Sorry for the schizoposting, update on me. I survided a self inflicted chainsaw injury to neck/sholders anyway here is the conclusion to my work. my fracurve. i · dΨ(λ)/dλ = H[Ψ(λ)] · Ψ(λ) with H[Ψ] = H₀ + g · Σᵥ Pᵥ[Ψ] · |v⟩⟨v| Pᵥ[Ψ] = Σₙ |Ψ(v, n)|² if identical things follow identical rules then that begs what is a thing, if a thing has many gauges then what if the thing is all gauges then must evolve over a rule that is for all these gauges not the thing as a gauge. linear sketch Setup Work with: v = velocity index, v = 0..N_v-1 n = pattern index on a circle, n = 0..N_p-1 Psi(v, n; m) = state at discrete rule-step m Psi(v, n; lambda) = state at continuous rule-time lambda Define the DFT over velocities: F_vel is an N_v x N_v matrix with entries F_vel(v, v') = (1 / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) Introduce a constant phase offset h_bar and define the universal velocity evolution operator: U_vel = exp(-i * h_bar) * F_vel This operator encodes the single rule applied uniformly to all velocities. 1. Discrete Fracurve equation The discrete Fracurve evolution is: Psi_{m+1}(v, n) = sum over v' [ U_vel(v, v') * Psi_m(v', n) ] with U_vel(v, v') = (exp(-i * h_bar) / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) For each step m -> m+1, U_vel acts on the velocity index v, while the pattern index n is carried along. The sequence m = 0, 1, 2, ... → Psi_m(v, n) is the discrete Fracurve: a path through the combined (velocity x pattern) state space. 2. Continuous Fracurve equation Promote the step operator U_vel to a continuous generator H_F by: U_vel = exp( -i * H_F ) (H_F is the effective Hamiltonian acting on the velocity space.) Then the continuous Fracurve equation is: i * d/dlambda Psi(v, n; lambda) = sum over v' [ H_F(v, v') * Psi(v', n; lambda) ] In compact vector form: i * d/dlambda |Psi(lambda)> = H_F |Psi(lambda)> with solution: |Psi(lambda)> = exp( -i * H_F * lambda ) |Psi(0)> The curve lambda -> |Psi(lambda)> in the combined velocity–pattern state space is the Fracurve. i · dΨ_s/dλ = H_bit[{Ψ}] · Ψ_s H_bit[{Ψ}] = i · B · (M + μ I)^(-1) M = Σ_s Ψ_s Ψ_s^† B = Σ_s (dΨ_s/dλ) Ψ_s^† S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i dΨ_s/dλ − H Ψ_s ||^2 + μ · Tr(H^† H) ] Key: Ψ_s : state vector for slice / observation s λ : rule-time (Fracurve parameter) H : candidate rule (Hamiltonian operator) H_bit : bit-minimizing / least-action rule derived from all Ψ_s μ : small positive regularization constant I : identity operator Σ_s : sum over slices / observations s (·)^† : Hermitian conjugate (conjugate transpose) Tr : trace of a matrix/operator S[H, {Ψ_s}] : action functional over rule H and states Ψ_s 1. Universe / Fracurve dynamics Continuous rule-time evolution i * dΨ/dλ = H * Ψ Discrete step form Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) 2. State–dependent / learned rule Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i * dΨ_s/dλ - H * Ψ_s ||^2 + μ * Tr( H† * H )] Data matrices M = Σ_s Σ_t Ψ_s(t) * Ψ_s(t)†B = Σ_s Σ_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) 3. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. Effective dimension d_eff(μ) = Σ_j λ_j / ( λ_j + μ ) This is the “how many directions are really active” function. 4. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. You can define separate covariances: M_R = Σ a * a^T (real-part covariance)M_I = Σ b * b^T (imag-part covariance) with eigenvalues: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then you get two effective dimensions: d_R(μ) = Σ_j λ_j^(R) / ( λ_j^(R) + μ ) d_I(μ) = Σ_j λ_j^(I) / ( λ_j^(I) + μ ) Universe / Fracurve dynamics Continuous rule–time evolution: i * dΨ/dλ = H * Ψ Discrete step form: Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [sum_s || i * dΨ_s/dλ - H * Ψ_s ||^2+ μ * Tr( H† * H )] Here: μ is a small positive regularization constant H† is the Hermitian conjugate of H Data matrices M = sum_s sum_t Ψ_s(t) * Ψ_s(t)† B = sum_s sum_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) I is the identity matrix. