Q1 50M Compulsory explain Quantum mechanics and atomic physics
(a) Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments. 10 marks
(b) Determine the ground state energy of an electron in an infinite potential well of width of 2 Å. 10 marks
(c) Draw the normal Zeeman pattern for ¹F₃—¹D₂ transition. 10 marks
(d) In case of pure rotational states, if the temperature will be doubled, then calculate the rotational quantum number corresponding to maximum population density. [Assume that temperature is high] 10 marks
(e) The quantum numbers of two electrons in a two-valence electron atom are n₁ = 6, l₁ = 3, s₁ = ½; n₂ = 5, l₂ = 1, s₂ = ½. Assuming L-S coupling, find the possible values of L and J. 10 marks
Answer approach & key points
Begin with part (a) explaining the fundamental distinction between Heisenberg's intrinsic quantum uncertainty and classical instrumental error, using clear conceptual distinction. For part (b), derive the ground state energy formula for infinite potential well and substitute a = 1 Å (half-width), showing E₁ = h²/(8ma²). Part (c) requires drawing the normal Zeeman triplet pattern with proper spacing and polarization labels. Part (d) involves deriving that J_max ≈ √(kT/2B) - ½, showing J_max increases by √2 when T doubles. Part (e) applies L-S coupling rules: L ranges from |l₁-l₂| to l₁+l₂ (i.e., 2,3,4), S=0 or 1, then J from |L-S| to L+S for each case. Allocate approximately 20% time to each part given equal marks distribution.
- Part (a): Distinguish Heisenberg uncertainty principle (intrinsic, ΔxΔp ≥ ℏ/2) from classical instrumental error (reducible with better apparatus); cite that uncertainty principle holds even with perfect instruments
- Part (b): Ground state energy E₁ = h²/(8ma²) = π²ℏ²/(2mL²) where L = 2a = 2 Å; calculate numerical value ~9.4 eV or 150 aJ
- Part (c): Normal Zeeman effect shows triplet pattern: π component (Δm=0, unshifted) and σ⁺, σ⁻ components (Δm=±1, shifted by ±eℏB/2m); draw energy level diagram with proper spacing and label polarizations
- Part (d): Rotational population N_J ∝ (2J+1)exp[-BJ(J+1)/kT]; find J_max by differentiation, yielding J_max ≈ √(kT/2B) - ½; when T→2T, J_max increases by factor of √2
- Part (e): L-S coupling: L = 2,3,4; S = 0 (singlet) or 1 (triplet); for S=0: J=L so J=2,3,4; for S=1: J ranges from |L-1| to L+1 giving appropriate values for each L
Q2 50M prove Quantum mechanics and statistical mechanics
(a) What is the density of states? For a relativistic particle of rest mass μ, prove that the density of states in the extreme relativistic limit (E ≫ μc²) is g(E) = V/π²ℏ³c³ E² where g(E) = density of states, V = volume of the system containing the particle, E = total energy, c = velocity of light and h = Planck's constant. 20 marks
(b) Obtain the expressions for reflection coefficient (R) and transmission coefficient (T) for reflected waves and transmitted waves from an infinite thin barrier. 15 marks
(c) For a potential with the boundary conditions V(x) = {0, x < -a; V, -a < x < a; 0, x > a}, solve the Schrödinger's equation in one dimension and find out the conditions for tunnelling. 15 marks
Answer approach & key points
This question demands rigorous mathematical derivation across all three parts. Begin with a brief conceptual introduction to density of states, then allocate approximately 40% of effort to part (a) given its 20 marks weightage, with 30% each to parts (b) and (c). For (a), start from relativistic energy-momentum relation and integrate over phase space; for (b), apply boundary conditions at a delta-function barrier; for (c), solve the Schrödinger equation in three regions and match wavefunctions at boundaries to derive tunnelling conditions. Conclude with physical significance of each result.
