Lamb Shift Estimator
Estimate the Lamb shift, the small QED energy splitting between otherwise-degenerate hydrogen-like atomic levels, using a calibrated Z⁴/n³ scaling law.
↕️ What is the Lamb Shift Estimator?
This Lamb shift estimator approximates the QED energy splitting between the 2S(1/2) and 2P(1/2) levels of hydrogen and hydrogen-like ions, the famous effect that the plain Dirac equation cannot explain. Enter a principal quantum number n and an atomic number Z, and it returns an estimated shift using a scaling law calibrated exactly to the precisely measured hydrogen value.
The Lamb shift is one of the most historically significant measurements in physics. In 1947, Willis Lamb and Robert Retherford used microwave spectroscopy to show that the hydrogen 2S(1/2) and 2P(1/2) levels, which the Dirac equation predicts should have exactly the same energy, are actually split by about 1057.8 MHz. This tiny discrepancy could not be explained by relativistic quantum mechanics alone, and it directly motivated Hans Bethe's first non-relativistic calculation of the effect and, ultimately, the full development of renormalized quantum electrodynamics.
Physically, the shift arises mostly from the electron's self-energy, its interaction with fluctuations of the vacuum electromagnetic field, which perturbs S states (where the electron has nonzero probability density at the nucleus) more strongly than P states. This calculator's Z⁴/n³ scaling law captures the leading-order dependence on quantum number and nuclear charge without requiring the tabulated Bethe logarithm values the exact QED calculation needs, which is why it is an estimator rather than a precision tool.
This calculator is useful for quantum mechanics and atomic physics students who want an intuitive feel for how the Lamb shift scales with n and Z, without needing to work through the full QED self-energy and vacuum polarization calculation.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Hydrogen 2S-2P (the classic measurement)
n=2, Z=1
Example 2 - Hydrogen Ground State (1S)
n=1, Z=1
Example 3 - Singly Ionized Helium (He+), 2S-2P
n=2, Z=2
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Lamb shift?
The Lamb shift is a small energy splitting between atomic levels that the Dirac equation predicts should be exactly degenerate, most famously the hydrogen 2S(1/2) and 2P(1/2) levels. It arises from quantum electrodynamics effects, chiefly the electron's interaction with vacuum fluctuations of the electromagnetic field, and was the first clear experimental signature that QED corrections beyond the Dirac equation were needed.
What formula does this calculator use?
It uses the scaling law ΔE(n,Z) = ΔE_ref × Z⁴ × (2/n)³, calibrated so that at n=2, Z=1 it exactly reproduces the precisely measured hydrogen 2S-2P Lamb shift of 1057.8 MHz. This captures the leading-order Z and n dependence of the full QED result without needing the tabulated Bethe logarithm values the exact calculation requires.
How accurate is this estimate?
It is exact by construction at n=2, Z=1 (hydrogen), and stays within a few percent for light hydrogen-like ions with Z up to about 3 to 5. Above that, the estimate increasingly overshoots the true value, because the full QED formula includes a logarithmic term that grows more slowly than the pure Z⁴ scaling assumed here. Treat results at higher Z as an order-of-magnitude guide, not a precision figure.
Why is the Lamb shift historically important?
Willis Lamb and Robert Retherford measured this splitting directly in 1947 using microwave spectroscopy, finding a nonzero value where the Dirac equation predicted exact degeneracy. The discovery motivated Hans Bethe's first non-relativistic calculation of the effect within weeks, and helped launch the full development of renormalized quantum electrodynamics by Feynman, Schwinger, and Tomonaga, work that earned the 1965 Nobel Prize in Physics.
Why does the Lamb shift only affect S states so strongly?
The dominant contribution to the Lamb shift comes from the electron self-energy interacting with the electron probability density at the nucleus, which is nonzero only for S states (l=0), where the wavefunction does not vanish at the origin. This is why the 2S(1/2) level is shifted upward relative to 2P(1/2), even though both have the same Dirac-equation energy.
What is the Lamb shift of the hydrogen ground state (1S)?
For n=1, Z=1 this calculator's scaling law gives roughly 8462 MHz, compared to the precisely measured value of about 8172.8 MHz, an overshoot of a few percent that illustrates the estimate's typical accuracy at the ground state.
How does the Lamb shift change for higher-Z hydrogen-like ions?
Because the estimate scales as Z⁴, the shift grows very quickly with atomic number. For He+ (Z=2) at n=2, this calculator gives about 16.9 GHz, versus a measured value closer to 14 GHz, showing the estimate's accuracy degrading somewhat as Z increases beyond hydrogen.
What units does this calculator report?
Results are shown in megahertz (MHz), matching how the Lamb shift is conventionally quoted in the precision spectroscopy literature, alongside gigahertz (GHz) and microelectronvolts (μeV) for comparison against other atomic energy scales.
Is the Lamb shift the same as fine structure?
No. Fine structure comes from relativistic corrections and spin-orbit coupling already contained in the Dirac equation, and it does not split 2S(1/2) from 2P(1/2), both have the same Dirac energy. The Lamb shift is a separate, purely QED effect on top of fine structure that lifts this remaining degeneracy.
Can this calculator be used for atomic number Z above 10?
The atomic number input is capped at 10 because the underlying Z⁴/n³ scaling law becomes increasingly unreliable for heavier hydrogen-like ions, where relativistic and higher-order QED corrections beyond this simple estimate become significant.