Quantum Mechanics 1Mark Questions

1. State and briefly explain two applications of quantum mechanics in modern technology.
   • Semiconductors: basis of modern electronics like computers, mobiles, and solar cells.
   • Lasers: based on stimulated emission, applied in medicine, communication, and industry.
2. Using the de-Broglie hypothesis, determine the wavelength of a particle given its momentum.
   The wavelength is given by \( \lambda = \dfrac{h}{p} \), where \( h \) is Planck’s constant and \( p \) is the momentum.
3. Explain Heisenberg's Uncertainty Principle.
   Heisenberg's Uncertainty Principle states that it is impossible to simultaneously know the exact position (\( x \)) and momentum (\( p \)) of a particle with perfect precision, mathematically \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), where \( \hbar \) is the reduced Planck's constant. This arises from the wave-particle duality.
4. What is wavefunction? Write two of its properties.
   The wavefunction \( \psi \) describes a particle's quantum state.
   • \( |\psi|^2 \) gives probability density.
   • \( \psi \) must be finite, single-valued, and continuous.
5. Draw the wavefunction and probabilityof the particle for energy levels (n = 1, 2, 3) in one-dimensional potential box.
   For a box of length \( L: \)
   \( \psi_n(x)=\sqrt{\tfrac{2}{L}}\sin\left(\tfrac{n\pi x}{L}\right) \) and \( P_n(x)=|\psi_n|^2 \).
   • n=1: one half-wave, no node inside.
   • n=2: two half-waves, one node inside.
   • n=3: three half-waves, two nodes inside.
   

   
6. Differentiate between eigenvalue and eigenfunction.
   • Eigenvalue: measurable number (e.g., energy).
   • Eigenfunction: wavefunction corresponding to that eigenvalue.
7. Why does a particle in a box never have zero energy?
   Zero energy means exact position and zero momentum, violating the uncertainty principle. Hence ground state energy is always > 0.
8. State the type of materials to which Bloch’s theorem applies.
   Bloch’s theorem applies to periodic crystalline materials.
9. Why are Bloch functions important for electron motion in solids?
   Bloch functions describe electron states in periodic potentials and explain band formation and conductivity.
10. Define Brillouin zone.
    It is the primitive cell in reciprocal (momentum) space of a crystal lattice.
11. Why do band gaps arise in solids according to Kronig-Penney model?
    Band gaps appear due to electron wave interference in periodic crystal potentials, making some energies forbidden.
12. Why is effective mass important in solid-state physics?
    It describes how electrons respond to forces in a crystal and simplifies transport calculations in devices.
13. Compare band gaps in conductors, semiconductors, and insulators.
    • Conductor: no gap, bands overlap.
    • Semiconductor: small gap (~1 eV).
    • Insulator: large gap (>3 eV).
14. How are solids classified based on band theory?
    Solids are classified as conductors, semiconductors, or insulators depending on their band gap size.
15. Differentiate between continuous and discrete energy spectrum.
    • Discrete: specific energy levels (atoms, particle in a box).
    • Continuous: unbroken range of energies (free particles).