Q.No:1 GATE-2012
The ground state of sodium atom \((^{11}\)Na) is a \({}^2 S_{1/2}\) state. The difference in energy levels arising in the presence of a weak external magnetic field \(B\), given in terms of Bohr magneton, \(\mu_B\), is
(A)
\(\mu_B B\)
(B)
\(2\mu_B B\)
(C)
\(4\mu_B B\)
(D)
\(6\mu_B B\)
Check Answer
Option B
Q.No:2 GATE-2012
The terms \(\{j_1, j_2\}_J\) arising from \(2s^1 3d^1\) electronic configuration in \(j\)-\(j\) coupling scheme are
(A)
\(\left\{\frac{1}{2}, \frac{3}{2}\right\}_{2, 1}\) and \(\left\{\frac{1}{2}, \frac{5}{2}\right\}_{3, 2}\)
(B)
\(\left\{\frac{1}{2}, \frac{1}{2}\right\}_{1, 0}\) and \(\left\{\frac{1}{2}, \frac{3}{2}\right\}_{2, 1}\)
(C)
\(\left\{\frac{1}{2}, \frac{1}{2}\right\}_{1, 0}\) and \(\left\{\frac{1}{2}, \frac{5}{2}\right\}_{3, 2}\)
(D)
\(\left\{\frac{3}{2}, \frac{1}{2}\right\}_{2, 1}\) and \(\left\{\frac{1}{2}, \frac{5}{2}\right\}_{3, 2}\)
Check Answer
Option A
Q.No:3 GATE-2013
The number of spectral lines allowed in the spectrum for the \(3 {}^2 D\to 3 {}^2 P\) transition in sodium is _______________.
Check Answer
Ans 3
Q.No:4 GATE-2013
In a normal Zeeman effect experiment, spectral splitting of the line at the wavelength \(643.8 nm\) corresponding to the transition \(5 {}^1 D_2\to 5 {}^1 P_1\) of cadmium atoms is to be observed. The spectrometer has a resolution of \(0.01 nm\). The minimum magnetic field needed to observe this is (\(m_e=9.1\times 10^{-31} kg, e=1.6\times 10^{-19}c, c=3\times 10^{8} m/s\))
(A)
\(0.26 T\)
(B)
\(0.52 T\)
(C)
\(2.6 T\)
(D)
\(5.2 T\)
Check Answer
Option B
Q.No:5 GATE-2014
The number of normal Zeeman splitting components of \({}^1 P\to {}^1 D\) transition is
(A)
\(3\)
(B)
\(4\)
(C)
\(8\)
(D)
\(9\)
Check Answer
Option A
Q.No:6 GATE-2014
The emission wavelength for the transition \({}^1 D_2\to {}^1 F_3\) is \(3122\) Angstrom. The ratio of populations of the final to the initial states at a temperature \(5000 K\) is (\(h=6.626\times 10^{-34} J.s, c=3\times 10^8 m/s, k_B=1.380\times 10^{-23} J/K\))
(A)
\(2.03\times 10^{-5}\)
(B)
\(4.02\times 10^{-5}\)
(C)
\(7.02\times 10^{-5}\)
(D)
\(9.83\times 10^{-5}\)
Check Answer
Option C
Q.No:7 GATE-2015
The number of permitted transitions from \({}^2 P_{3/2}\to {}^2 S_{1/2}\) in the presence of a weak magnetic field is ______________.
Check Answer
Ans 6
Q.No:8 GATE-2016
There are four electrons in the \(3d\) shell of an isolated atom. The total magnetic moment of the atom in units of Bohr magneton is ______________.
Check Answer
Ans 0
Q.No:9 GATE-2016
Which of the following transitions is NOT allowed in the case of an atom, according to the electric dipole radiation selection rule?
(A)
\(2s\)-\(1s\)
(B)
\(2p\)-\(1s\)
(C)
\(2p\)-\(2s\)
(D)
\(3d\)-\(2p\)
Check Answer
Option A
Q.No:10 GATE-2016
The number of spectroscopic terms resulting from the \(\mathbf{L}\cdot \mathbf{S}\) coupling of a \(3p\) electron and a \(3d\) electron is _____________.
Check Answer
Ans 12,12-6,6
Q.No:11 GATE-2017
The electronic ground state energy of the Hydrogen atom is \(-13.6 eV\). The highest possible electronic energy eigenstate has an energy equal to
(A)
\(0\)
(B)
\(1 eV\)
(C)
\(+13.6 eV\)
(D)
\(\infty\)
Check Answer
Option D
Q.No:12 GATE-2017
Using Hund's rule, the total angular momentum quantum number \(J\) for the electronic ground state of the nitrogen atom is
(A)
\(1/2\)
(B)
\(3/2\)
(C)
\(0\)
(D)
\(1\)
Check Answer
Option B
Q.No:13 GATE-2017
Positronium is an atom made of an electron and a positron. Given the Bohr radius for the ground state of the Hydrogen atom to be \(0.53\) Angstroms, the Bohr radius for the ground state of positronium is ____________Angstroms. (up to two decimal places).
