---
title: "Hydrogen Orbitals & Spectra Interactive Lab | Sandbox Physics"
description: "Why can a hydrogen spectral line be missing even when the energy gap matches? Matching the energy is not enough; the light must also connect the two orbitals."
canonical_url: "https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hydrogen-orbitals-spectra/"
last_modified: "2026-10-27"
---

Q011 · Orbital shape, phase and selection rules

# Hydrogen Orbitals & Spectra

**Why can a hydrogen spectral line be missing even when the energy gap matches?** Matching the energy is not enough; the light must also connect the two orbitals. A polarized probe drives a transition only when the dipole matrix element between them is nonzero. Angular-momentum selection rules make it vanish for some pairs, so those transitions are forbidden even at the right energy.

Prepare an orbital, sample where the electron is found, and scan a weak polarized probe. Compare an allowed transition with a forbidden one at exactly the same energy.

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

The model is a single electron in a nonrelativistic Coulomb field, using the proton reduced mass for every nuclear charge. It simulates ideal position samples and state-selective counts from a weak probe. Fine structure, the Lamb shift, spin, ionization and instrument calibration are not included.

02 / Record first, then explain

## Build the spectrum from yes-or-no outcomes.

$\delta E/\mathrm{meV},\quad \widehat P_f$ · Blue: current sample. Amber: the most recent other saved sample. Drag to choose a probe setting.

Show the model prediction

$\text{—}$

Each cell is one recent yes-or-no result at this detuning. The whiskers are pointwise 95% Wilson intervals. No readout error is added.

03 / Connect the shape to the result

## A node stays empty no matter how many positions you record.

$r/\mathrm{nm},\quad \widehat p_r/\mathrm{nm}^{-1}$ · Blue bars use independent recorded positions. The reference is the normalized radial density.

$\text{—}$

Model view, separate from the counts

## Follow the phase, then test the overlap.

Section plane $x/\mathrm{nm},\quad z/\mathrm{nm}$

Model phase: cyan near zero, amber near half a cycle; hue follows phase. Brightness is the square root of density relative to the section maximum. A node stays dark. This is not a detector image.

$\text{—}$

$n,\quad E/\mathrm{eV}$

04 / Save the evidence

## Save the settings with the records.

You can keep up to eight frozen samples. Restore one to see its original settings. Acquiring after a save starts a new sample. The CSV contains every zero outcome as well as every successful detection.

Model limits, units & sources

Each position record is an ideal destructive measurement on a separately prepared atom. Spectroscopy uses a weak Gaussian spectral envelope and resolves the selected final state ideally. The dimensionless coupling is scaled by the square of the initial principal number. The controls stay within the first-order weak-excitation range. Degenerate magnetic superpositions share one radial function and one energy. No real atom-imaging or state-selection hardware is simulated.

Nonrelativistic, one electron, fixed proton reduced mass. All wavelengths are vacuum values. Finite radial sampling extends to sixteen times the squared principal number in scaled Bohr radii; the omitted probability is reported. Fine structure, Lamb shifts and multiphoton transitions are omitted. A forbidden dipole line does not mean the state is eternally stable.

[Fitzpatrick · Hydrogen](https://farside.ph.utexas.edu/teaching/qm/Quantum/node44.html) · [Electric dipole transitions](https://farside.ph.utexas.edu/teaching/qm/Quantum/node86.html) · [NIST · Hydrogen lines](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable3.htm)

Physics tutorial

## Hydrogen Orbitals & Spectra

**Background** Prepare an orbital, sample where the electron is found, and scan a weak polarized probe. Compare an allowed transition with a forbidden one at exactly the same energy.

**Why it matters** Symmetry restricts the amplitude. Measure what that restriction does, then change one part of the preparation.

Start with the essentials

- **Focus question**: Why can a hydrogen spectral line be missing even when the energy gap matches?
- **One-sentence intuition**: Matching the energy is necessary, but whether the line appears depends on the spatial overlap of the two states through the polarized dipole operator.

### Core mathematical model

#### The prepared normalized state

$$
\psi=R_{nl}(r)[\cos\alpha\,Y_l^m+e^{i\phi}\sin\alpha\,Y_l^{-m}]
$$

When the magnetic number is nonzero, the two components are orthogonal and have the same energy. When it is zero, only one spherical harmonic is used and the mixing controls have no effect.

#### Energy levels in a Coulomb field

$$
E_n=-13.605693\,\frac{\mu}{m_e}\frac{Z^2}{n^2}\,\mathrm{eV}
$$

The proton reduced mass stays fixed when you change the nuclear charge, so the higher charges are hypothetical hydrogen-like centers, not calibrated isotopes.

#### Transition probability for a weak probe

$$
P_f=A^2\frac{|\langle f|\boldsymbol\epsilon\cdot\mathbf r/a_Z|i\rangle|^2}{n^4}\exp[-\delta E^2/(2\sigma_E^2)]
$$

This is the first-order transition probability for a probe with a Gaussian spectrum. The coupling is scaled with the initial principal quantum number so that every available setting stays weak; saturated Rabi dynamics are not modeled.

