(a) An operator P describing the interaction of two spin 1/2 particles is P = a + bσ⃗₁·σ⃗₂, where a and b are constants, and σ⃗₁…

(a) Simultaneous measurability of P, S² and S_z

Observables can be measured together if they commute, [A, B] = 0, because they then share a complete set of eigenstates. With S = (ħ/2)(σ₁ + σ₂) and σ₁² = σ₂² = 3, S² = (ħ²/4)(σ₁² + σ₂² + 2σ₁·σ₂) = (ħ²/4)(6 + 2σ₁·σ₂) = (ħ²/2)(3 + σ₁·σ₂), so σ₁·σ₂ = 2S²/ħ² − 3. Hence P = a + bσ₁·σ₂ = (a − 3b) + (2b/ħ²)S², a function of S² alone. Therefore [P, S²] = 0. Also [P, S_z] = (2b/ħ²)[S², S_z] = 0, because S² commutes with every component of the total spin, [S², S_z] = 0 (the Sᵢ satisfy the angular-momentum algebra). So P, S² and S_z form a mutually commuting set and can be measured simultaneously. Common eigenstates: the singlet (S = 0, S_z = 0) has P = a − 3b; the triplet (S = 1, S_z = ħ, 0, −ħ) has P = a + b.

(b) Reflection at a potential step

Let k = √(2mE)/ħ. For x ≤ 0: ψ = A e^ikx + B e^−ikx. For x > 0 the kinetic energy is E − 3E/4 = E/4, so q = √(2m·E/4)/ħ = k/2 and ψ = C e^iqx (only a transmitted wave). Continuity of ψ and ψ′ at x = 0: A + B = C, k(A − B) = qC = (k/2)C. So A − B = (A + B)/2, which gives A = 3B and B/A = 1/3. Reflection coefficient: R = |B/A|² = ((k − q)/(k + q))² = 1/9. The transmitted fraction is T = 1 − R = 8/9, equal to (q/k)|C/A|² = (1/2)(4/3)² = 8/9. Partial reflection occurs even though E > V₀, a purely wave effect with no classical counterpart.

(c) Particle in a one-dimensional box: electron and a 1 g sphere

(a) Eₙ = n²π²ħ²/(2ma²), with ħ = 1.0546×10⁻³⁴ J s. Electron: m = 9.109×10⁻³¹ kg, a = 10⁻¹⁰ m. E₁ = π²(1.0546×10⁻³⁴)²/(2 × 9.109×10⁻³¹ × 10⁻²⁰) = 6.03×10⁻¹⁸ J = 37.6 eV. E₂ = 4E₁ = 150 eV (first excited) and E₃ = 9E₁ = 339 eV (second excited). Sphere: m = 10⁻³ kg, a = 0.10 m. E₁ = π²(1.0546×10⁻³⁴)²/(2 × 10⁻³ × 10⁻²) = 5.5×10⁻⁶³ J (3.4×10⁻⁴⁴ eV). E₂ = 4E₁ = 2.2×10⁻⁶² J and E₃ = 9E₁ = 4.9×10⁻⁶² J. (b) For the electron, the level spacing is of order 100 eV and the de Broglie wavelength is comparable to the box, so the spectrum is strongly quantised and quantum effects dominate. For the sphere, the spacing of about 10⁻⁶² J is vastly smaller than kT ≈ 4×10⁻²¹ J at 300 K, so the levels are effectively continuous and quantum effects are negligible. (c) Take Δx ≈ a, so Δp ≈ ħ/a and v ≈ ħ/(ma). Electron: v ≈ 1.05×10⁻³⁴/(9.109×10⁻³¹ × 10⁻¹⁰) ≈ 1×10⁶ m/s, non-relativistic (the ground-state rms speed √(2E₁/m) ≈ 3.6×10⁶ m/s agrees in order of magnitude). Sphere: v ≈ 1.05×10⁻³⁴/(10⁻³ × 0.10) ≈ 1×10⁻³⁰ m/s, far too small to observe.