---
title: "Lamb Shift Microwave Bench Interactive Lab | Sandbox Physics"
description: "How can microwaves find the Lamb shift in hydrogen? Look for the frequency where fewer atoms survive."
canonical_url: "https://sandboxphysics.com/labs/lamb-shift-microwave/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/lamb-shift-microwave/"
last_modified: "2026-10-27"
---

Quantum evidence · missing atoms

# Lamb Shift Microwave Bench

**How can microwaves find the Lamb shift in hydrogen?** Look for the frequency where fewer atoms survive. At the Lamb-shift frequency, the microwaves push metastable hydrogen atoms into a nearby short-lived state, which decays almost at once, so fewer atoms reach the detector. Scan the frequency and compare each count with the count when the microwaves are off; the dip in survivors marks the resonance.

Use microwaves to drive a beam of metastable hydrogen atoms into a short-lived state, which removes atoms from the beam. Count the survivors at each frequency, compare them with counts taken with the microwaves off, and locate the effective Lamb resonance.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model uses a rate equation for a single resonance under weak drive, centered on a rounded reference frequency. Hyperfine and Zeeman components, coherent transients, the spread of atom velocities and QED corrections are left out. The resonance center is an input value; it is not calculated from first-principles QED and does not come from historical data.

Schematic geometry with enlarged gaps and parts. The colors and moving markers do not show visible light or measured trajectories. Drag to orbit.

Exposure: signal and reference counted separately

## Give the spectrum a record.

Each coordinate has a signal exposure and a separate reference: microwaves off. Exposure time controls Poisson noise; scan playback is accelerated.

Ready. Predict, then acquire.

Recorded points

$0$

Metastable survival · model

$0$

Effective energy gap · source

$0$

Measured spectrum in counts per second

### Which structure survives measurement?

$\nu/\mathrm{MHz}$ · $N_i/\tau_i\;(\mathrm{s^{-1}})$

Teal: source expectation. Gold: acquired rates with fitted Poisson standard deviations, once a fit exists. Purple dashes: inference from records. Gray dots: saved run. Background is included.

Click or drag to move the probe; its slider supports keyboard input. The vertical scale follows the data.

Inference with a joint Poisson likelihood

### Estimate from both exposures.

Inferred center

$\text{—}$

Inferred source rate

$\text{—}$

Deviance per degree

$\text{—}$

Flat alternative

$\text{—}$

$r_i=(N_i-\widehat\mu_i)/\sqrt{\widehat\mu_i}$

Gold: signal residuals. Blue: reference residuals. Known response, splitting/width controls and background; unknown source rate and spectral parameter. At least sixteen distinct probe positions are required. Descriptive residuals and deviance do not establish identifiability, confidence intervals or discovery significance.

### Keep the original counts.

Save a full or partial run, with its settings and seed, for as long as this page stays open.

| # | $\nu/\mathrm{MHz}$ | Signal | Microwaves off | $\tau/\mathrm{s}$ |
| --- | --- | --- | --- | --- |

The latest six points are shown. The CSV has every raw count, exposure, setting and unit, plus the seed. Moving the probe never changes data already taken.

Scientific scope, geometry & primary sources

Prepared metastable hydrogen travels from the source through a microwave interaction gap without static fields to a metal collector. Only surviving metastable atoms contribute to the signal. Atoms transferred to the short-lived state radiatively decay before collection; the collector is not a photon detector.

An effective isolated transition uses a rounded 1057.8 MHz reference center and a 100 MHz natural full width. The source-center control also represents a hypothetical frequency offset. Weak-drive adiabatic elimination gives exponential loss during the selected interaction time. The largest drive is four percent of the decay width; the shortest interaction is much longer than the short-lived-state lifetime.

The model leaves out hyperfine and Zeeman structure, velocity distributions, coherent transients, stray-field quenching and collector calibration. The reference exposure is taken with the microwaves off. The Lab does not reconstruct the historical magnetic-field experiment or calculate radiative corrections. Fitting a resonance that was supplied as input cannot independently verify QED.

Independent Poisson exposures, fixed calibrated background and stable source. No drift, dead time or calibration uncertainty. The source follows the displayed model; successful recovery validates the simulator, not the historical discovery.