- Part (a): Definition of density of states as number of quantum states per unit energy interval; derivation starting from relativistic dispersion relation E² = p²c² + μ²c⁴ and phase space volume element
- Part (a): Transformation to spherical coordinates in momentum space, integration over angular variables yielding factor 4π, and proper handling of extreme relativistic limit E ≫ μc² where E ≈ pc
- Part (b): Setup of Schrödinger equation with delta-function potential V(x) = V₀δ(x); wavefunction ansatz in regions x<0 and x>0 with incident, reflected, and transmitted components
- Part (b): Application of continuity condition at x=0 and discontinuity condition for derivative from integration across delta barrier; derivation of R = |B/A|² and T = |C/A|² with R + T = 1
- Part (c): General solution forms in three regions: plane waves e^(±ikx) for |x|>a with k = √(2mE)/ℏ, and decaying/growing exponentials e^(±κx) for |x|<a with κ = √[2m(V-E)]/ℏ
- Part (c): Matching conditions at x = ±a for wavefunction and first derivative; derivation of transcendental equation for bound states and tunnelling condition when E < V involving real κ
- Physical interpretation: Connection between density of states and blackbody radiation/fermion systems; tunnelling applications to nuclear alpha decay and scanning tunnelling microscopy
Q3 50M calculate Atomic and molecular spectroscopy
(a) Find out the difference in frequencies of Lyman-alpha line in hydrogen and deuterium atoms. 15 marks
(b) The Stern-Gerlach experiment is a landmark experiment in quantum mechanics. Discuss about the most important findings of this experiment. 15 marks
(c) (i) From the pure rotational absorption spectra of a diatomic molecule (HF), the wave number difference between the consecutive rotational lines is found to be Δν̄ = 4050 m⁻¹. Calculate the following:
(1) Rotational constant
(2) Moment of inertia
(3) Distance between two atoms (bond length)
[Given, M_H = 1 u, M_F = 19 u] 10 marks
(ii) The force constant of HCl molecule is 4.8×10⁵ dyne/cm. Calculate the wave numbers of Stokes and anti-Stokes lines, when excited with a radiation of wavelength 4358 Å.
[Given, μ_HCl = 1.61×10⁻²⁴ g] 10 marks
Answer approach & key points
Begin with the directive to calculate and discuss across four sub-parts: spend ~30% time on (a) isotope shift calculation using reduced mass correction; ~25% on (b) discussing Stern-Gerlach findings with experimental schematic; ~25% on (c)(i) rotational spectroscopy of HF; and ~20% on (c)(ii) Raman spectroscopy of HCl. Structure as: brief theory → step-by-step derivation → numerical substitution → final result with units → physical significance.
- (a) Reduced mass calculation for H (μ_H) and D (μ_D), Rydberg formula with reduced mass correction, frequency difference Δν = ν_H − ν_D ≈ 4.53×10¹¹ Hz or equivalent
- (b) Experimental setup with inhomogeneous magnetic field, silver atom beam splitting into two discrete components, direct evidence of space quantization and electron spin (intrinsic angular momentum ℏ/2)
- (c)(i) Rotational constant B = Δν̄/2 = 2025 m⁻¹, moment of inertia I = h/(8π²cB), bond length r₀ = √(I/μ) ≈ 0.92 Å for HF
- (c)(ii) Vibrational frequency ω = (1/2π)√(k/μ), Raman shift Δν̄ = ±(ν₀ ∓ ν_vib), Stokes and anti-Stokes lines at ν̄₀ − ν̄_vib and ν̄₀ + ν̄_vib respectively
- Proper unit conversions throughout: CGS to SI for (c)(ii), unified atomic mass to kg, wavenumber to frequency where needed
- Physical significance: isotope shift tests QED predictions, Stern-Gerlach validates quantum mechanics vs classical expectations, spectroscopic constants determine molecular structure
Q4 50M state Quantum mechanics and atomic physics
(a) State how for spin-half particles, the spin (σ) can be expressed by its three components σ_x, σ_y and σ_z. 20 marks
(b) By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy. 15 marks
(c) Distinguish between fluorescence and phosphorescence. Explain the mechanisms responsible for these phenomena. Discuss the applications of fluorescence and phosphorescence in the fields such as biochemistry, material science, etc. 15 marks
Answer approach & key points
The directive 'state' in part (a) demands precise, formal presentation of spin-half algebra with Pauli matrices, while parts (b) and (c) require 'determine' and 'distinguish/explain/discuss' respectively. Allocate approximately 40% of effort to part (a) given its 20 marks: present σ = (σ_x, σ_y, σ_z) with explicit 2×2 Pauli matrices, their commutation relations, and eigenvalue properties. Spend ~30% on part (b): solve radial Schrödinger equation for n=1, l=0, derive E₁ = -13.6 eV and identify zero-point energy from kinetic term or uncertainty principle. Allocate remaining ~30% to part (c): construct clear comparison table, explain singlet vs. triplet states and intersystem crossing, cite Indian applications such as fluorescence microscopy at IISc Bengaluru or phosphorescent safety signage in Indian Railways. Conclude with integrated remarks on quantum phenomena spanning atomic to molecular scales.
