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

(a) The deuteron is the simplest bound two-nucleon system. It tests the neutron–proton interaction, spin dependence, tensor force, iso-singlet character, and the fact that pp and nn are unbound. Its binding energy, magnetic moment, quadrupole moment and size constrain nuclear-force models and are important in stellar fusion and nuclear astrophysics.

In the central-force approximation, the deuteron ground state is L = 0, S = 1, J^π = 1⁺. Let u(r) = rR(r). With reduced mass μ = m_p m_n/(m_p + m_n) ≈ M/2, the radial Schrödinger equation is −ℏ²/(2μ) d²u/dr² + V(r)u = E u, E = −B, where B = 2.224 MeV is the binding energy. Take a square well: V(r) = −V₀ for r < R, V(r) = 0 for r > R. For r < R: d²u/dr² = −k²u, k² = 2μ(V₀ − B)/ℏ², so u(r) = A sin(kr), since u(0) = 0. For r > R: d²u/dr² = κ²u, κ² = 2μB/ℏ², so u(r) = C exp(−κr). Continuity of u and u′ at r = R gives the eigenvalue condition k cot(kR) = −κ. Normalising ∫₀^∞ |u(r)|² dr = 1 gives the deuteron wavefunction ψ(r) = [u(r)/r] χ(S=1,M), with u(r) = A sin(kr) for r < R and u(r) = A sin(kR) exp[−κ(r − R)] for r > R. Here A = [∫₀^R sin²(kr) dr + sin²(kR) ∫_R^∞ exp(−2κ(r − R)) dr]^(−1/2).

To show loose binding, use μc² ≈ 469.5 MeV, B = 2.224 MeV, ℏc = 197.3 MeV fm: κ = √(2μc²B)/(ℏc) = √(2×469.5×2.224)/(197.3) fm⁻¹ ≈ 0.232 fm⁻¹. Hence the exponential tail falls over 1/κ ≈ 4.3 fm, while the nuclear-force range is only R ≈ 2 fm. Also V₀ ≈ 35 MeV, so B/V₀ ≈ 0.064. Binding per nucleon is only about 1.11 MeV, much smaller than the usual ~8 MeV. The deuteron has no bound excited state. These facts show it is a loosely bound system. The actual deuteron also has a small D-state admixture due to the tensor force; the central-force model above gives the dominant S-state.

(b) In the nuclear shell model, nucleons move in an isotropic harmonic oscillator potential V(r) = ½ Mω²r². The energy levels are E_N = (N + 3/2)ℏω, where N = 2n + l. Thus the spacing between major oscillator shells is ℏω.

Including spin and isospin, the degeneracy of level N is g_N = 2(N + 1)(N + 2). The number of nucleons up to the highest filled shell N is A = Σ(n=0 to N) g_n = (2/3)(N + 1)(N + 2)(N + 3) ≈ 2N³/3. Therefore N ≈ (3A/2)^(1/3).

For a state with N oscillator quanta, ⟨r²⟩_N = (N + 3/2)ℏ/(Mω) = (N + 3/2)b², where b² = ℏ/(Mω). Averaging over all filled shells, ⟨r²⟩ = [Σ(n=0 to N) g_n (n + 3/2)b²]/A. Since Σ n g_n/A = 3N/4 exactly for filled shells, we get ⟨r²⟩ ≈ (3N/4)b².

Equate this to a uniform nuclear sphere: ⟨r²⟩ = (3/5)R² = (3/5)R₀²A^(2/3), with R₀ ≈ 1.2 fm. So (3N/4)b² = (3/5)R₀²A^(2/3). Using N = (3A/2)^(1/3), b² = (4/5)R₀²(2/3)^(1/3)A^(1/3) ≈ 1.006A^(1/3) fm². Then ℏω = ℏ²/(M b²) = (ℏc)²/(Mc² b²). With ℏc = 197.3 MeV fm and Mc² ≈ 938 MeV, ℏω ≈ (197.3)²/(938×1.006A^(1/3)) MeV ≈ 41A^(−1/3) MeV. Hence the major oscillator shell spacing is approximately ℏω ≈ 41A^(−1/3) MeV.

(c) In the standard three-neutrino oscillation framework, there are three active neutrino flavours: ν_e, ν_μ and ν_τ, with their corresponding antineutrinos. The invisible Z-boson width gives the number of light active neutrino species as N_ν = 2.984 ± 0.008, confirming three. Sterile neutrinos may exist, but they are not required and are not active.

These flavour states are not mass eigenstates. They are quantum superpositions of three mass eigenstates ν₁, ν₂ and ν₃. Neutrino oscillations measure mass-squared differences:

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

Thus the mass eigenstates differ by very small amounts. The absolute masses are not yet known, only differences. The sign of Δm²₃₁ is also unknown, giving two possible orderings:

• Normal hierarchy: m₃ > m₂ > m₁.
• Inverted hierarchy: m₂ > m₁ > m₃.

If the lightest mass is very small, the heavier states are around √(2.5×10⁻³) eV ≈ 0.05 eV. Cosmology gives Σm_ν < 0.12 eV, so quasi-degenerate masses must each be less than about 0.04 eV; direct beta-decay experiments give an effective electron-neutrino mass below about 0.8 eV. Neutrino masses are extremely small compared with charged-lepton masses.