Check Answer
Ans 0.99-1.10
Q.No:14 GATE-2018
Match the physical effects and order of magnitude of their energy scales given below, where \(\alpha=\frac{e^2}{4\pi \epsilon_0 \hbar c}\) is fine structure constant; \(m_e\) and \(m_p\) are electron and proton mass, respectively.
(A)
P-3, Q-1, R-2, S-4
(B)
P-2, Q-3, R-1, S-4
(C)
P-4, Q-2, R-1, S-3
(D)
P-2, Q-4, R-1, S-3
Check Answer
Option C
Q.No:15 GATE-2018
Which one of the following is an allowed electric dipole transition?
(A)
\({}^1 S_0\to {}^3 S_1\)
(B)
\({}^2 P_{3/2}\to {}^2 D_{5/2}\)
(C)
\({}^2 D_{5/2}\to {}^2 P_{1/2}\)
(D)
\({}^3 P_0\to {}^5 D_0\)
Check Answer
Option B
Q.No:16 GATE-2018
The term symbol for the electronic ground state of oxygen atom is
(A)
\({}^1 S_0\)
(B)
\({}^1 D_2\)
(C)
\({}^3 P_0\)
(D)
\({}^3 P_2\)
Check Answer
Option D
Q.No:17 GATE-2018
An atom in its singlet state is subjected to a magnetic field. The Zeeman splitting of its \(650 nm\) spectral line is \(0.03 nm\). The magnitude of the field is ___________ Tesla (up to two decimal places).
(\(e=1.60\times 10^{-19} C, m_{e}=9.11\times 10^{-31} kg, c=3.0\times 10^8 {-1}\))
Check Answer
Ans 1.51-1.53
Q.No:18 GATE-2019
The ground state electronic configuration of the rare-earth ion \(({Nd^{3+}})\) is \([Pd]\) \(4f^3 5s^2 5p^6\). Assuming LS coupling, the Lande \(g\)-factor of this ion is \(\frac{8}{11}\). The effective magnetic moment in units of Bohr magneton \(\mu_B\) (rounded off to two decimal places) is ____________.
Check Answer
Ans 3.61-3.63
Q.No:19 GATE-2020
A hydrogenic atom is subjected to a strong magnetic field. In the absence of spin-orbit coupling, the number of doubly degenerate states created out of the \(d\)-level is ________________.
Check Answer
Ans 3
Q.No:20 GATE-2021
Among the term symbols
\[
{}^4 S_1, {}^2 D_{7/2}, {}^3 S_1 \text{ and } {}^2 D_{5/2}
\]
choose the option(s) possible in the \(LS\) coupling notation.
(A)
\({}^4 S_1\)
(B)
\({}^2 D_{7/2}\)
(C)
\({}^3 S_1\)
(D)
\({}^2 D_{5/2}\)
Check Answer
Option C_D
Q.No:21 GATE-2021
The transition line, as shown in the figure, arises between \({}^2 D_{3/2}\) and \({}^2 P_{1/2}\) states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is ________________.
Check Answer
Ans 6
Q.No:22 GATE-2021
The spin-orbit effect splits the \({}^2 P\to {}^2 S\) transition (wavelength, \(\lambda=6521\) Angstrom in. Lithium into two lines with separation of \(\Delta \lambda=0.14\) Angstrom. 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 _________.
\({\it Given:}\) Planck's constant, \(h=4.125\times 10^{-15} eV s\)
Speed of light, \(c=3\times 10^8 ms^{-1}\))
Check Answer
Ans 3-5
Q.No:23 GATE-2022
An excited state of \({Ca}\) atom is \([ce{Mg}]3p^5 4s^2 3d^1\). The spectroscopic terms corresponding to the total orbital angular momentum are
(a)
S, P, and D
(b)
P, D, and F
(c)
P and D
(d)
S and P
Check Answer
Option b
Q.No:24 GATE-2022
For normal Zeeman lines observed \(\parallel\) and \(\perp\) to the magnetic field applied to an atom, which of the following statements are true?
(a)
Only \(\pi\)-lines are observed \(\parallel\) to the field
(b)
\(\sigma\)-lines \(\perp\) to the field are plane polarized
(c)
\(\pi\)-lines \(\perp\) to the field are plane polarized
(d)
Only \(\sigma\)-lines are observed \(\parallel\) to the field
Check Answer
Option b,c,d
Q.No:25 GATE-2023
Which one of the following is a dimensionless constant?
