GATE Physics Atomic and Molecular Physics Syllabus

Atomic and Molecular Physics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Atomic and Molecular Physics in GATE Physics, not a summary of it.

1. 1

   Spectra of one-and many-electron atoms

   5 topics

   • Spin-orbit interaction
   • LS and jj couplings
   • Fine and hyperfine structures
   • Zeeman and Stark effects
   • Electric dipole transitions and selection rules
2. 2

   Rotational and vibrational spectra of diatomic molecules

   overview

   Examined as a single unit within Atomic and Molecular Physics — no further topic split in the official outline.

3. 3

   Electronic transitions in diatomic molecules

   1 topic

   • Franck-Condon principle
4. 4

   Raman effect

   overview

   Examined as a single unit within Atomic and Molecular Physics — no further topic split in the official outline.

5. 5

   EPR, NMR, ESR, X-ray spectra

   overview

   Examined as a single unit within Atomic and Molecular Physics — no further topic split in the official outline.

6. 6

   Lasers

   3 topics

   • Einstein coefficients
   • Population inversion
   • Two and three level systems

Atomic and Molecular Physics flashcards for GATE Physics

23 of 50 cards from the Atomic and Molecular Physics deck — real questions with worked answers.

1. What is the physical origin of the spin-orbit interaction in an atom?

   In the electron's rest frame the nucleus orbits it, producing a magnetic field $\vec{B}$ that couples to the electron's spin magnetic moment. The interaction energy is $H_{SO} = \xi(r)\,\vec{L}\cdot\vec{S}$, where $\xi(r) = \frac{1}{2m_e^2c^2}\frac{1}{r}\frac{dV}{dr}$.

2. Write the spin-orbit energy in terms of the quantum numbers $j$, $l$, and $s$.

   $$E_{SO} = \frac{\zeta_{nl}}{2}\left[j(j+1) - l(l+1) - s(s+1)\right]$$ obtained from $\vec{L}\cdot\vec{S} = \frac{1}{2}(J^2 - L^2 - S^2)$.

3. How does the spin-orbit coupling constant scale with nuclear charge $Z$?

   It scales roughly as $Z^{4}$ (for the energy splitting of a given level, $\propto Z^4/n^3$), so spin-orbit effects become dominant in heavy atoms.

4. In LS (Russell-Saunders) coupling, how are the angular momenta combined?

   Individual orbital momenta couple first to give $\vec{L} = \sum \vec{l_i}$ and individual spins couple to give $\vec{S} = \sum \vec{s_i}$; then $\vec{L}$ and $\vec{S}$ couple to give $\vec{J} = \vec{L} + \vec{S}$.

5. In jj coupling, what is the order of angular-momentum coupling?

   Each electron's own $\vec{l_i}$ and $\vec{s_i}$ couple first to give $\vec{j_i} = \vec{l_i} + \vec{s_i}$; then the individual $\vec{j_i}$ couple to give total $\vec{J} = \sum \vec{j_i}$.

6. When is LS coupling valid versus jj coupling?

   LS coupling holds for light atoms (low $Z$) where electrostatic (residual Coulomb) interaction $\gg$ spin-orbit interaction. jj coupling holds for heavy atoms (high $Z$) where spin-orbit interaction $\gg$ residual Coulomb interaction.

7. What is the spectroscopic term symbol notation for an atomic level?

   $^{2S+1}L_J$, where $2S+1$ is the spin multiplicity, $L$ is the total orbital angular momentum letter (S,P,D,F for $L=0,1,2,3$), and $J$ is the total angular momentum quantum number.

8. State Hund's rules for the ground-state term of an atom.

   (1) Maximum $S$ (largest multiplicity) is lowest in energy. (2) For given $S$, maximum $L$ is lowest. (3) For a shell less than half-filled the lowest $J = |L-S|$ (normal multiplet); for more than half-filled the lowest $J = L+S$ (inverted multiplet).

