Skip to main content
Sandbox Physics

Q011 · Shape / phase / selection

Hydrogen Orbitals & Spectra

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

Interactive modelHydrogen Orbitals & Spectra
Recorded outcomes—\text{—}
Sampled transition fraction—\text{—}

02 / RECORD BEFORE EXPLAINING

Build the spectrum from yes-or-no outcomes.

δE/meV,P^f\delta E/\mathrm{meV},\quad \widehat P_f · Blue: active. Amber: most recent other saved sample. Drag to select a probe setting.

—\text{—}

Each cell is one recent binary result at this detuning. Whiskers: pointwise 95% Wilson intervals. No readout error is added.

03 / CONNECT SHAPE TO THE RESULT

A node is an absence that persists.

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

—\text{—}

MODEL INSPECTION · SEPARATE FROM COUNTS

Follow the phase, then test the overlap.

x/nm,z/nmx/\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,E/eVn,\quad E/\mathrm{eV}

04 / KEEP THE EVIDENCE

Save the settings with the records.

Up to eight frozen samples. Restore to inspect the original settings. New acquisition after saving starts another sample. CSV contains all zero outcomes as well as successful detections.

Model limits, units & sources

Position records are ideal destructive measurements on separately prepared atoms. Spectroscopy uses a weak Gaussian spectral envelope with an ideally selected final state. The dimensionless coupling is scaled by the square of the initial principal number. The supplied controls stay in 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 · Electric dipole transitions · NIST · Hydrogen lines

Physics tutorial

Hydrogen Orbitals & Spectra

BackgroundPrepare an orbital, sample electron positions, and scan a weak polarized probe. Compare an allowed transition with a forbidden one at exactly the same energy.

Why it mattersA symmetry constrains the amplitude. Measure its consequences, then change one part of the preparation.

Start with the essentials

Focus question
The energy gap is right. Why is the line missing?
One-sentence intuition
An energy match is necessary, but the spatial overlap with the polarized dipole operator decides whether this transition appears.

Core mathematical model

One normalized state

ψ=Rnl(r)[cos⁡α Ylm+eiϕsin⁡α Yl−m]\psi=R_{nl}(r)[\cos\alpha\,Y_l^m+e^{i\phi}\sin\alpha\,Y_l^{-m}]

For nonzero magnetic number the components are orthogonal and degenerate. For zero magnetic number only one harmonic is used, independent of the mixing controls.

Coulomb energies

En=−13.605693 μmeZ2n2 eVE_n=-13.605693\,\frac{\mu}{m_e}\frac{Z^2}{n^2}\,\mathrm{eV}

The proton reduced mass is held fixed even when charge changes; those charge comparisons are hypothetical hydrogenic centers, not calibrated isotopes.

A weak selective probe

Pf=A2∣⟨f∣ϵ⋅r/aZ∣i⟩∣2n4exp⁡[−δE2/(2σE2)]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)]

First-order transition probability under a declared Gaussian spectral probe. Coupling is scaled by the initial principal number to keep all exposed settings weak; this is not saturated Rabi dynamics.

Common difficulties

Representation is not a trajectory

Typical misconceptionThe cloud is a photograph of one electron.

Better mental modelThe gray cloud samples the model probability density; blue dots are independent ideal position records. The nucleus and cloud are magnified schematically.

Model boundaries

Typical misconceptionA numerical pass proves a real apparatus has been reproduced.

Better mental modelOne-electron nonrelativistic Coulomb model with proton reduced mass at every charge. Ideal position samples and state-selective weak-probe counts are simulated. No fine structure, Lamb shift, spin, ionization or instrumental calibration.

Run the experiment

  1. 01

    Predict before counting

    Scan the default allowed transition. Save it, choose the forbidden contrast, and repeat without changing the energy gap.

    What to observe: Empty charts are empty evidence. Reference curves stay separate until explicitly enabled.
  2. 02

    Change one physical cause

    Prepare the coherent orbital, rotate its phase, and compare the x-polarized transition. Sample positions on independently prepared copies.

    What to observe: The phase changes angular density and the dipole overlap together. Position sampling and spectroscopy use different copies.
  3. 03

    Inspect a failure mode

    Change polarization or choose a target with forbidden orbital angular momentum. A vanishing dipole channel is not the absence of all possible transitions.

    What to observe: Export the raw records, including zero outcomes, and reproduce the displayed estimator.