(A)
Permittivity of free space
(B)
Permeability of free space
(C)
Bohr magneton
(D)
Fine structure constant
Check Answer
Option D
Q.No:26 GATE-2023
The atomic number of an atom is 6. What is the spectroscopic notation of its ground state, according to Hund’s rules?
(A)
\(^3 P_0\)
(B)
\(^3 P_1\)
(C)
\(^3 D_3\)
(D)
\(^3 S_1\)
Check Answer
Option A
Q.No:27 GATE-2023
Consider two non-identical spin \(\frac{1}{2}\) particles labelled 1 and 2 in the spin product state \(| \frac{1}{2}, \frac{1}{2} \rangle \) \(|\frac{1}{2}, -\frac{1}{2}\rangle\). The Hamiltonian of the system is
\[H=\frac{4\lambda}{\hbar^2} \vec{S}_1 \cdot \vec{S}_2,\]
where \(\vec{S}_1\) and \(\vec{S}_2\) are the spin operators of particles 1 and 2, respectively, and \(\lambda\) is a constant with appropriate dimensions. What is the expectation value of \(H\) in the above state?
(A)
\(-\lambda\)
(B)
\(-2\lambda\)
(C)
\(\lambda\)
(D)
\(2\lambda\)
Check Answer
Option A
Q.No:28 GATE-2024
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
(A) \( \frac{1}{2} k\hbar^2 \)
(B) \( \frac{3}{2} k\hbar^2 \)
(C) \( \frac{3}{4} k\hbar^2 \)
(D) \( 2k\hbar^2 \)
Check Answer
Option B
Q.No:29 GATE-2024
An atom is subjected to a weak uniform magnetic field \(\vec{B}\). The number of lines in its Zeeman spectrum for transition from \(n = 2\), \(l = 1\) to \(n = 1\), \(l = 0\) is
(A) 8
(B) 10
(C) 12
(D) 5
Check Answer
Option B
Q.No:30 GATE-2024
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
\[ H' = \frac{1}{2m^2c^2} \frac{1}{r} \frac{dV}{dr} \vec{L} \cdot \vec{S} \]
added to \(H_0\), which of the following statement is/are true?
(A) \(H'\) commutes with \(L^2\)
(B) \(H'\) commutes with \(L_z\) and \(S_z\)
(C) For a given value of principal quantum number \(n\) and orbital angular momentum quantum number \(l\), there are \(2(2l + 1)\) degenerate eigenstates of \(H_0\)
(D) \(H_0\), \(L^2\), \(S^2\), \(L_z\), and \(S_z\) have a set of simultaneous eigenstates
Check Answer
Option A,C,D
Q.No:31 GATE-2025
Which of the following option(s) is/are correct for the ground state of a hydrogen atom?
A) Linear Stark effect is zero
B) It has definite parity
C) Spin–orbit coupling is zero
D) Hyperfine splitting is zero
Check Answer
Option A,B,C
Q.No:32 GATE-2025
The effective magnetic moment (in units of Bohr magneton) for the ground state
of an isolated 4f ion with 6 unpaired electrons in the 4f shell according to
Hund’s rules is (in integer) ______.
Check Answer
ANS 0
Q.No:33 GATE-2025
The screened nuclear charge of a neutral Helium atom is given as \(1.7e\),
where \(e\) is the magnitude of the electronic charge. Assuming the Bohr model
of the atom, for which the energy levels are
\[
E_n = -\frac{Z^2}{2n^2} \quad \text{(atomic units)}
\]
(where \(Z\) is the atomic number), the first ionization potential of Helium
in atomic units is
A) 0.89
B) 1.78
C) 0.94
D) 3.16
Check Answer
Option A
Q.No:34 GATE-2025
A system of three non-identical spin \( \frac{1}{2} \) particles has the Hamiltonian
\[
H = \frac{A}{\hbar^2}\,(\vec S_1 + \vec S_2)\cdot \vec S_3,
\]
where \( \vec S_1, \vec S_2 \) and \( \vec S_3 \) are the spin operators of particles
labelled 1, 2 and 3 respectively, and \(A\) is a constant with appropriate dimensions.
The set of possible energy eigenvalues of the system is
A) \(0,\; \frac{A}{2},\; -A\)
B) \(0,\; \frac{A}{2},\; -\frac{A}{2}\)
C) \(0,\; \frac{3A}{2},\; -\frac{A}{2}\)
D) \(0,\; -\frac{3A}{2},\; \frac{A}{2}\)