9. Determine the ground-state term symbol of carbon ($2p^2$).

   For $2p^2$: max $S=1$ ($2S+1=3$), max $L=1$ (P), shell less than half-filled so $J=|L-S|=0$. Ground term is $^{3}P_0$.

10. What are the allowed $J$ values for an LS term with $L=2$ and $S=1$?

    $J$ ranges from $|L-S|$ to $L+S$ in integer steps: $J = 1, 2, 3$, giving levels $^{3}D_1, {}^{3}D_2, {}^{3}D_3$.

11. State the Landé interval rule for fine-structure multiplets.

    The energy separation between adjacent fine-structure levels $J$ and $J-1$ is proportional to the larger $J$: $$E_J - E_{J-1} = \zeta\, J.$$

12. What is the physical cause of atomic fine structure?

    Fine structure arises from relativistic corrections: the spin-orbit interaction, the relativistic kinetic-energy correction, and the Darwin term. It splits levels of the same $n,l$ according to $j$.

13. What is the order of magnitude (energy scale) of fine-structure splitting relative to gross atomic energies?

    Fine structure is smaller by a factor of $\alpha^2 \approx (1/137)^2 \approx 5\times10^{-5}$, where $\alpha$ is the fine-structure constant.

14. Give the fine-structure energy correction formula for hydrogen.

    $$E_{fs} = -\frac{(E_n)^2}{2m_ec^2}\left(\frac{2n}{j+\tfrac{1}{2}} - \frac{3}{2}\right)$$ The energy depends only on $n$ and $j$ (not $l$).

15. What causes hyperfine structure in atomic spectra?

    The coupling of the nuclear spin $\vec{I}$ to the total electronic angular momentum $\vec{J}$ (magnetic dipole and electric quadrupole interactions of the nucleus with the electrons), defining total atomic angular momentum $\vec{F} = \vec{I} + \vec{J}$.

16. How much smaller is hyperfine structure compared to fine structure, and why?

    Hyperfine splitting is smaller by roughly the electron-to-proton mass ratio $\sim m_e/m_p \approx 1/1836$, because the nuclear magnetic moment is $\sim 1000$ times smaller than the electron's.

17. What is the hyperfine interaction energy in terms of $F$, $I$, $J$?

    $$E_{hf} = \frac{A}{2}\left[F(F+1) - I(I+1) - J(J+1)\right]$$ where $A$ is the hyperfine constant and $F = |I-J|,\dots,I+J$.

18. What famous astronomical line arises from a hyperfine transition?

    The 21 cm (1420 MHz) line of neutral hydrogen, from the hyperfine transition between $F=1$ and $F=0$ of the $1s$ ground state (electron and proton spins parallel vs. antiparallel).

19. What is the normal Zeeman effect and when does it occur?

    Splitting of a spectral line into three components (a Lorentz triplet) in a magnetic field, occurring for transitions between singlet states ($S=0$), where spin plays no role. Spacing is $\Delta E = \mu_B B$.

20. What is the anomalous Zeeman effect?

    The splitting of spectral lines into more than three components in a magnetic field when total spin $S \neq 0$. The splitting depends on the Landé $g$-factor and gives unequally spaced multiplets.

21. Write the Landé g-factor formula for LS coupling.

    $$g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$$

22. What is the energy shift of a level with magnetic quantum number $m_J$ in the (weak-field) anomalous Zeeman effect?

    $$\Delta E = g_J\, \mu_B\, B\, m_J$$ where $\mu_B = \frac{e\hbar}{2m_e}$ is the Bohr magneton.

23. What is the Paschen-Back effect?

    The strong-field limit of the Zeeman effect, where the external magnetic field is strong enough to decouple $\vec{L}$ and $\vec{S}$ (spin-orbit coupling broken). $\vec{L}$ and $\vec{S}$ then precess independently about $\vec{B}$, with energy $\Delta E = (m_L + 2m_S)\mu_B B$.

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