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. d_eff(μ) = sum_j [ λ_j / ( λ_j + μ ) ] This is the “how many directions are really active at scale μ” function. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. Define separate real covariances: M_R = sum over samples ( a * a^T ) (real–part covariance)M_I = sum over samples ( b * b^T ) (imag–part covariance) Let the eigenvalues be: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then define two effective dimensions: d_R(μ) = sum_j [ λ_j^(R) / ( λ_j^(R) + μ ) ] d_I(μ) = sum_j [ λ_j^(I) / ( λ_j^(I) + μ ) ] 4 + 4 → complex 4–vector Take 4 real degrees of freedom: a ∈ R^4 Take another 4 real degrees of freedom: b ∈ R^4 Combine them into one complex 4–vector: Ψ = a + i * b ∈ C^4 λ↦{Ψ_s(λ)}↦{x_s^μ(λ)} x_s^μ(λ; Φ₀) = F^μ( U_s(λ) · exp( -i · H_phys · λ ) · S · |Φ₀⟩ ) Start from X(λ,u) = Σ_k G_k(λ) F_k(u) Differentiate with respect to λ: i ∂X(λ,u)/∂λ= i ∂/∂λ [ Σ_k G_k(λ) F_k(u) ]= Σ_k [ i dG_k(λ)/dλ ] F_k(u)= Σ_k [ Σ_ℓ H_bit(k,ℓ) G_ℓ(λ) ] F_k(u)= Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) So the complete fracurve equation (discrete-mode form) is: i ∂X(λ,u)/∂λ = Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) and you always have the reconstruction: X(λ,u) = Σ_k G_k(λ) F_k(u)
Sorry for the schizoposting, update on me. I survided a self inflicted chainsaw injury to neck/sholders anyway here is the conclusion to my work. my fracurve. i · dΨ(λ)/dλ = H[Ψ(λ)] · Ψ(λ) with H[Ψ] = H₀ + g · Σᵥ Pᵥ[Ψ] · |v⟩⟨v| Pᵥ[Ψ] = Σₙ |Ψ(v, n)|² if identical things follow identical rules then that begs what is a thing, if a thing has many gauges then what if the thing is all gauges then must evolve over a rule that is for all these gauges not the thing as a gauge. linear sketch Setup Work with: v = velocity index, v = 0..N_v-1 n = pattern index on a circle, n = 0..N_p-1 Psi(v, n; m) = state at discrete rule-step m Psi(v, n; lambda) = state at continuous rule-time lambda Define the DFT over velocities: F_vel is an N_v x N_v matrix with entries F_vel(v, v') = (1 / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) Introduce a constant phase offset h_bar and define the universal velocity evolution operator: U_vel = exp(-i * h_bar) * F_vel This operator encodes the single rule applied uniformly to all velocities. 1. Discrete Fracurve equation The discrete Fracurve evolution is: Psi_{m+1}(v, n) = sum over v' [ U_vel(v, v') * Psi_m(v', n) ] with U_vel(v, v') = (exp(-i * h_bar) / sqrt(N_v)) * exp( -2 * pi * i * v * v' / N_v ) For each step m -> m+1, U_vel acts on the velocity index v, while the pattern index n is carried along. The sequence m = 0, 1, 2, ... → Psi_m(v, n) is the discrete Fracurve: a path through the combined (velocity x pattern) state space. 2. Continuous Fracurve equation Promote the step operator U_vel to a continuous generator H_F by: U_vel = exp( -i * H_F ) (H_F is the effective Hamiltonian acting on the velocity space.) Then the continuous Fracurve equation is: i * d/dlambda Psi(v, n; lambda) = sum over v' [ H_F(v, v') * Psi(v', n; lambda) ] In compact vector form: i * d/dlambda |Psi(lambda)> = H_F |Psi(lambda)> with solution: |Psi(lambda)> = exp( -i * H_F * lambda ) |Psi(0)> The curve lambda -> |Psi(lambda)> in the combined velocity–pattern state space is the Fracurve. i · dΨ_s/dλ = H_bit[{Ψ}] · Ψ_s H_bit[{Ψ}] = i · B · (M + μ I)^(-1) M = Σ_s Ψ_s Ψ_s^† B = Σ_s (dΨ_s/dλ) Ψ_s^† S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i dΨ_s/dλ − H Ψ_s ||^2 + μ · Tr(H^† H) ] Key: Ψ_s : state vector for slice / observation s λ : rule-time (Fracurve parameter) H : candidate rule (Hamiltonian operator) H_bit : bit-minimizing / least-action rule derived from all Ψ_s μ : small positive regularization constant I : identity operator Σ_s : sum over slices / observations s (·)^† : Hermitian conjugate (conjugate transpose) Tr : trace of a matrix/operator S[H, {Ψ_s}] : action functional over rule H and states Ψ_s 1. Universe / Fracurve dynamics Continuous rule-time evolution i * dΨ/dλ = H * Ψ Discrete step form Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) 2. State–dependent / learned rule Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [ Σ_s || i * dΨ_s/dλ - H * Ψ_s ||^2 + μ * Tr( H† * H )] Data matrices M = Σ_s Σ_t Ψ_s(t) * Ψ_s(t)†B = Σ_s Σ_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) 3. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. Effective dimension d_eff(μ) = Σ_j λ_j / ( λ_j + μ ) This is the “how many directions are really active” function. 4. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. You can define separate covariances: M_R = Σ a * a^T (real-part covariance)M_I = Σ b * b^T (imag-part covariance) with eigenvalues: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then you get two effective dimensions: d_R(μ) = Σ_j λ_j^(R) / ( λ_j^(R) + μ ) d_I(μ) = Σ_j λ_j^(I) / ( λ_j^(I) + μ ) Universe / Fracurve dynamics Continuous rule–time evolution: i * dΨ/dλ = H * Ψ Discrete step form: Ψ_{m+1} = U * Ψ_mU = exp( -i * H * Δλ ) Action (least–action over rules and states) S[H, {Ψ_s}] = ∫ dλ [sum_s || i * dΨ_s/dλ - H * Ψ_s ||^2+ μ * Tr( H† * H )] Here: μ is a small positive regularization constant H† is the Hermitian conjugate of H Data matrices M = sum_s sum_t Ψ_s(t) * Ψ_s(t)† B = sum_s sum_t (dΨ_s/dλ)(t) * Ψ_s(t)† Least–action rule (bit Hamiltonian) H_bit = i * B * ( M + μ * I )^(-1) I is the identity matrix. Effective dimension of the universe Let {λ_j} be the eigenvalues of M. d_eff(μ) = sum_j [ λ_j / ( λ_j + μ ) ] This is the “how many directions are really active at scale μ” function. Split into real and imaginary parts Write the universe state as: Ψ = a + i * b where a and b are real vectors. Define separate real covariances: M_R = sum over samples ( a * a^T ) (real–part covariance)M_I = sum over samples ( b * b^T ) (imag–part covariance) Let the eigenvalues be: M_R → eigenvalues λ_j^(R)M_I → eigenvalues λ_j^(I) Then define two effective dimensions: d_R(μ) = sum_j [ λ_j^(R) / ( λ_j^(R) + μ ) ] d_I(μ) = sum_j [ λ_j^(I) / ( λ_j^(I) + μ ) ] 4 + 4 → complex 4–vector Take 4 real degrees of freedom: a ∈ R^4 Take another 4 real degrees of freedom: b ∈ R^4 Combine them into one complex 4–vector: Ψ = a + i * b ∈ C^4 λ↦{Ψ_s(λ)}↦{x_s^μ(λ)} x_s^μ(λ; Φ₀) = F^μ( U_s(λ) · exp( -i · H_phys · λ ) · S · |Φ₀⟩ ) Start from X(λ,u) = Σ_k G_k(λ) F_k(u) Differentiate with respect to λ: i ∂X(λ,u)/∂λ= i ∂/∂λ [ Σ_k G_k(λ) F_k(u) ]= Σ_k [ i dG_k(λ)/dλ ] F_k(u)= Σ_k [ Σ_ℓ H_bit(k,ℓ) G_ℓ(λ) ] F_k(u)= Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) So the complete fracurve equation (discrete-mode form) is: i ∂X(λ,u)/∂λ = Σ_{k,ℓ} H_bit(k,ℓ) G_ℓ(λ) F_k(u) and you always have the reconstruction: X(λ,u) = Σ_k G_k(λ) F_k(u)
