy allows first-order mixing and produces the large linear Stark effect for hydrogen n=2.
Parabolic quantum numbers (n1, n2, m) arise when the Schrodinger equation is solved in parabolic (rather than spherical) coordinates, which are the natural coordinates for hydrogen in an electric field.
A rapidly oscillating (AC) electric field such as a laser beam also shifts atomic energy levels, called the AC Stark effect or light shift. It is the basis of optical dipole traps used to confine cold atoms.
Stark spectroscopy uses the electric-field dependence of rotational lines to determine molecular electric dipole moments and polarizabilities with high precision.
A device using time-varying electric fields to decelerate a beam of polar molecules. The alternating field gradient exerts a force that slows molecules, enabling their trapping for precision spectroscopy.
Rydberg atoms (very high n) are extremely sensitive to electric fields because the electron orbit radius scales as n^2. Even weak fields produce large Stark shifts, making them useful for field sensing.
Yes. Molecular rotational levels split in electric fields (rotational Stark effect). This is used in microwave spectroscopy to measure molecular constants.
In dense stellar atmospheres or high-pressure laboratory discharges, electric fields from neighboring ions and electrons shift energy levels by varying amounts, broadening spectral lines. This is called Stark broadening or pressure broadening.
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