Level 1 — RecognitionModern Physics

Modern Physics

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level: 1 — Recognition (MCQ, Matching, True/False with justification) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. The photoelectric effect demonstrates the ______ nature of light. (a) wave (b) particle (c) relativistic (d) magnetic

Q2. The energy of a photon of frequency $f$ is: (a) $hf$ (b) $h/f$ (c) $hf^2$ (d) $f/h$

Q3. The de Broglie wavelength of a particle of momentum $p$ is: (a) $hp$ (b) $h/p$ (c) $p/h$ (d) $h/p^2$

Q4. The energy of the ground state ( $n=1$ ) of the hydrogen atom is: (a) $-3.4\text{ eV}$ (b) $+13.6\text{ eV}$ (c) $-13.6\text{ eV}$ (d) $-1.51\text{ eV}$

Q5. The Heisenberg uncertainty principle for position and momentum states: (a) $\Delta x\,\Delta p \le \hbar/2$ (b) $\Delta x\,\Delta p \ge \hbar/2$ (c) $\Delta x\,\Delta p = 0$ (d) $\Delta x + \Delta p \ge \hbar$

Q6. The Davisson–Germer experiment provided evidence for: (a) photon momentum (b) nuclear structure (c) wave nature of electrons (d) time dilation

Q7. In the particle-in-a-box model, the allowed energy levels are proportional to: (a) $n$ (b) $n^2$ (c) $1/n$ (d) $1/n^2$

Q8. The physical meaning of $|\psi|^2$ is: (a) energy of the particle (b) probability density (c) momentum (d) wavelength

Q9. The Balmer spectral series of hydrogen corresponds to electron transitions ending at: (a) $n=1$ (b) $n=2$ (c) $n=3$ (d) $n=\infty$

Q10. The Pauli exclusion principle states that no two electrons in an atom can have: (a) the same energy (b) the same spin (c) the same set of all four quantum numbers (d) opposite spins

Q11. Relativistic momentum is given by: (a) $mv$ (b) $\gamma mv$ (c) $mv/\gamma$ (d) $\gamma m$

Q12. In the BE-per-nucleon curve, the peak (most stable nucleus) occurs near: (a) hydrogen (b) helium (c) iron (d) uranium

Section B — Matching (1 mark each, 6 marks)

Q13. Match each quantity/concept in Column I with its correct description in Column II.

Column I	Column II
(i) $E=hf$	(P) wavelength shift in photon–electron scattering
(ii) Compton effect	(Q) $N=N_0 e^{-\lambda t}$
(iii) Radioactive decay law	(R) photon energy
(iv) Time dilation	(S) mass defect converted to energy
(v) Binding energy	(T) $\Delta t = \gamma\,\Delta t_0$
(vi) Fusion	(U) combining light nuclei to release energy

Section C — True/False WITH Justification (2 marks each: 1 for T/F, 1 for justification)

Q14. Increasing the intensity of incident light always increases the maximum kinetic energy of emitted photoelectrons. (True/False + justify)

Q15. A more massive particle moving at the same speed as a lighter one has a shorter de Broglie wavelength. (True/False + justify)

Q16. In alpha decay, the mass number of the nucleus decreases by 4. (True/False + justify)

Q17. According to special relativity, the speed of light in vacuum is the same in all inertial frames. (True/False + justify)

Q18. Quantum tunneling allows a particle to pass through a barrier even when its energy is less than the barrier height. (True/False + justify)

Q19. The half-life of a radioactive sample depends on the initial number of nuclei present. (True/False + justify)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (b) particle — photoelectric effect is explained by photons (quanta), not wave theory. Q2. (a) $hf$ — Planck/Einstein relation for photon energy. Q3. (b) $h/p$ — de Broglie hypothesis. Q4. (c) $-13.6\text{ eV}$ — $E_n=-13.6/n^2$ with $n=1$ . Q5. (b) $\Delta x\,\Delta p \ge \hbar/2$ — standard uncertainty relation. Q6. (c) wave nature of electrons — electron diffraction from a Ni crystal. Q7. (b) $n^2$ — $E_n = n^2\pi^2\hbar^2/(2mL^2)$ . Q8. (b) probability density — Born interpretation. Q9. (b) $n=2$ — Balmer series terminates at level 2. Q10. (c) the same set of all four quantum numbers. Q11. (b) $\gamma mv$ . Q12. (c) iron — $^{56}$ Fe region has maximum BE per nucleon (~8.8 MeV).

Section B (1 mark each)

Q13.

(i) → (R) — $E=hf$ is photon energy.
(ii) → (P) — Compton scattering gives wavelength shift.
(iii) → (Q) — exponential decay law.
(iv) → (T) — dilated time = $\gamma \times$ proper time.
(v) → (S) — binding energy from mass defect.
(vi) → (U) — fusion combines light nuclei.

Section C (2 marks each)

Q14. False (1). Max KE depends on frequency, not intensity: $KE_{max}=hf-\phi$ . Intensity increases the number of electrons, not their maximum energy (1).

Q15. True (1). $\lambda=h/p=h/(mv)$ ; for equal $v$ , larger $m$ ⇒ larger $p$ ⇒ smaller $\lambda$ (1).

Q16. True (1). An alpha particle is $^4_2$ He; emitting it reduces mass number $A$ by 4 (and $Z$ by 2) (1).

Q17. True (1). Second postulate of special relativity: $c$ is invariant in all inertial frames (1).

Q18. True (1). Tunneling: the wavefunction is nonzero beyond a finite barrier, giving a nonzero transmission probability even for $E<V_0$ (1).

Q19. False (1). Half-life $t_{1/2}=\ln 2/\lambda$ depends only on the decay constant, an intrinsic property, not on $N_0$ (1).

[
  {"claim":"Hydrogen ground state energy is -13.6 eV (n=1)","code":"E=-13.6/1**2; result = (E == -13.6)"},
  {"claim":"Balmer series terminates at n=2; example transition n=3->2 gives positive photon energy","code":"E=-13.6; dE = E/3**2 - E/2**2; result = (dE > 0)"},
  {"claim":"Half-life relation t_half = ln2/lambda independent of N0","code":"lam=symbols('lam',positive=True); t_half=ln(2)/lam; result = (t_half.free_symbols == {lam})"},
  {"claim":"Particle in box energy scales as n^2: E2/E1 = 4","code":"n1,n2=1,2; ratio=n2**2/n1**2; result = (ratio == 4)"}
]