Identical Particles and Many-Electron Atoms

The exchange operator P swaps the labels of two particles, PΨ(r₁, r₂) = Ψ(r₂, r₁); because P² = 1 its only eigenvalues are +1, for symmetric states, and −1, for antisymmetric ones. Use the supplied topic model Y_out = (1000*sind(2*th))/1000, where Y_out is EXCHANGE MEAN ⟨P⟩ (before display rounding) (dimensionless). 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 c (Particle detected at c) = 0.25 L; n (Upper level n) = 2 fixed. For this exercise use only 141.975 ° ≤ th ≤ 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 = -0.79855147, with a bounded reading uncertainty ±0.025793. 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.

The symmetrisation postulate requires a state of identical particles to be unchanged by every pairwise exchange (bosons) or to change sign under every one (fermions); the spin-statistics connection, quoted from field theory, makes integer-spin particles bosons and half-integer-spin ones fermions. Use the supplied topic model Y_out = s*(1-m) + m*cosd(2*th-120), where Y_out is ⟨P₂₃⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. th is Direction θ in the mixed plane, measured in °. Keep m (Weight m in the mixed plane) = 1; s (Allowed sector s: +1 sym, −1 antisym) = 1 fixed. For this exercise use only 19.125 ° ≤ th ≤ 56.925 ° 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 = 0.81275518, with a bounded reading uncertainty ±0.027223. 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 0.00378 °, 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 normalised antisymmetric state of N fermions in N orthonormal spin-orbitals, (1/√(N!)) det[ψ_aⱼ(xᵢ)]; it vanishes whenever two of the spin-orbitals coincide, and that vanishing is the Pauli exclusion principle. Use the supplied topic model Y_out = (1000*0.5642*exp(-xi^2)*(cosd(theta) + sqrt(2)*sind(theta)*xi))/1000, where Y_out is BARE PRODUCT AT ξ₁ = ξ₂ (before display rounding) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; exp(x) means eˣ; sind/cosd/tand use degrees. theta is Angle θ between orbitals a and b, measured in °. Keep xi (Particle 2 at ξ₂ = x₂/ℓ) = 0.5; eta (Exchange sign η (−1 same-spin fermions, +1 bosons)) = -1 fixed. For this exercise use only 37.9125 ° ≤ 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 = 0.47254076, with a bounded reading uncertainty ±0.0097555. 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 0.0042525 °, 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.

An exchange effect is a difference between identical and distinguishable particles that comes from the symmetry of the state alone, with no interaction in the Hamiltonian; for positions it appears as a correlation between where the two particles are found. Use the supplied topic model Y_out = sqrt((1/6-(1+1/m^2)/(2*pi^2))-64*m^2*(1+cos(pi*m))/(pi^4*(m^2-1)^4)), where Y_out is RMS x₁ − x₂, BOSONS (a). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. m is Second level m (first is n = 1), dimensionless. Keep p (Particle 1 found at x₁) = 0.3 a fixed. For this exercise use only 2 ≤ m ≤ 5, 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 = 0.33529761 a, with a bounded reading uncertainty ±0.011322 a. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of m 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.