Physics Quiz

1. The degeneracy of energy levels in a hydrogen atom for a given \( n \) is (don't consider spin degeneracy):

\( n^2 \)
\( 2n^2 \)
\( n \)
\( 2n - 1 \)

2. The radial wavefunction \( R(r) \) for the hydrogen atom is dependent on:

\( n \) only
\( l \) only
Both \( n \) and \( l \)
Neither \( n \) nor \( l \)

3. The azimuthal quantum number \( l \) defines the:

Total energy of the electron
Probability density at the nucleus
Magnitude of orbital angular momentum
Number of nodes in the wavefunction

4. In the spectra of alkali metals, the deviation from the hydrogen spectrum is due to:

Increased nuclear charge
Core penetration and shielding
Higher angular momentum states
Absence of \( d \)-orbitals

5. For a hydrogen atom, the probability of finding an electron at the nucleus is:

Maximum for \( l = 0 \)
Zero for \( l = 0 \)
Maximum for \( l = 1 \)
Independent of \( l \)

6. The spin-orbit coupling energy for a hydrogen-like atom is proportional to:

\( \frac{Z}{n^2} \)
\( \frac{Z^2}{n} \)
\( \frac{n}{Z} \)
\( \frac{1}{Z \cdot n^2} \)

7. How can the component of \( \mathbf{L} \) in the direction of \( \mathbf{J} \) be obtained?

By projecting \( \mathbf{L} \) along \( \mathbf{S} \) using \( \mathbf{S} \cdot \mathbf{J} \)
Using the commutator of \( \mathbf{L} \) and \( \mathbf{J} \)
By using the relation \( L_J = \frac{\mathbf{L} \cdot \mathbf{J}}{|\mathbf{J}|} \)
By solving the eigenvalue equation for \( \mathbf{L} \) in the basis of \( \mathbf{J} \)

8. How can \( \mathbf{L} \cdot \mathbf{J} \) be obtained?

By using the equation \( \mathbf{J} = \mathbf{L} + \mathbf{S} \) and calculating \( (\mathbf{J} - \mathbf{L}) \cdot (\mathbf{J} - \mathbf{L}) = \mathbf{S} \cdot \mathbf{S} \) where \(\mathbf{J.J}=\mathbf{J^2}=\mathbf{J(J+1)},\mathbf{L.L}=\mathbf{L^2}=\mathbf{L(L+1)},\mathbf{S.S}=\mathbf{S^2}=\mathbf{S(S+1)}\) and then isolating \(\mathbf{L.J}\)
By expanding \( \mathbf{J} \cdot \mathbf{J} = \mathbf{L} \cdot \mathbf{L} + \mathbf{S} \cdot \mathbf{S} + 2 \mathbf{L} \cdot \mathbf{S} \) and isolating \( \mathbf{L} \cdot \mathbf{J} \)
By directly measuring the projection of \( \mathbf{L} \) on \( \mathbf{J} \) in experimental data
By solving eigenvalue equations for \( \mathbf{L} \) and \( \mathbf{J} \)

9. The magnetic quantum numbers \( m_l \) and \( m_s \) define the:

Total angular momentum \( J_z \)
Orbital angular momentum \( L \)
Components of angular momentum \( L_z \) and spin angular momentum \( S_z \)
Radial probability distribution

10. Fine structure splitting in a sodium atom is more prominent than in hydrogen due to:

Higher nuclear charge and penetration effects
Stronger Zeeman splitting
Larger \( s \)-orbitals
Weak spin-orbit interaction

11. In the spin-orbit interaction, the effective magnetic field experienced by the electron is due to:

The nucleus
Orbital motion of the electron
External magnetic field
Hyperfine interactions

12. Fine structure in sodium \( D \)-lines is characterized by:

Equal energy spacing between \( D_1 \) and \( D_2 \)
Unequal energy spacing due to \( j = l \pm s \)
Splitting into multiple components
Zeeman effect dominance

13. The normal Zeeman effect occurs when:

Spin angular momentum \( s = 0 \)
\( j = l + s \)
Spin-orbit coupling dominates
Nuclear spin contributes significantly

14. In the anomalous Zeeman effect, the number of spectral lines observed depends on:

\( n \) and \( m_l \)
\( j \), \( m_j \), and \( g \)-factor
\( l \) and \( s \)
The Landé interval rule

15. For a hydrogen atom in a magnetic field, the energy shift is given by:

\( \mu_B m_l B \)
\( \mu_B l B \)
\( g \mu_B j B \)
\( g \mu_B m_j B \)

16. The Landé \( g \)-factor for a \( p \)-electron is:

1
2
\( 1 + \frac{3}{4} \)
\( \frac{1}{2} \)

17. The Zeeman effect is reduced to the Paschen–Back effect when:

The spin-orbit coupling is negligible compared to the magnetic field strength
The magnetic field is weak
Quadratic terms dominate
The nucleus has spin

18. In the Paschen–Back effect, the \( m_l \) and \( m_s \) values:

Are coupled strongly
Become independent of each other
Are zero
Contribute equally

19. The splitting of sodium \( D \)-lines in the Paschen–Back effect results in:

A single line
Four equally spaced lines
Two doublets separated by a constant energy gap
A continuum spectrum

20. The Stark effect splits spectral lines into components due to:

Spin-orbit coupling
Interaction with an external electric field
Zeeman splitting
Hyperfine interaction

21. The quadratic Stark effect is significant for:

Ground state energy levels
States with \( n \gg 1 \)
Linear polarization of light
Weak electric fields

22. The \( L \)-\( S \) coupling scheme is valid when:

Spin-orbit coupling dominates
Orbital and spin angular momenta are separately conserved
Nuclear spin dominates
External magnetic fields are strong

23. For two equivalent \( s \)-electrons, the allowed spectroscopic terms are:

\( ^1S \) only
\( ^3P \) and \( ^1P \)
\( ^1S \) and \( ^3P \)
\( ^3S \) and \( ^1P \)

24. The energy difference between two \( j \)-levels in \( j \)-\( j \) coupling is proportional to:

\( j \)
\( \frac{1}{j(j + 1)} \)
\( j(j + 1) \)
\( \sqrt{j(j + 1)} \)

25. Hund’s rule predicts the ground state configuration by maximizing:

Orbital angular momentum
Total spin and orbital angular momenta
Energy splitting
Fine structure levels

26. Hyperfine structure in hydrogen is due to:

Nuclear spin coupling with electron spin
Relativistic effects
Quadratic Stark effect
Fine structure splitting

27. The hyperfine splitting in alkali metals is larger than in hydrogen because:

Higher nuclear spin and charge density
Stronger spin-orbit coupling
Penetrating orbits dominate
Weak nuclear spin

28. In the hyperfine structure, the total angular momentum \( F \) is given by:

\( F = I + J \)
\( F = J + L \)
\( F = L + S \)
\( F = J - S \)

29. The hyperfine structure splitting in a spectrum is proportional to:

Nuclear charge and \( S \)
\( I \cdot J \)
\( n^2 / Z \)
\( m_s \cdot m_l \)

30. The hydrogen hyperfine transition at 21 cm is caused by:

Quadrupole splitting
Magnetic dipole interaction
Spin-orbit coupling
Zeeman effect