Bohr’s quantization condition for angular momentum is
A L=nhL = nhL=nh
B L=nh/2L = nh/2L=nh/2
C L=nℏL = n\hbarL=nℏ
D L=ℏ/nL = \hbar/nL=ℏ/n
Bohr postulated mvr=nℏmvr = n\hbarmvr=nℏ.
Bohr model fails for multi-electron atoms because
A It ignores nuclear charge
B It ignores electron–electron interactions
C It assumes elliptical orbits
D It assumes electrons are massless
The radius of the nth Bohr orbit is proportional to
A n
B n²
C 1/n
D n³
rn=a0n2r_n = a_0 n^2rn=a0n2.
Energy levels of hydrogen vary as
A En∝nE_n \propto nEn∝n
B En∝1/n2E_n \propto 1/n^2En∝1/n2
C En∝n2E_n \propto n^2En∝n2
D En∝n3E_n \propto n^3En∝n3
A Rydberg atom refers to
A Hydrogen in ground state
B Highly excited atom with electron in high n
C Ionized atom
D Atom losing nucleus
In hydrogen, degeneracy of levels with same n is lifted by
A Magnetic field (Zeeman effect)
B Fine structure
C Electric field (Stark effect)
D All of the above
The radial probability density of hydrogen is maximum at
A r = 0
B Bohr radius a₀ for ground state
C Infinity
D Negative r
The classical electron radius is obtained by
A Equating rest energy with electrostatic energy
B Coulomb’s law
C Schrödinger equation
D Bohr quantization
The commutator [Lx,Ly][L_x, L_y][Lx,Ly] equals
A 0
B iℏLzi\hbar L_ziℏLz
C −iℏLz-i\hbar L_z−iℏLz
D ℏ\hbarℏ
The commutator [L2,Lz][L^2, L_z][L2,Lz] equals
A 0
B iℏLxi\hbar L_xiℏLx
C iℏLyi\hbar L_yiℏLy
D ℏ2\hbar^2ℏ2
L2L^2L2 commutes with any component.
A spherical harmonic YlmY_{lm}Ylm is eigenfunction of
A Lz^\hat{L_z}Lz^ only
B L^2\hat{L}^2L^2 and Lz^\hat{L_z}Lz^
C Schrödinger time equation
D Momentum operator only
Parity of a hydrogen orbital depends on
A n only
B l only
C m only
D s only
Parity = (−1)l(-1)^l(−1)l.
Total angular momentum quantum number j can be
A j=l+sj = l + sj=l+s or j=l−sj = l − sj=l−s
B Only l + s
C Only integer values
D Random
Spin of a proton is
A 0
B 1/2
C 1
D 2
In LS coupling, total spin S is obtained by
A Multiplying all individual spins
B Vector addition of individual electron spins
C Subtracting orbital and spin
D Using relativistic correction only
In jj-coupling scheme, strongest interaction is
A Spin–orbit
B Spin–spin
C Electron–electron Coulomb
D Photon momentum
The normal Zeeman effect applies for
A Atoms with S = 0
B Atoms with S ≠ 0
C All atoms
D Only hydrogen
When spin contribution vanishes → triplet splitting.
In anomalous Zeeman effect, splitting depends on
A L only
B S only
C L, S, and j
D Magnetic field only
Lande g-factor determines
A Frequency of photons emitted
B Energy splitting in magnetic field
C Number of electrons
D Nuclear force
Stern–Gerlach experiment required
A Uniform magnetic field
B Non-uniform magnetic field (∂B/∂z ≠ 0)
C Electric field only
D No field
A particle with spin 1 has how many spin projections?
A 1
B 2
C 3
D Infinite
m_s = −1, 0, +1.
Electron orbital magnetic moment is proportional to
A L
B S
C L + S
D L × S
When a photon is absorbed, selection rule for orbital angular momentum is
A Δl = 0
B Δl = ±1
C Δl = ±2
D Any value
Magnetic quantum number selection rule is
A Δm = 0 only
B Δm = ±1, 0
C Δm = ±2
D Δm arbitrary
A free electron cannot absorb a photon because
A Energy is too small
B Momentum and energy conservation cannot both be satisfied
C Electron mass changes
D Electron must be bound
Compton scattering involves
A Bound electrons only
B Free or weakly bound electrons
C Only photons of visible light
D Neutrons
The de Broglie wavelength of a 1 MeV electron is
A Extremely large
B Comparable to atomic spacing
C Very small (relativistic small λ)
D Undefined
Wave packets represent
A Energy eigenstates only
B Localized particles formed by superposition of waves
C Waves that cannot move
D Static patterns
Dispersion relation for a free particle is
A ω∝k\omega \propto kω∝k
B ω=ℏk22m\omega = \frac{\hbar k^2}{2m}ω=2mℏk2
C ω=ck\omega = ckω=ck
D None
The group velocity of a wave packet is
A vg=dω/dkv_g = d\omega/dkvg=dω/dk
B vg=ω/kv_g = \omega/kvg=ω/k
C 0
D Infinite
A rigid box potential generates
A Continuous energy levels
B Discrete quantized energy levels
C No levels
D Negative-only energies
The radial part of hydrogen wavefunction contains Laguerre polynomials because
A They describe spherical waves
B They solve differential equation with Coulomb potential
C They describe standing sound waves
D They are arbitrary choice
In the hydrogen atom, quantum number l can take values
A 0 to n
B 0 to n−1
C 1 to n
D 1 to n−1
Quantum tunneling allows
A Violation of energy conservation
B Temporary access to classically forbidden region
C Stable infinite energies
D Classical escape
The zero-point energy of a particle in a box arises from
A Temperature
B Uncertainty principle
C Interaction with walls
D Photons
The LS coupling quantum numbers L and S combine to form
A j = L ± S
B j = L + S only
C j = S only
D j = 0 only
The fine-structure splitting is proportional to
A Z²
B Z⁴
C Z
D Z¹⁰
Relativistic & spin–orbit ~ Z4Z^4Z4 dependence.
In the Paschen–Back effect, magnetic field is
A Weak
B Very strong, breaking LS coupling
C Absent
D Reversed
Electron magnetic moment is approximately
A μ_B (Bohr magneton)
B 1 C·m
C Zero
D 137 μ_B
Spin–orbit coupling vanishes when
A l = 0
B s = 0
C m = 0
D n = 1
For s-orbitals orbital angular momentum is zero → no coupling.
Hyperfine structure arises mostly due to
A Nuclear spin interaction with electrons
B Photons
C Compton effect
D Tunneling
Quantum numbers (n, l, m, s) uniquely define
A Classical orbit
B Electron spin only
C A unique electron state in atom
D No physical meaning
A photon carries
A Charge
B Spin 1
C Spin 1/2
D Mass
Degeneracy breaking in electric field is
A Zeeman effect
B Stark effect
C Lamb shift
D Darwin correction
If two electrons have same n, l, m but opposite spin, the configuration is
A Forbidden
B Allowed
C Degenerate only
D Infinite energy
The probability current density for free particle plane wave is
A Zero
B Constant
C Infinite
D Imaginary
A superposition of energy eigenstates leads to
A Stationary wave
B Time-dependent probability density
C No wave
D Constant phase
When measuring energy in QM, the outcome must be
A Any real number
B Only discrete eigenvalues (for bound systems)
C Only classical energy
D Undefined
The wavefunction collapse is
A Required for normalization
B A change to an eigenstate of measured observable
C A mathematical error
D A relativistic effect
Quantum numbers allowed for electron in n = 3 level include
A l = 0, 1, 2
B l = 0 only
C l = 1 only
D l = 0, 2 only