[Lamb & Retherford · 1947](https://physics.umd.edu/grt/taj/624b/Lamb1947.pdf) [Lamb & Retherford · 1950](https://journals.aps.org/pr/abstract/10.1103/PhysRev.79.549) [Washington · optical comparison](https://courses.washington.edu/phys432/lamb_shift/lamb_shift.php)

[Read the historical evidence](https://sandboxphysics.com/stories/quantum-evidence/lamb-shift-microwave-evidence/) [Stark Splitting Spectrometer](https://sandboxphysics.com/labs/stark-splitting-spectrometer/) [Mössbauer Doppler Observatory](https://sandboxphysics.com/labs/mossbauer-doppler-resonance/)

Physics tutorial

## Lamb Shift Microwave Bench

**Background** Use microwaves to drive a beam of metastable hydrogen atoms into a short-lived state, which removes atoms from the beam. Count the survivors at each frequency, compare them with counts taken with the microwaves off, and locate the effective Lamb resonance.

**Why it matters** A spectral line becomes evidence only when it is measured and you know how the instrument's response depends on the quantity you want.

Start with the essentials

- **Focus question**: How can microwaves find the Lamb shift in hydrogen?
- **One-sentence intuition**: Fit the signal counts and the reference counts together with one Poisson likelihood, and estimate the source rate from the same data.

### Core mathematical model

#### Loss rate of the metastable atoms

$$
R(\nu)=\frac{\Omega^2\Gamma}{\Gamma^2+4[2\pi(\nu-\nu_0)]^2},\qquad \Gamma=2\pi\,(100\,\mathrm{MHz})
$$

In this equation the Rabi frequency and the decay rate are angular frequencies. The frequency controls show ordinary frequencies.

#### Fraction of metastable atoms that survive

$$
g(\nu)=e^{-R(\nu)t},\qquad \Delta E=h\nu_0
$$

Eliminating the destination state under weak drive assumes that it decays quickly and that the interaction lasts much longer than its lifetime. The metal collector counts the metastable atoms that survive.

#### Two independent exposures

$$
N_i\sim\operatorname{Poisson}\!\left[\tau_i(Sg_i+b)\right],\qquad M_i\sim\operatorname{Poisson}\!\left[\tau_i(S+b)\right]
$$

The background rate is known in advance. The source rate and the spectral parameter are fitted together, and the scatter in the reference counts is included. Counts are drawn directly from the Poisson distribution, not rounded from a Gaussian approximation.

### Common difficulties

#### A good fit does not test the law

**Typical misconception** If the fit matches the data, that independently confirms the physical law built into the simulator.

**Better mental model** The simulator generates the data from a response it is given. Recovering the parameter shows that this simulated measurement and analysis work together. The deviance and residuals describe how well the curve fits; they are not confidence intervals, significance tests or an outside scientific certification.

### Run the experiment

1. 01

   #### Find the dip

   Scan the microwave frequency. Compare each signal exposure with its own separate microwave-off exposure.

   **What to observe:** Fewer metastable atoms reach the collector at the effective resonance. The instrument measures how many atoms survive, not how much microwave power they absorb directly.

2. 02

   #### Lengthen the interaction

   Save the default run. Increase the interaction time, scan again, and look at the depth of the dip and its wings.

   **What to observe:** Because the loss is exponential, the shape of the dip changes too. The fit uses this exponential response instead of an arbitrary Lorentzian dip.

3. 03

   #### Turn the microwaves off

   Choose the Microwaves off preset and scan. Then restore the saved run and export the raw CSV.

   **What to observe:** With no microwave coupling, the counts carry no information about the resonance. No center is reported, even if random counts happen to form a dip.

Physical conclusion

The model uses a rate equation for a single resonance under weak drive, centered on a rounded reference frequency. Hyperfine and Zeeman components, coherent transients, the spread of atom velocities and QED corrections are left out. The resonance center is an input value; it is not calculated from first-principles QED and does not come from historical data.

## Related experiments and articles

Quantum evidence · Reading path

### Fields, spins and precision spectra

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How missing hydrogen atoms revealed the Lamb shift**](https://sandboxphysics.com/stories/quantum-evidence/lamb-shift-microwave-evidence/)
2. [Lab **Hydrogen Orbitals & Spectra** Compare the hydrogen energy levels before the Lamb correction](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)