(a) What is the importance of study of deuteron ? Obtain the solution of Schrödinger equation for ground state of deuteron and…

(a) Importance: the deuteron is the only bound two-nucleon system, so it gives direct information on the n–p force: its range, spin dependence (S = 1 bound, S = 0 unbound), tensor force (from the quadrupole moment and D-state admixture) and charge independence (pp and nn are unbound). It also anchors stellar fusion.

Ground state: L = 0, S = 1, J^π = 1⁺. With u(r) = rR(r), reduced mass μ ≈ M/2 and E = −B (B = 2.224 MeV), take a square well V = −V₀ for r < R and 0 for r > R.

For r < R: u″ = −k²u, with k² = 2μ(V₀ − B)/ħ², and u(0) = 0, so u = A sin kr. For r > R: u″ = κ²u, with κ² = 2μB/ħ², so u = C e^(−κr).

Continuity of u and u′ at r = R gives k cot kR = −κ. Since the right side is negative, kR lies between π/2 and π. Normalisation gives u = A sin kr (r < R) and u = A sin kR e^(−κ(r−R)) (r > R), with ψ = (u/r)χ(S = 1).

Loose binding: with μc² ≈ 469.5 MeV and ħc = 197.3 MeV fm, κ = √(2μc²B)/ħc ≈ 0.232 fm⁻¹, so 1/κ ≈ 4.3 fm. The wavefunction therefore extends well beyond the force range R ≈ 2 fm; for R = 2 fm the condition gives V₀ ≈ 37 MeV, so B/V₀ ≈ 0.06. Binding per nucleon is only 1.11 MeV against about 8 MeV for typical nuclei, and there is no bound excited state. Hence the deuteron is loosely bound. (The small D-state admixture from the tensor force is neglected here.)

(b) Nucleons move in V = ½Mω²r², with levels E_N = (N + 3/2)ħω, N = 2n + l. Adjacent major shells therefore differ by ħω.

Including spin and isospin, the degeneracy of shell N is 2(N + 1)(N + 2). Filling shells up to N gives A = (2/3)(N + 1)(N + 2)(N + 3) ≈ 2N³/3, so N ≈ (3A/2)^(1/3).

For a nucleon in shell n, ⟨r²⟩ = (n + 3/2)b², with b² = ħ/(Mω). Averaging over the filled shells (for large N, Σ n g_n/A ≈ 3N/4), ⟨r²⟩ ≈ (3N/4)b².

A uniform sphere of radius R = R₀A^(1/3) has ⟨r²⟩ = (3/5)R₀²A^(2/3). Equating, with R₀ = 1.2 fm, b² = (4/5)R₀²(2/3)^(1/3)A^(1/3) ≈ 1.006 A^(1/3) fm².

Then ħω = ħ²/(Mb²) = (ħc)²/(Mc² b²) = (197.3)²/(938 × 1.006 A^(1/3)) MeV ≈ 41A^(−1/3) MeV.

(c) Three active light neutrino flavours exist: ν_e, ν_μ and ν_τ (with antineutrinos). The invisible width of the Z boson fixes the number of light active species at N_ν ≈ 3 (LEP: 2.984 ± 0.008). Sterile neutrinos are not required by this count.

The flavour states are superpositions of three mass eigenstates ν₁, ν₂, ν₃. Oscillation experiments measure only mass-squared differences:

• Solar: Δm²₂₁ = m₂² − m₁² ≈ 7.5 × 10⁻⁵ eV².
• Atmospheric: |Δm²₃₁| ≈ 2.5 × 10⁻³ eV².

So the masses differ by very little, and the absolute scale is unknown. The sign of Δm²₃₁ is also unknown, giving two orderings: normal (m₃ > m₂ > m₁) and inverted (m₂ > m₁ > m₃).

Limits on the absolute scale: tritium beta decay (KATRIN) gives an effective electron-neutrino mass below 0.45 eV, and cosmology limits the sum of masses to roughly 0.1 eV or less (model dependent). In every case neutrino masses are far smaller than those of charged leptons.