m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Atomic physics: bound states, spectra and interactions

Use quantum mechanics to derive atomic structure and spectra, then connect interacting atoms to solids, liquids, gases and plasmas.

Subject library · 51 guides · derivations & worked examples

Fundamental physics

Space and flight

Materials and transport

Systems and mathematics

Marine and road vehicles

Electrical and computing

Companion site

Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Coulomb bound states

Definitions & inputs. μ reduced electron–nucleus mass,Z nuclear charge,n principal quantum number,l orbital quantum number,m its projection.

  1. Separate relative motion from center-of-mass motion.

    H=−ℏ2∇2/(2μ)−Ze2/(4πϵ0r)H=-\hbar^2\nabla^2/(2\mu)-Ze^2/(4\pi\epsilon_0r)

  2. Spherical symmetry separates the radial and angular equations; normalizability quantizes the radial solution.

    ψnlm=Rnl(r)Ylm(θ,ϕ),n=nr+l+1\psi_{nlm}=R_{nl}(r)Y_{lm}(\theta,\phi),\quad n=n_r+l+1

  3. Requiring the radial series to terminate yields the discrete energy spectrum.

    En=−μZ2e42(4πϵ0)2ℏ2n2E_n=-\frac{\mu Z^2e^4}{2(4\pi\epsilon_0)^2\hbar^2n^2}

Interpretation. Quantum orbitals are probability amplitudes, not classical planetary trajectories.

↑ Return to definitions and contents

2. Spectra and dipole transitions

Definitions & inputs. ν photon frequency,λ wavelength,d electric dipole operator.

  1. Energy conservation relates a downward level transition to emitted light.

    hν=Ei−Ef,λ=hc/(Ei−Ef)h\nu=E_i-E_f,\quad\lambda=hc/(E_i-E_f)

  2. Transition strength depends on a matrix element, not merely an energy difference.

    ⟨f∣d∣i⟩=−e∫ψf∗rψi d3r\langle f|\mathbf d|i\rangle=-e\int\psi_f^*\mathbf r\psi_i\,d^3r

  3. Angular parity and the vector operator give electric-dipole selection rules in the orbital basis.

    Δl=±1,Δm=0,±1\Delta l=\pm1,\quad\Delta m=0,\pm1

Interpretation. Spin, total angular momentum and coupling scheme add selection rules; forbidden dipole transitions may occur through higher multipoles or multiple photons.

↑ Return to definitions and contents

3. Many-electron atoms and external fields

Definitions & inputs. H electron Hamiltonian,rij electron separation,μB Bohr magneton,gJ Landé factor,mJ angular projection.

  1. Electron–electron repulsion prevents the simple hydrogenic separation.

    H=∑i[−ℏ2∇i2/(2me)−Ze2/(4πϵ0ri)]+∑i<je2/(4πϵ0rij)H=\sum_i[-\hbar^2\nabla_i^2/(2m_e)-Ze^2/(4\pi\epsilon_0r_i)]+\sum_{i<j}e^2/(4\pi\epsilon_0r_{ij})

  2. Antisymmetry enforces fermionic exchange and Pauli exclusion; use spin orbitals.

    Ψ(…i…j…)=−Ψ(…j…i…)\Psi(\ldots i\ldots j\ldots)=-\Psi(\ldots j\ldots i\ldots)

  3. First-order perturbation in a weak magnetic field splits angular-momentum states.

    ΔEZ=μBgJmJB\Delta E_Z=\mu_Bg_Jm_JB

Interpretation. Hartree–Fock, configuration interaction and density functional methods offer different approximations; screening with an effective Z is only a rough model.

↑ Return to definitions and contents

4. From isolated atoms to matter

Definitions & inputs. R internuclear spacing,t hopping energy,k wavevector,a lattice spacing,ne electron density.

  1. Overlapping atomic orbitals form a many-site electronic model.

    H=∑iϵaci†ci−t∑⟨i,j⟩(ci†cj+cj†ci)H=\sum_i\epsilon_a c_i^\dagger c_i-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)

  2. Fourier-transform a one-dimensional periodic chain to obtain a band from one atomic level.

    E(k)=ϵa−2tcos⁡(ka)E(k)=\epsilon_a-2t\cos(ka)

  3. When atoms ionize into a suitable weakly coupled plasma, collective screening introduces a different length scale.

    λD=ϵ0kBTe/(nee2)\lambda_D=\sqrt{\epsilon_0k_BT_e/(n_ee^2)}

Interpretation. Solids have ordered or disordered bonded structure; liquids have persistent short-range correlations; gases may have atoms or molecules; plasmas add free charges. Quantum foundations apply throughout, not just at the atomic stage.

