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.

Recorded outcomes $\text{—}$

Sampled transition fraction $\text{—}$

01 / Choose what to test

Same energy. A different overlap.

Scan the allowed transition and save it. Switch to the forbidden target: the energy stays the same. Does a line still appear?

Orbital model · gray probability samples; blue recorded positions · nucleus enlarged · scale adapts to the chosen orbital · no electron paths

Probe detuning

\delta E/\mathrm{meV}

0

Each spectral point uses 2,000 newly prepared atoms. Position samples use separate copies.

Try a comparison

Principal quantum number

n

1

Orbital angular momentum

l

0

Magnetic component

|m|

0

Nuclear charge

Z

1

Coherent preparation

Mix opposite magnetic components

\alpha/\mathrm{deg}

0

Relative phase

\phi/\mathrm{deg}

0

You can mix states only when the magnetic number is nonzero. The two components have opposite magnetic numbers and the same energy.

Selected final state & probe

Target principal number

n_f

2

Target angular momentum

l_f

1

Target magnetic number

m_f

0

Weak probe coupling

A

0.25

Probe spectral width

\sigma_E/\mathrm{meV}

30

Probe polarization

$a_Z=a_0/[(\mu/m_e)Z]$ · The selected final state is resolved ideally. This channel can be dark while other allowed transitions still occur.

Detection & sampling

Sampling seed

s_0

137

Changing the preparation clears the current data, so save a sample first. The probe only selects a measurement setting.

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 · Electric dipole transitions · NIST · Hydrogen lines

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 misconceptionThe cloud is a photograph of one electron.

Better mental modelThe 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 misconceptionIf the numerical checks pass, a real apparatus has been reproduced.

Better mental modelThe 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

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.
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.
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.

Hydrogen Orbitals & Spectra

Build the spectrum from yes-or-no outcomes.

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

Follow the phase, then test the overlap.

Save the settings with the records.