- Part (a): Pauli spin matrices σ_x = [[0,1],[1,0]], σ_y = [[0,-i],[i,0]], σ_z = [[1,0],[0,-1]]; spin operator S = (ℏ/2)σ; commutation relations [σ_i, σ_j] = 2iε_ijk σ_k; eigenvalues ±1 for each component
- Part (a): Spin as vector operator σ = σ_x î + σ_y ĵ + σ_z k̂; total spin magnitude S² = s(s+1)ℏ² with s=1/2; connection to SU(2) representation
- Part (b): Radial Schrödinger equation for hydrogen ground state: -ℏ²/2μ ∇²ψ - e²/4πε₀r ψ = Eψ; separation into radial and angular parts; ground state wavefunction ψ₁₀₀ = (1/√πa₀³) exp(-r/a₀)
- Part (b): Zero-point energy derivation: E₁ = -μe⁴/8ε₀²ℏ² = -13.6 eV; kinetic energy expectation value ⟨T⟩ = +13.6 eV; potential energy ⟨V⟩ = -27.2 eV; zero-point energy identified as positive kinetic energy contribution or via ΔxΔp ≥ ℏ/2
- Part (c): Distinction table: fluorescence (spin-allowed, S₁→S₀, 10⁻⁹-10⁻⁷ s) vs phosphorescence (spin-forbidden, T₁→S₀, 10⁻³-10³ s); Jablonski diagram with radiative and non-radiative transitions
- Part (c): Mechanisms: fluorescence via prompt emission without change in spin multiplicity; phosphorescence requires intersystem crossing (ISC) with spin-orbit coupling, delayed emission through forbidden transition
- Part (c): Applications: fluorescence—green fluorescent protein (GFP) tagging, flow cytometry, Indian biomedical research (CCMB Hyderabad); phosphorescence—organic light-emitting diodes (OLEDs), persistent luminescent materials for emergency exit signs, security inks in Indian currency
Q5 50M Compulsory derive Quantum states, particle physics, solid state physics, electronics
(a) Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0). 10 marks
(b) ρ⁰ and K⁰ mesons both decay mostly to π⁺ and π⁻. Why the mean lifetime of ρ⁰ is 10⁻²³ s, whereas that of K⁰ is 0·89 × 10⁻¹⁰ s? 10 marks
(c) Find the radius of the interstitial sphere which can just fit into the void at the body centre of the fcc structure coordinated by the facial atoms. 10 marks
(d) In powder diffraction method pattern for lead with radiation of wavelength λ = 1·54 Å, the (220) Bragg reflection angle is θ = 32°. Find the radius of the atom. 10 marks
(e) (i) What are the differences in electrical characteristics of FET (JFET) and MOSFET? 7 marks
(ii) How does n-channel FET differ from p-channel FET? 3 marks
Answer approach & key points
Begin with a brief introduction acknowledging the diverse physics domains covered (quantum mechanics, particle physics, solid state, and electronics). For part (a), derive the energy splitting using exchange interaction and spin wavefunctions; for (b), explain using strong vs. weak decay selection rules; for (c), derive the octahedral void geometry in FCC; for (d), apply Bragg's law and unit cell calculation; for (e)(i)-(ii), tabulate comparative characteristics. Allocate approximately 20% time to (a), 15% to (b), 20% to (c), 20% to (d), and 25% to (e) combined, reflecting mark distribution and derivation complexity.