↑ Return to definitions and contents

Graphical worked example

Hydrogen 1s radial probability density 4(r/a0)²exp(−2r/a0); maximum at r=a0 and integral one. X axis: Scaled radius r/a0 (dimensionless). Y axis: Radial probability density in scaled radius.
Hydrogen 1s radial probability density 4(r/a0)²exp(−2r/a0); maximum at r=a0 and integral one. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Hydrogen ground energy

Definitions & inputs. Z1,n1.

  1. Choose the governing model and isolate the requested quantity.

    En=−13.6Z2/n2E_n=-13.6Z^2/n^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −13.6-13.6

  3. Evaluate the expression; the result uses the units shown.

    Result=−13.6 eV\mathrm{Result}=-13.6\ \mathrm{eV}

Interpretation. Rounded infinite-mass Coulomb benchmark.

↑ Return to definitions and contents
Example 02. Hydrogen second level

Definitions & inputs. Z1,n2.

  1. Choose the governing model and isolate the requested quantity.

    E2=−13.6/22E_2=-13.6/2^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −13.6/4-13.6/4

  3. Evaluate the expression; the result uses the units shown.

    Result=−3.4 eV\mathrm{Result}=-3.4\ \mathrm{eV}

Interpretation. Excludes fine structure.

↑ Return to definitions and contents
Example 03. Hydrogen third level

Definitions & inputs. n3.

  1. Choose the governing model and isolate the requested quantity.

    E3=−13.6/32E_3=-13.6/3^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −13.6/9-13.6/9

  3. Evaluate the expression; the result uses the units shown.

    Result=−1.511111 eV\mathrm{Result}=-1.511111\ \mathrm{eV}

Interpretation. Single-electron approximation.

↑ Return to definitions and contents
Example 04. Ground ionization energy

Definitions & inputs. E1=−13.6eV,continuum0.

  1. Choose the governing model and isolate the requested quantity.

    I=0−E1I=0-E_1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    0−(−13.6)0-(-13.6)

  3. Evaluate the expression; the result uses the units shown.

    Result=13.6 eV\mathrm{Result}=13.6\ \mathrm{eV}

Interpretation. Ionization removes the electron to zero kinetic energy at infinity.

↑ Return to definitions and contents
Example 05. Excited-state ionization

Definitions & inputs. n2.

  1. Choose the governing model and isolate the requested quantity.

    I2=13.6/4I_2=13.6/4

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3.43.4

  3. Evaluate the expression; the result uses the units shown.

    Result=3.4 eV\mathrm{Result}=3.4\ \mathrm{eV}

Interpretation. Smaller binding at largern.

↑ Return to definitions and contents
Example 06. Lyman-alpha energy

Definitions & inputs. Transition2→1.

  1. Choose the governing model and isolate the requested quantity.

    ΔE=13.6(1−1/4)\Delta E=13.6(1-1/4)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    13.6(.75)13.6(.75)

  3. Evaluate the expression; the result uses the units shown.

    Result=10.2 eV\mathrm{Result}=10.2\ \mathrm{eV}

Interpretation. Actual allowed2p→1s transition.

↑ Return to definitions and contents
Example 07. Lyman-alpha wavelength

Definitions & inputs. Photon10.2eV.

  1. Choose the governing model and isolate the requested quantity.

    λ=hc/ΔE\lambda=hc/\Delta E

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1239.841984/10.21239.841984/10.2

  3. Evaluate the expression; the result uses the units shown.

    Result=121.5531 nm\mathrm{Result}=121.5531\ \mathrm{nm}

Interpretation. Rounded model, not a precision reference wavelength.

↑ Return to definitions and contents
Example 08. Balmer-alpha energy

Definitions & inputs. 3→2.

  1. Choose the governing model and isolate the requested quantity.

    ΔE=13.6(1/4−1/9)\Delta E=13.6(1/4-1/9)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    13.6(1/4−1/9)13.6(1/4-1/9)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.888889 eV\mathrm{Result}=1.888889\ \mathrm{eV}

Interpretation. Allowed orbital sublevels must also be chosen.

↑ Return to definitions and contents
Example 09. Balmer-alpha wavelength

Definitions & inputs. Same photon.

  1. Choose the governing model and isolate the requested quantity.

    λ=hc/ΔE\lambda=hc/\Delta E

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1239.841984/[13.6(1/4−1/9)]1239.841984/[13.6(1/4-1/9)]

  3. Evaluate the expression; the result uses the units shown.

    Result=656.3869 nm\mathrm{Result}=656.3869\ \mathrm{nm}

Interpretation. Vacuum model wavelength.

↑ Return to definitions and contents
Example 10. Hydrogenic helium energy

Definitions & inputs. HeplusZ2,n1.

  1. Choose the governing model and isolate the requested quantity.

    E1=−13.6Z2E_1=-13.6Z^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −13.6(4)-13.6(4)

  3. Evaluate the expression; the result uses the units shown.

    Result=−54.4 eV\mathrm{Result}=-54.4\ \mathrm{eV}

Interpretation. Does not describe neutral helium.

