(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

Question

(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

UPSC Answer Check · Accepted Answer

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|

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that density of states counts accessible microstates; properly distinguishes between non-relativistic (g(E) ∝ E^(1/2)) and extreme relativistic (g(E) ∝ E²) regimes; for (c) correctly identifies tunnelling requires E < V with evanescent wave in barrier region	Basic definition of density of states correct but confuses relativistic limits; identifies wavefunction forms in each region for (b) and (c) but makes errors in boundary condition application	Fundamental misunderstanding of density of states concept; treats all particles as non-relativistic; fails to recognize quantum tunnelling phenomenon or applies classical forbidden region incorrectly
Derivation rigour	25%	12.5	Complete step-by-step derivations: (a) proper Jacobian transformation from p-space to E-space with d³p = 4πp²dp; (b) explicit delta-function integration yielding ψ'(0⁺) - ψ'(0⁻) = (2mV₀/ℏ²)ψ(0); (c) full determinant calculation for matching conditions leading to even/odd parity solutions	Major steps present but skips intermediate algebra; correct final expressions but unclear on delta-function matching conditions; sets up secular equation for (c) without complete solution	Missing critical steps like angular integration or boundary conditions; jumps to final formulas without justification; algebraic errors in wavefunction matching
Diagram / FBD	15%	7.5	Clear schematic for (a) showing momentum space spherical shell; for (b) wavefunction diagram showing incident, reflected, transmitted waves with phase shifts at barrier; for (c) potential well diagram with three regions and qualitative sketch of exponentially decaying wavefunction in barrier	At least two relevant diagrams present but lacking detail; basic potential diagram for (c) without wavefunction sketches; no momentum space visualization	No diagrams or completely irrelevant sketches; fails to illustrate key physical situations across all three parts
Numerical accuracy	20%	10	Correct prefactors throughout: (a) achieves exact coefficient V/(π²ℏ³c³) including factor of 2 for spin degeneracy if mentioned; (b) derives R = V₀²/(V₀² + 4k²ℏ⁴/m²) for delta barrier; (c) obtains correct transcendental equations tan(ka) = κ/k or -cot(ka) = κ/k for bound states	Correct functional dependencies but numerical prefactors off by factors of 2 or π; correct structure of R and T but algebraic errors in final simplification	Major dimensional errors; incorrect powers of E in density of states; violates probability conservation R + T ≠ 1; transcendental equations with wrong sign structure
Physical interpretation	20%	10	For (a), connects E² dependence to photon gas/blackbody radiation and explains relevance to early universe cosmology; for (b), discusses perfect transmission at specific energies (Ramsauer-Townsend effect analogue); for (c), explains Gamow factor for nuclear alpha decay and exponential sensitivity to barrier width with Indian context (Bhabha's contributions to cosmic ray physics)	Brief mention of applications without quantitative detail; recognizes tunnelling importance but no specific examples; mentions blackbody radiation without clear connection	Purely mathematical treatment with no physical insight; fails to explain why results matter; no connection to experimental phenomena or Indian scientific contributions

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