Second Quantisation and the Hartree-Fock Method

An occupation-number state lists, for an ordered set of orthonormal orbitals, how many identical particles occupy each; a creation operator raises one occupation, with a factor √(n + 1) for bosons and, for fermions, a −1 for each occupied orbital earlier in the order. Use the supplied topic model Y_out = (1-abs(op-floor(s/2^(k-1))%2))*(-1)^((s%2^(k-1))-floor((s%2^(k-1))/2)-floor((s%2^(k-1))/4)-floor((s%2^(k-1))/8)), where Y_out is FERMION COEFFICIENT (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|. k is Orbital acted on, k, dimensionless. Keep s (Pattern s = n₁ + 2n₂ + 4n₃ + 8n₄ + 16n₅) = 5; op (Operator: 0 = c†ₖ, 1 = cₖ) = 0 fixed. For this exercise use only 2 ≤ k ≤ 4, at the specified test settings separated by 1. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0, with a bounded reading uncertainty ±0.16. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of k consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The field operator ψ(x) = Σᵢ φᵢ(x)cᵢ removes a particle at position r with spin σ; it is the same operator whichever complete orthonormal orbitals expand it, and its matrix elements between the vacuum and an N-particle state are that state's wavefunction. Use the supplied topic model Y_out = sqrt(2)*(sin(pi/4)*(cosd(mix)*sin(2*pi*xp) + sind(mix)*sin(3*pi*xp)) - (cosd(mix) + sin(pi/4)*sind(mix))*sin(pi*xp)), where Y_out is Ψ(L/4, x₂) × L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; sind/cosd/tand use degrees; unmarked trigonometric functions use radians. mix is Mixing with empty φ₃, β, measured in °. Keep th (Occupied rotation θ) = 0 °; xp (Second electron x₂) = 0.3 L fixed. For this exercise use only 9.5625 ° ≤ mix ≤ 66.2625 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = -0.48679633, with a bounded reading uncertainty ±0.0072932. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown mix by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00567 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

A one-body operator is the sum of one single-particle operator f over all the particles; in second quantisation it is F = Σᵢⱼ ⟨i|f|j⟩ c†ᵢ cⱼ, each term emptying orbital j and filling orbital i with amplitude ⟨i|f|j⟩. Use the supplied topic model Y_out = cosd(theta), where Y_out is OVERLAP ⟨u|v⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Mixing angle θ, measured in °. Keep k (Admixed box level k) = 2 fixed. For this exercise use only 9.5625 ° ≤ theta ≤ 80.4375 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.60465125, with a bounded reading uncertainty ±0.057218. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0070875 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

A two-body operator in second quantisation removes a pair of particles from any two orbitals and puts them into any two, the same or different, weighted by the interaction's matrix element between the pairs: V̂ = ½Σ⟨ij|v|kl⟩c†ᵢc†ⱼcₗcₖ, with the annihilators in reverse order. Use the supplied topic model Y_out = ord*(11.422-1.1945*sind(2*th)), where Y_out is ⟨V̂⟩ IN ψ(θ) (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. th is Mixing angle θ, measured in °. Keep ord (Annihilator order: +1 cₗcₖ, −1 cₖcₗ) = 1 fixed. For this exercise use only 51.975 ° ≤ th ≤ 70.875 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 10.46813 eV, with a bounded reading uncertainty ±0.03081 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00189 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.