- (a) Derivation of triplet-singlet energy splitting using symmetric/antisymmetric spin wavefunctions and exchange integral J, showing E_triplet = E_0 - J and E_singlet = E_0 + J for two-electron system
- (b) Explanation of ρ⁰ decay via strong interaction (OZI-allowed, resonant, Γ ~ 150 MeV) versus K⁰ decay via weak interaction (ΔS = 1, strangeness changing, CP violation context with K_S and K_L)
- (c) Geometric derivation: octahedral void radius r = 0.414R where R is atomic radius, using FCC geometry with face-center to body-center distance relationship
- (d) Application of Bragg's law nλ = 2d sinθ, calculation of d_220 = a/√8, determination of lattice parameter a, and atomic radius r = a√2/4 for FCC lead
- (e)(i) Distinction between JFET (depletion-mode only, pn-junction gate, higher input impedance ~10⁹ Ω) and MOSFET (enhancement/depletion modes, insulated gate, higher input impedance ~10¹² Ω, threshold voltage concept)
- (e)(ii) Carrier type (electrons vs holes), mobility differences, threshold voltage polarity, and drain current direction in n-channel versus p-channel FETs
Q6 50M calculate Nuclear physics, shell model, leptonic conservation
(a) The total binding energies of ¹⁵₈O, ¹⁶₈O and ¹⁷₈O are 111·96 MeV, 127·62 MeV and 131·76 MeV respectively. Determine the energy gap between 1p₁/₂ and 1d₅/₂ neutron shells for the nuclide whose mass number is close to 16. 15 marks
(b) State the basic assumption of single-particle shell model. How do the centrifugal and spin-orbit terms remove the degeneracy of three-dimensional spherical harmonic oscillator? 10+10=20 marks
(c) Explain the various leptonic family members. What is leptonic number conservation? Based on this conservation law, tell whether the following reactions are possible or not:
(i) π⁻ → μ⁻ + ν̄ₜ
(ii) n → p⁺ + e⁻ + ν̄ₑ 15 marks
Answer approach & key points
Begin with a clear statement of the shell model assumptions for part (b), then proceed to calculate the energy gap in part (a) using binding energy differences—this carries the highest marks (15) and requires careful identification of neutron shell transitions. Allocate approximately 35% effort to (a), 40% to (b) given its theoretical depth (20 marks), and 25% to (c). Structure as: (b) theoretical foundation → (a) numerical application → (c) particle physics application with conservation law verification.
- Part (a): Correct identification that ¹⁶O has closed shells (Z=N=8), and that ¹⁵O has a 1d₅/₂ neutron hole while ¹⁷O has a 1d₅/₂ neutron particle; calculation of energy gap using BE(¹⁶O) - BE(¹⁵O) and BE(¹⁷O) - BE(¹⁶O) with proper averaging
- Part (b): Statement of independent particle motion in a mean potential; explanation of how l(l+1)ħ²/2mr² centrifugal term lowers energy for higher l at same n, and spin-orbit coupling ξ(r)L·S splits j = l ± 1/2 states with inverted ordering for natural parity
- Part (b): Clear derivation or explanation of the spin-orbit term origin from Dirac equation or phenomenological potential, showing how it creates the shell structure magic numbers 2, 8, 20, 28, 50, 82, 126
- Part (c): Enumeration of three lepton families (e, μ, τ) with their neutrinos and antiparticles; definition of lepton number Lₑ, Lᵤ, Lₜ with L = +1 for leptons, -1 for antileptons, 0 for hadrons
- Part (c)(i): Analysis showing π⁻ → μ⁻ + ν̄ₜ violates tau lepton number conservation (Lₜ: 0 → 0 + (-1)), hence forbidden; correct allowed decay is π⁻ → μ⁻ + ν̄ᵤ
- Part (c)(ii): Verification that n → p⁺ + e⁻ + ν̄ₑ conserves baryon number, charge, and all lepton numbers (Lₑ: 0 → 0 + 1 + (-1) = 0), hence allowed as standard beta decay
Q7 50M calculate Nuclear physics and magnetic materials
(a) What is the minimum energy required to break a ₂He⁴ nucleus into free protons and neutrons?
[ Given, m_H = 1·007825 amu, m_n = 1·008665 amu, m_e = 0·00055 amu and m_He = 4·002603 amu ]
15
(b) (i) Consider a uranium nucleus (₉₂U²³⁶) breaking up spontaneously into two equal parts. Estimate the reduction of electrostatic energy of the nucleus considering uniform charge distribution.
[ Assume that nuclear radius is 1·2×10⁻¹³ A¹/³ cm ]
15
(ii) Is it possible for a photon to transfer all its energy to a free electron? Give reasons.
5
(c) Explain the cause of hysteresis phenomenon in ferromagnetic materials. What does the area of the hysteresis loop signify?