### Common difficulties

#### A picture is not a trajectory

**Typical misconception** The cloud is a photograph of one electron.

**Better mental model** The gray cloud is a sample of the model probability density, and the blue dots are independent ideal position records. The nucleus and cloud are enlarged schematically.

#### What the model leaves out

**Typical misconception** If the numerical checks pass, a real apparatus has been reproduced.

**Better mental model** The model is a single electron in a nonrelativistic Coulomb field, using the proton reduced mass for every nuclear charge. It simulates ideal position samples and state-selective counts from a weak probe. Fine structure, the Lamb shift, spin, ionization and instrument calibration are not included.

### Run the experiment

1. 01

   #### Predict before counting

   Scan the default allowed transition and save it. Then choose the forbidden comparison and repeat the scan at the same energy gap.

   **What to observe:** An empty chart means there is no evidence yet. Reference curves appear only when you turn them on, and they stay separate from the counts.

2. 02

   #### Change one physical cause

   Prepare the coherent orbital, rotate its phase and compare the transition driven by x-polarized light. Position samples come from separately prepared copies.

   **What to observe:** Changing the phase changes both the angular density and the dipole overlap. Position sampling and spectroscopy use different copies of the state.

3. 03

   #### Look at a case where the signal vanishes

   Change the polarization, or choose a target state whose orbital angular momentum is forbidden. One dipole channel going to zero does not rule out every other transition.

   **What to observe:** Export the raw records, zero outcomes included, and you can recompute the displayed estimate yourself.

Physical conclusion

The model is a single electron in a nonrelativistic Coulomb field, using the proton reduced mass for every nuclear charge. It simulates ideal position samples and state-selective counts from a weak probe. Fine structure, the Lamb shift, spin, ionization and instrument calibration are not included.

## Related experiments and articles

Quantum evidence · Reading path

### Laws and the formation of quantum mechanics

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

1. [Article **Atomic spectra: evidence for states, not photographs of orbits**](https://sandboxphysics.com/stories/quantum-evidence/atomic-spectra-stationary-states/)
2. [Lab **Hydrogen–Deuterium Isotope Spectrometer** Measure an isotope shift](https://sandboxphysics.com/labs/hydrogen-deuterium-isotope-spectrometer/)
3. [Lab **Two-Isotope Doppler Thermometer** Separate heat from instrumental blur](https://sandboxphysics.com/labs/two-isotope-doppler-thermometer/)
4. [Article **de Broglie: a wavelength for matter, a speed for the packet**](https://sandboxphysics.com/stories/quantum-evidence/de-broglie-matter-waves-phase-packets/)
5. [Lab **Wave Packet & Uncertainty** Watch a wave packet move and spread](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
6. [Article **Matrices and waves: two routes to the same predictions**](https://sandboxphysics.com/stories/quantum-evidence/matrix-wave-mechanics-observables/)
7. [Article **Dirac’s electron: a successful equation with an unfinished story**](https://sandboxphysics.com/stories/quantum-evidence/dirac-electron-spin-relativity/)
8. [Lab **Dirac Trembling Motion** Watch a free Dirac wave packet tremble](https://sandboxphysics.com/labs/dirac-trembling-motion/)
9. [Lab **Klein Wave-Packet Tunnelling** Send a wave packet into a very high barrier (Klein tunnelling)](https://sandboxphysics.com/labs/klein-wavepacket-tunneling/)
10. [Lab **Coupled Spin Spectroscopy** Compare spin spectra from an effective model](https://sandboxphysics.com/labs/coupled-spin-spectroscopy/)

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 an electric field splits hydrogen’s spectral lines**](https://sandboxphysics.com/stories/quantum-evidence/stark-effect-electric-field-spectra/)
2. [Lab **Stark Splitting Spectrometer** Change the electric field and separate polarized hydrogen lines](https://sandboxphysics.com/labs/stark-splitting-spectrometer/)
3. [Article **How missing hydrogen atoms revealed the Lamb shift**](https://sandboxphysics.com/stories/quantum-evidence/lamb-shift-microwave-evidence/)
4. [Lab **Lamb Shift Microwave Bench** Find a microwave resonance in simulated surviving-atom counts](https://sandboxphysics.com/labs/lamb-shift-microwave/)

Quantum evidence · Reading path

### Charge, excitation and photon recoil

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

1. [Article **Franck–Hertz: what a drop in current can show**](https://sandboxphysics.com/stories/quantum-evidence/franck-hertz-discrete-excitation/)
2. [Lab **Franck–Hertz Excitation** Scan electron-impact excitation](https://sandboxphysics.com/labs/franck-hertz-excitation/)
3. [Lab **Photoemission: Retarding Current & Energy Analysis** Compare photoemission energy accounting](https://sandboxphysics.com/labs/photoemission-analyzer/)

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)