Physics:Quantum Spectral lines and series

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← Back to Atomic and spectroscopy Spectral lines are discrete wavelengths of light emitted or absorbed by atoms and molecules, arising from transitions between quantized energy levels, and they provided some of the earliest direct evidence for quantum theory.^[1]

Spectral lines arise from discrete electronic transitions in atoms, forming characteristic series such as Lyman, Balmer, and Paschen.

Origin of spectral lines

In quantum mechanics, electrons in atoms occupy discrete energy eigenstates. When an electron transitions between two states, a photon is emitted or absorbed with energy given by:

$E = h ν = E_{i} - E_{f}$

where:

$h$ is Planck’s constant
$ν$ is the frequency of the radiation
$E_{i}$ , $E_{f}$ are the initial and final energy levels

This leads to sharply defined spectral lines rather than a continuous spectrum.^[2]

Hydrogen spectral series

The hydrogen atom provides the simplest and most important example of spectral line structure. Its energy levels are given by:

$E_{n} = - \frac{13.6 eV}{n^{2}}$

Transitions between these levels produce series of spectral lines described by the Rydberg formula:

$\frac{1}{λ} = R (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$

where:

$R$ is the Rydberg constant
$n_{i} > n_{f}$

This relation accurately predicts observed hydrogen spectral lines.^[3]

Major series

Lyman series ( $n_{f} = 1$ ) – ultraviolet region
Balmer series ( $n_{f} = 2$ ) – visible region
Paschen series ( $n_{f} = 3$ ) – infrared region
Brackett series ( $n_{f} = 4$ ) – infrared
Pfund series ( $n_{f} = 5$ ) – far infrared

Each series corresponds to transitions ending at a fixed lower energy level.^[4]

Fine structure and splitting

Real spectral lines are not perfectly sharp. They exhibit splitting due to additional physical effects:

Fine structure — relativistic corrections and spin–orbit coupling
Zeeman effect — splitting in an external magnetic field
Stark effect — splitting in an electric field

These effects reveal deeper structure in atomic energy levels.^[5]^[6]

Selection rules

Not all transitions are allowed. Selection rules determine which spectral lines appear:

$Δ l = \pm 1$
$Δ m = 0, \pm 1$

These arise from conservation of angular momentum and symmetry properties of atomic wavefunctions.^[7]

Spectroscopy and applications

Spectral lines are fundamental in many areas of physics and astronomy:

Identifying chemical elements in stars and galaxies
Measuring Doppler shifts and cosmic expansion
Determining temperatures and densities of plasmas
Laser technology and atomic clocks

Each element has a unique spectral “fingerprint.”^[8]

See also

Table of contents (184 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Quantum Book II

→ Matter by scale

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

Quantum Book III

→ Methods and tools

Full contents

1. Foundations (11)↑

2. Conceptual and interpretations (13)↑

3. Mathematical structure and systems (13)↑

4. Atomic and spectroscopy (14)↑

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum number
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum defect
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule
Physics:Quantum beats

5. Wavefunctions and modes (9)↑

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states
Physics:Quantum carpet

6. Quantum dynamics and evolution (17)↑

7. Measurement and information (9)↑

8. Quantum information and computing (10)↑

9. Quantum optics and experiments (5)↑

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics
Physics:Quantum optics

Template:Quantum optics operators

10. Open quantum systems (9)↑

11. Quantum field theory (20)↑

Structural dependency map of quantum field theory.

12. Statistical mechanics and kinetic theory (9)↑

13. Condensed matter and solid-state physics (13)↑

14. Plasma and fusion physics (8)↑

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Plasma (fusion context)
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

15. Timeline (8)↑

16. Advanced and frontier topics (16)↑

References

Author: Harold Foppele

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