List of top Atomic and Molecular Physics Questions asked in GATE PH

An atom is subjected to a weak uniform magnetic field \( B \). The number of lines in its Zeeman spectrum for the transition from \( n = 2, l = 1 \) to \( n = 1, l = 0 \) is
The mean distance between the two atoms of an HD molecule is \( r \), where H and D denote hydrogen and deuterium, respectively. The mass of the hydrogen atom is \( m_H \). The energy difference between the two lowest lying rotational states of HD in multiples of \(\frac{h^2}{m_H r^2}\) is:
Apart from the acoustic modes, 9 optical modes are identified from the measurements of phonon dispersions of a solid with the chemical formula \( A_nB_m \), where \( A \) and \( B \) denote the atomic species, and \( n \) and \( m \) are integers. Which of the following combinations of \( n \) and \( m \) is/are possible?
The non-relativistic Hamiltonian for a single electron atom is
\[ H_0 = \frac{p^2}{2m} - V(r) \]
where \( V(r) \) is the Coulomb potential and \( m \) is the mass of the electron.Considering the spin-orbit interaction term:
\[ H' = \frac{1}{2m^2c^2} \frac{1}{r} \frac{dV}{dr} \overrightarrow{L} \cdot \overrightarrow{S} \]
added to \( H_0 \), which of the following statements is/are true?
The spin-orbit interaction in a hydrogen-like atom is given by the Hamiltonian \( H' = -k \vec{L} \cdot \vec{S} \), where \( k \) is a real constant. The splitting between levels \( ^2P_{3/2} \) and \( ^2P_{1/2} \) due to this interaction is _______.
Atomic numbers of V, Cr, Fe and Zn are 23, 24, 26 and 30, respectively. Which one of the following materials does NOT show an electron spin resonance (ESR) spectra?
Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant a. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures.

The electron concentrations in A and B are \( n_A \) and \( n_B \), respectively. The value of \((n_B / n_A)\) is:

The spin-orbit effect splits the \( ^2p \rightarrow ^2s \) transition (wavelength, \( \lambda = 6521 \, \text{Å} \)) in Lithium into two lines with separation of \( \Delta \lambda = 0.14 \, \text{Å} \). The corresponding positive value of energy difference between the above two lines, in eV, is \( m \times 10^{-5} \). The value of \( m \) (rounded off to the nearest integer) is \(\underline{\hspace{2cm}}\). (Given: Planck's constant, \( h = 4.125 \times 10^{-15} \, \text{eV s} \), speed of light, \( c = 3 \times 10^8 \, \text{m/s} \))

Consider the atomic system as shown in the figure, where the Einstein A coefficients for spontaneous emission for the levels are \( A_{2 \to 1} = 2 \times 10^7 \, \text{s}^{-1} \) and \( A_{1 \to 0} = 10^8 \, \text{s}^{-1} \). If \( 10^{14} \, \text{atoms/cm}^3 \) are excited from level 0 to level 2 and a steady state population in level 2 is achieved, then the steady state population at level 1 will be \( x \times 10^{13} \, \text{cm}^{-3} \). The value of \( x \) (in integer) is \(\underline{\hspace{2cm}}\). 

To sustain lasing action in a three-level laser as shown in the figure, necessary condition(s) is(are) 

The spin \( \vec{S} \) and orbital angular momentum \( \vec{L} \) of an atom precess about \( \vec{J} \), the total angular momentum. \( \vec{J} \) precesses about an axis fixed by a magnetic field \( \vec{B_1} = 2B_0 \hat{z} \), where \( B_0 \) is a constant. Now the magnetic field is changed to \( \vec{B_2} = B_0 (\hat{x} + \sqrt{2} \hat{y} + \hat{z}) \). Given the orbital angular momentum quantum number \( l = 2 \) and spin quantum number \( s = 1/2 \), \( \theta \) is the angle between \( \vec{B_1} \) and \( \vec{J} \) for the largest possible values of total angular quantum number \( j \) and its z-component \( j_z \). The value of \( \theta \) (in degree, rounded off to the nearest integer) is \(\underline{\hspace{2cm}}\).
Among the term symbols \(4S_1\), \(2D_{7/2}\), \(3S_1\), and \(2D_{5/2}\), choose the option(s) possible in the LS coupling notation.