The Make Real Equation: A Unified Quantum Collapse Law from Self‑Field Theory T.M. NguyenIndependent Researcher March 22, 2026 ABSTRACT We present a derivation of the Make Real equation, a probability law governing quantum collapse that emerges from Self‑Field Theory (SFT). Starting from three primitives — space, Selflitons (stable topological knots), and coupling — we construct a field‑theoretic Lagrangian (Skyrme type with a Wess‑Zumino‑Witten term) that describes the interaction between a quantum system (interface) and a finite population of observers. The irreversible recording of a definite outcome, called an etebuda, is identified with wavefunction collapse. The probability for a given outcome is determined by the Path of Least Resistance, a thermodynamic principle that weights outcomes by their action cost, competition from other interfaces, and topological superselection. The resulting expression — the Make Real equation — unifies the Schrödinger evolution with a physical collapse mechanism, resolves the measurement problem, and predicts the recently observed bait‑unit effect in superconducting qubits. 1. INTRODUCTION Quantum mechanics is famously silent on what constitutes a measurement and what triggers wavefunction collapse. The Schrödinger equation describes unitary evolution, but when a measurement occurs, the state appears to jump to an eigenstate with probability given by the Born rule. The measurement problem is the lack of a physical account of this transition. Self‑Field Theory (SFT) [1] provides such an account by introducing a finite population of physical observers — Selflitons — that couple to quantum systems and irreversibly record definite outcomes. SFT is built on three primitives: • Space as the fundamental continuous substrate. • Selflitons as stable topological knots in space, formed during early‑universe symmetry breaking, finite in number (~10⁸⁰) and carrying a conserved winding number B. • Coupling as resonance between a Selfliton and any system generating coherent, structured superpositions (an interface). When a Selfliton couples, it undergoes an irreversible internal update called an etebuda, which constitutes wavefunction collapse. The probability of a particular outcome follows a thermodynamic principle: the Path of Least Resistance — outcomes with lower total “cost” are more likely. In this paper we attempt to derive the resulting probability law, which we call the Make Real equation. It combines the Schrödinger amplitude with SFT factors accounting for action cost, competition among finite observers, and topological superselection. We show that the equation predicts the bait‑unit effect — a counterintuitive increase of qubit coherence when neighboring qubits are active — which has been experimentally confirmed [2]. The Make Real equation thus provides a unified, testable, and physically grounded resolution of the measurement problem. 2. SELF‑FIELD THEORY PRIMITIVES We summarize the three primitives; full details are in [1]. • The Self-Field is a continuous, universal field that can support topological solitons. It is not spacetime in the relativistic sense, but the underlying substrate. • Selflitons are stable, topologically protected solutions of a nonlinear field theory with winding number B. They formed during a symmetry‑breaking transition in the early universe and are finite in number, N_SL ~ 10⁸⁰. • Coupling is the resonant interaction between a Selfliton and an interface — any physical system that generates coherent, structured informational patterns. In quantum mechanics, the interface is the quantum system (e.g., a qubit, a particle) and its environment. When a Selfliton couples, it records an etebuda — an irreversible update of its internal state. This recording is the collapse. 