Multi-Electron Atoms

The exchange operator P̂ᵢⱼ swaps the labels of two identical particles in a state, and because it is Hermitian and squares to the identity its only eigenvalues are +1 for a symmetric state and −1 for an antisymmetric one. Use the supplied topic model Y_out = 2*(1+eta)*sin(pi*x0)*sin(nb*pi*x0)/max(abs(2*sin(nb*pi*x0))+abs(2*eta*sin(pi*x0)),0.35), where Y_out is Ψ AT x₁ = x₂ (SCALED) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|; max(a,b) selects the larger value; unmarked trigonometric functions use radians. eta is Exchange mixing η, dimensionless. Keep x0 (Position x₂ of particle 2) = 0.23 L; nb (Second orbital n_b) = 2 fixed. For this exercise use only -0.7875 ≤ eta ≤ 0.7875 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 = 0.69555898, with a bounded reading uncertainty ±0.011944. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown eta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575, 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.

Electron spin is a two-dimensional complex vector space that every electron carries alongside its wavefunction, acted on by S = (ħ/2)σ, whose measurement along any axis returns only +ħ/2 or −ħ/2. Use the supplied topic model Y_out = 0.5*cosd(b-a), where Y_out is ⟨S⋅n⟩ IN STATE m (ħ). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. a is State direction θm from +z, measured in °. Keep b (Analyser direction θn from +z) = 120 ° fixed. For this exercise use only 123.075 ° ≤ a ≤ 160.875 ° 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 = 0.44888702 ħ, with a bounded reading uncertainty ±0.007559 ħ. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00378 °, 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 Slater determinant is the antisymmetrised product of N orthonormal spin-orbitals, Ψ = (1/√N!) det[φ_i(x_j)] — the simplest N-fermion state, and the one whose occupation numbers are all exactly 0 or 1. Use the supplied topic model Y_out = 2*sin(pi*p)^2*sin(2*pi*p)^2/max(sin(pi*p)^2+sin(2*pi*p)^2,0.05), where Y_out is DIRECT TERM at x₂ = x₁ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; unmarked trigonometric functions use radians. p is Reference electron x₁ / L, measured in L. Keep s (Exchange weight s (−1 = fermion)) = -1 fixed. For this exercise use only 0.323 L ≤ p ≤ 0.4175 L 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 = 0.60469016, with a bounded reading uncertainty ±0.023362. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00000945 L, 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.

The variational bound on helium is the lowest expectation value ⟨ψ|H|ψ⟩ reachable from a chosen family of trial states; it is guaranteed to lie at or above the true ground-state energy, never below it. Use the supplied topic model Y_out = 27.2114*zp*(zp-4+0.625*lam), where Y_out is TRIAL ENERGY E(Z′) (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. zp is Trial nuclear charge Z′, dimensionless. Keep lam (Repulsion switched on, λ) = 1 fixed. For this exercise use only 1.71063 ≤ zp ≤ 2.34063 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 = -72.971753 eV, with a bounded reading uncertainty ±0.67536 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown zp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000063, 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.