10+5=15
Answer approach & key points
This is a multi-part numerical-cum-descriptive question requiring precise calculations for (a) and (b)(i), conceptual reasoning for (b)(ii), and explanatory analysis for (c). Allocate approximately 35% time to part (a) for careful mass-energy conversion, 35% to part (b) including both calculation and reasoning, and 30% to part (c) with a clear hysteresis diagram. Begin with the binding energy calculation using atomic masses correctly, proceed through electrostatic energy estimation with proper radius scaling, address the photon-electron collision physics with reference to Compton scattering constraints, and conclude with domain theory explanation and energy dissipation interpretation.
- Part (a): Correct identification that atomic mass of hydrogen includes electron, so use m_p = m_H - m_e or appropriate atomic mass accounting; calculation of mass defect Δm = [2m_H + 2m_n - m_He] or equivalent; conversion to energy using 1 amu = 931.5 MeV/c² yielding ~28.3 MeV
- Part (b)(i): Application of electrostatic energy formula U = (3/5)(Z²e²)/(4πε₀R) for uniform sphere; correct radius scaling R ∝ A^(1/3) with R_He = R₀(4)^(1/3) and R_U = R₀(236)^(1/3); calculation of energy ratio and reduction factor considering two fragments each with Z/2 and A/2
- Part (b)(ii): Recognition that photon-electron energy transfer requires momentum conservation; explanation that free electron cannot absorb photon completely due to simultaneous energy-momentum conservation violation; reference to Compton scattering or need for bound electron/third body
- Part (c): Explanation of hysteresis via domain wall movement, irreversible domain rotation, and pinning by impurities/crystal defects; clear distinction between reversible and irreversible magnetization processes; area interpretation as energy dissipated per unit volume per cycle (hysteresis loss)
- Part (c): Qualitative or quantitative sketch of B-H loop showing saturation, remanence, coercivity; labeling of key points and proper loop orientation
Q8 50M explain Magnetic materials, XRD and microprocessors
(a) Classify diamagnetic, paramagnetic and ferromagnetic materials in terms of their magnetic susceptibility (χ). Plot and explain the variation of 1/χ with temperature for the three materials.
20
(b) What is X-ray diffraction? How is an XRD pattern used to determine the crystal structure of the material?
15
(c) (i) What is a microprocessor? Describe the internal functioning of a microprocessor with block diagrams.
(ii) How do thermistors and solar cells differ in structure and operation?
10+5=15
Answer approach & key points
Begin with a brief introduction acknowledging the interdisciplinary nature of the question spanning solid-state physics and electronics. For part (a), spend approximately 40% of your effort (20 marks) explaining magnetic susceptibility classifications with clear Curie-Weiss law derivations and properly labeled 1/χ vs T plots. Allocate 30% (15 marks) to part (b), describing Bragg's law and indexing XRD peaks for crystal structure determination. Distribute the remaining 30% (15 marks) between (c)(i) microprocessor architecture with block diagrams and (c)(ii) comparative analysis of thermistors versus solar cells, using Indian examples like ISRO's microprocessor developments or solar cell applications in rural electrification.
- Part (a): Classification of diamagnetic (χ < 0, temperature-independent), paramagnetic (χ > 0, χ = C/T), and ferromagnetic (χ >> 0, follows Curie-Weiss law χ = C/(T-Tc)) with correct sign conventions and temperature dependence
- Part (a): Accurate plots showing 1/χ vs T: horizontal line for diamagnetic, straight line through origin for paramagnetic, and linear with positive intercept (Tc) for ferromagnetic above Curie temperature
- Part (b): Bragg's law (nλ = 2d sinθ) explanation, powder method/Bragg spectrometer, and systematic procedure for determining crystal structure through d-spacing calculation and comparison with standard tables
- Part (c)(i): Definition of microprocessor as CPU on single chip, block diagram showing ALU, control unit, registers, address/data/control buses, and fetch-decode-execute cycle explanation
- Part (c)(ii): Structural differences (thermistors: metal oxide semiconductors with negative temperature coefficient; solar cells: p-n junction with depletion region) and operational differences (resistance vs. photovoltaic effect)
- Part (c)(ii): Indian context: mention indigenous microprocessors like Shakti processor (IIT-Madras) or ISRO's Vikram series, and solar applications in National Solar Mission