↑ Return to definitions and contents
Example 11. Most-probable1s radius

Definitions & inputs. Z2,a0=.0529177nm.

  1. Choose the governing model and isolate the requested quantity.

    rmp=a0/Zr_{mp}=a_0/Z

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .0529177/2.0529177/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02645885 nm\mathrm{Result}=0.02645885\ \mathrm{nm}

Interpretation. Peak of the radial probability distribution.

↑ Return to definitions and contents
Example 12. Mean1s radius

Definitions & inputs. Hydrogen a0=.0529177nm.

  1. Choose the governing model and isolate the requested quantity.

    ⟨r⟩=3a0/2\langle r\rangle=3a_0/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.5(.0529177)1.5(.0529177)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.07937655 nm\mathrm{Result}=0.07937655\ \mathrm{nm}

Interpretation. Differs from most-probable radius.

↑ Return to definitions and contents
Example 13. Orbital degeneracy

Definitions & inputs. Principal n3,ignore spin.

  1. Choose the governing model and isolate the requested quantity.

    g=n2g=n^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    323^2

  3. Evaluate the expression; the result uses the units shown.

    Result=9 \mathrm{Result}=9\ {}

Interpretation. Coulomb degeneracy before perturbations.

↑ Return to definitions and contents
Example 14. Shell capacity

Definitions & inputs. n3,including two spin states.

  1. Choose the governing model and isolate the requested quantity.

    Nmax=2n2N_{max}=2n^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2(32)2(3^2)

  3. Evaluate the expression; the result uses the units shown.

    Result=18 \mathrm{Result}=18\ {}

Interpretation. Pauli spin-orbital count.

↑ Return to definitions and contents
Example 15. Angular momentum magnitude

Definitions & inputs. l1.

  1. Choose the governing model and isolate the requested quantity.

    L/ℏ=l(l+1)L/\hbar=\sqrt{l(l+1)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2\sqrt2

  3. Evaluate the expression; the result uses the units shown.

    Result=1.414214 \mathrm{Result}=1.414214\ {}

Interpretation. Magnitude differs from maximal projection.

↑ Return to definitions and contents
Example 16. Weak Zeeman shift

Definitions & inputs. μB5.7883818×10⁻⁵eV/T,gJ2,mJ.5,B1T.

  1. Choose the governing model and isolate the requested quantity.

    ΔE=μBgJmJB\Delta E=\mu_Bg_Jm_JB

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5.7883818×10−5(2)(.5)(1)5.7883818\times10^{-5}(2)(.5)(1)

  3. Evaluate the expression; the result uses the units shown.

    Result=5.788382×10−5 eV\mathrm{Result}=5.788382\times10^{-5}\ \mathrm{eV}

Interpretation. Weak-field perturbative example.

↑ Return to definitions and contents
Example 17. Natural lifetime scale

Definitions & inputs. DecayA10⁸/s.

  1. Choose the governing model and isolate the requested quantity.

    τ=1/A\tau=1/A

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/1081/10^8

  3. Evaluate the expression; the result uses the units shown.

    Result=1×10−8 s\mathrm{Result}=1\times10^{-8}\ \mathrm s

Interpretation. One dominant spontaneous channel.

↑ Return to definitions and contents
Example 18. Lifetime-limited linewidth

Definitions & inputs. τ10ns.

  1. Choose the governing model and isolate the requested quantity.

    Δν=1/(2πτ)\Delta\nu=1/(2\pi\tau)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/(2π10−8)1/(2\pi10^{-8})

  3. Evaluate the expression; the result uses the units shown.

    Result=1.591549×107 Hz\mathrm{Result}=1.591549\times10^{7}\ \mathrm{Hz}

Interpretation. Lorentzian FWHM for exponential population decay without extra dephasing.

↑ Return to definitions and contents
Example 19. Thermal population ratio

Definitions & inputs. Equal degeneracies,ΔE1eV,kBT.5eV.

  1. Choose the governing model and isolate the requested quantity.

    N2/N1=e−ΔE/(kBT)N_2/N_1=e^{-\Delta E/(k_BT)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e−2e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1353353 \mathrm{Result}=0.1353353\ {}

Interpretation. Thermal equilibrium; not arbitrary laser-driven populations.

↑ Return to definitions and contents
Example 20. Atomic-level band width

Definitions & inputs. 1D nearest-neighbor t1eV.

  1. Choose the governing model and isolate the requested quantity.

    W=4∣t∣W=4|t|

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4(1)4(1)

  3. Evaluate the expression; the result uses the units shown.

    Result=4 eV\mathrm{Result}=4\ \mathrm{eV}

Interpretation. One-orbital tight-binding chain.

↑ Return to definitions and contents

Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.