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Quantum Physics overview

Topic 7 of 7

Atomic transitions and spectra

An isolated atom has discrete electronic energy levels. Emission and absorption involve differences between those levels, with the actual initial state determining which transitions are available.

Atomic hydrogen is a real example of discrete bound electronic states with spatial wavefunctions. Their energies need not be equally spaced or follow the infinite well's n2 pattern. The box is a model of confinement, not an exact model of hydrogen.

For a transition that emits a photon, the photon carries the decrease in atomic energy. Absorption requires a photon that supplies an available upward gap. Photon energy is positive:

Ephoton = |Efinal - Einitial|
= hf = hc/λ

Read a supplied three-level atom

Consider levels at -6.00, -3.00 and -1.00 eV relative to a zero ionisation reference. All three pairwise transitions are permitted in this supplied model. It is not a labelled hydrogen model. The following comparison considers only these bound-bound transitions.

Use the actual starting level and the allowed energy gap

Use the actual starting level and the allowed energy gapThis supplied atom has bound levels minus one, minus three and minus six electronvolts relative to the zero ionisation reference. It is not labelled hydrogen. The energy axis is linear: y equals sixty minus forty times energy in electronvolts, so zero and the three levels are at sixty, one hundred, one hundred and eighty, and three hundred. All three pairwise bound transitions are permitted in this model. Three brown downward emission lanes show gaps two, three and five electronvolts. Separate blue upward lanes start at the ground level and end at minus three and minus one electronvolts, requiring three and five electronvolt absorption. The five-electronvolt arrows pass the intermediate energy without ending there. Arrow width is schematic; photon energy is the positive magnitude of each level change.Energy / eV0Ionisation reference-1-3-62 eV3 eV5 eV3 eV5 eVAll three pairs are permitted here.EmissionAbsorption

The three downward transitions can emit photons when their upper states are occupied. A ground-state absorber can use the upward 3 eV or 5 eV transition; it cannot use the 2 eV gap from an unoccupied excited state. Zero is the ionisation reference, not another bound level.

The level positions use one linear energy scale, with zero as the ionisation reference. Separate arrow lanes identify the 2, 3 and 5 eV gaps without making a direct transition appear to stop at an intermediate level.

Worked emitted photon

Subtract the levels before using hf

For emission from -1.00 to -3.00 eV, the atom loses 2.00 eV. Using 1 eV = 1.60 × 10-19 J, h = 6.63 × 10-34 J s and c = 3.00 × 108 m/s:

Ephoton = 3.20 × 10-19 J
f = Ephoton/h = 4.82655 × 1014 Hz
λ = c/f = 6.215625 × 10-7 m
≈ 622 nm
Emitted photons for the three permitted downward transitions
Levels / eVPhoton energy / eVλ / nm
-1 to -32.00622
-3 to -63.00414
-1 to -65.00249

For the 3.00 and 5.00 eV photons, the frequencies are about 7.24 × 1014 and 1.21 × 1015 Hz. With the supplied constants, their wavelength calculations give 414.375 and 248.625 nm before rounding.

A direct upper-to-ground transition emits one 5.00 eV photon. A two-step cascade emits one 2.00 eV photon and one 3.00 eV photon, with the same total energy. It does not emit one photon at an average of their wavelengths.

Distinguish emission and absorption spectra

An excited low-density gas can produce separate emission lines against a dark background. Their wavelengths correspond to permitted downward gaps, with suitable excited-state populations. The level diagram alone does not determine each line's strength.

A cooler gas in front of a continuous source can remove selected wavelengths from the transmitted beam, producing absorption lines. The relevant lower levels must be occupied. Absorbed energy has not disappeared; later emission can send it in other directions.

An excited population can show all three emission lines

An excited population can show all three emission linesBoth spectra use the same numerical wavelength axis from two hundred to seven hundred nanometres, with x equal to fifty plus one half times wavelength minus two hundred. The exact three model wavelengths are 248.625,414.375 and621.5625 nanometres, at x74.3125,157.1875 and260.78125. The shortest line is ultraviolet, not visible violet. Three narrow bright markers stand on a dark background when suitable excited levels are occupied. The ultraviolet marker is neutral rather than coloured violet. The visible shorter and longer markers denote approximately violet and red wavelengths, without asserting relative line intensity. Marker widths and any displayed brightness are schematic, not supplied spectral-resolution or population data.Suitable excited-state populations249 nmUV414 nm622 nm200300400500600700Vacuum wavelength / nmLine widths and strengths are schematic.

An appropriately excited population can supply the 2 eV, 3 eV and 5 eV downward transitions. The numerical axis includes ultraviolet. Equal marker height or width does not mean equal emitted intensity.

Ground-state absorption removes only the available gaps

Ground-state absorption removes only the available gapsBoth spectra use the same numerical wavelength axis from two hundred to seven hundred nanometres, with x equal to fifty plus one half times wavelength minus two hundred. The exact three model wavelengths are 248.625,414.375 and621.5625 nanometres, at x74.3125,157.1875 and260.78125. The shortest line is ultraviolet, not visible violet. A neutral light band represents the supplied incident continuum; dark notches occur only at 248.625 and414.375 nanometres for a population entirely in the ground state. The 621.5625-nanometre position is unnotched because the lower state of that two-electronvolt transition is unoccupied. The band is not a literal colour display of ultraviolet. Marker widths and any displayed brightness are schematic, not supplied spectral-resolution or population data.All atoms initially in the ground state249 nmUV414 nm622 nmpasses200300400500600700Vacuum wavelength / nmNeutral band represents a continuum.

For this all-ground-state bound-transition model, 249 nm and 414 nm are removed from the beam; 622 nm is not. Absorbed energy is not destroyed, and subsequent emission can be in other directions. A different lower-state population changes the possible absorption lines.

The two spectra share a numerical wavelength axis. Suitable excited populations can produce all three emission lines. The stated ground-state absorber removes only the 249 and 414 nm components in this model. The 249 nm line is ultraviolet, not visible violet; line widths and strengths are schematic.

If all atoms start at -6.00 eV, photons of 3.00 or 5.00 eV can excite the listed upper states. A 2.00 eV photon cannot make either of those ground-state transitions. It could produce the -3.00 to -1.00 eV transition if that lower excited state were occupied.

These photons are below the 6.00 eV ground-state ionisation threshold. Start from the actual occupied level and the transitions permitted in the question. A mismatch with these specified gaps is not a universal claim that the photon cannot interact with matter in any other way.

Optional check A supplied atom has levels -6.00, -3.00 and -1.00 eV, with the three pairwise transitions permitted. It starts in the -6.00 eV ground state. Can a 2.00 eV photon produce one of these bound-bound absorptions?
A supplied atom has levels -6.00, -3.00 and -1.00 eV, with the three pairwise transitions permitted. It starts in the -6.00 eV ground state. Can a 2.00 eV photon produce one of these bound-bound absorptions?