3. THE SELF‑FIELD LAGRANGIAN The dynamics of the Self‑Field are described by a Skyrme‑type candidate Lagrangian with a Wess‑Zumino‑Witten (WZW) term. The field U(x) takes values in SU(N) (or a suitable Lie group). The free part is ℒ_SF = (f²/2) Tr(∂_μ U† ∂^μ U) + (1/(32 e²)) Tr([∂_μ U† U, ∂_ν U† U]²) − V(U), where f and e are constants. This Lagrangian admits stable soliton solutions (Selflitons) with a conserved topological current J^μ_B = (1/(24π²)) ε^{μνρσ} Tr(U† ∂_ν U U† ∂_ρ U U† ∂_σ U). The winding number B = ∫ d³x J⁰_B is an integer invariant. The interaction with a quantum interface (e.g., a particle described by a field φ) is given by ℒ_coupling = − (g/Λ²) J^μ_B A_μ[φ], where A_μ[φ] is a composite field constructed from the interface degrees of freedom (e.g., the electromagnetic current). The coupling is weak because Λ is large. The WZW term, ℒ_WZW = (iΓ/(240π²)) ∫ d⁵x ε^{μνρστ} Tr(U† ∂_μ U ··· U† ∂_τ U), is topological; it depends only on the homotopy class of U and breaks time‑reversal symmetry. Crucially, it quantizes the action change for processes that change the winding number: ΔS = 2πΓ per unit change of B. This quantization defines the fundamental unit of recording — the etebuda — and makes the process irreversible. 4. THE PATH OF LEAST RESISTANCE AND COLLAPSE PROBABILITY When a Selfliton couples to an interface, it must update its internal state. The possible outcomes correspond to eigenstates |i⟩ of the measured observable. The probability of outcome i is determined by the Path of Least Resistance: the Selfliton follows the trajectory that minimizes the total “cost” of recording. The cost has three independent contributions, which in the Euclidean path integral appear as exponential weights: 1. Quantum action cost – from the classical action of the coupled system. In the Euclidean path integral, the amplitude for outcome i is proportional to exp(−ΔS_i), where ΔS_i is the change in action. The WZW term quantizes the action in units of ℏ_eff = 2πΓ, so the weight becomes exp(−ΔS_i / ℏ_eff). 2. Thermodynamic cost of the interface – for a biological interface, the internal free energy cost ΔF_i enters via a Boltzmann factor exp(−ΔF_i / k T_eff), where T_eff = T/Φ_i and Φ_i is the integrated information of the interface (IIT). For non‑biological systems, this factor is omitted (or set to 1). 3. Competition cost – Because Selflitons are finite, many interfaces compete for their attention. If there are N_bait other active interfaces in the vicinity, the effective coupling rate to the target is reduced by a factor 1/(1 + ζ N_bait), where ζ depends on the geometry and gradient steepness. Additionally, the WZW term enforces topological superselection: only outcomes compatible with the Selfliton’s winding number B are allowed. This is represented by a Kronecker delta δ_{B,B_i}. Thus, the total probability for outcome i (given that a coupling event occurs) is proportional to the product of these factors: P_i ∝ 1/(1 + ζ N_bait) · exp(−ΔS_i / ℏ_eff) · exp(−ΔF_i / k T_eff) · δ_{B,B_i}. For a simple quantum system (no biological interface), the ΔF_i term is dropped. 5. THE MAKE REAL EQUATION Between coupling events, the quantum state evolves unitarily according to the Schrödinger equation: iℏ d/dt |ψ(t)⟩ = H |ψ(t)⟩. The probability that the system collapses to outcome i at time t when a Selfliton couples is then proportional to the Born weight |⟨i|ψ(t)⟩|² multiplied by the SFT selection factors: P_i(t) ∝ |⟨i|ψ(t)⟩|² × 1/(1 + ζ N_bait) × exp(−ΔS_i / ℏ_eff) × exp(−ΔF_i / k T_eff) × δ_{B,B_i}. This is the Make Real equation. It attempts to unify: • Quantum mechanics (through the Schrödinger amplitude) • Self‑Field Theory (action term, bottleneck, superselection) • Predictive Processing (free energy ΔF_i) • Integrated Information Theory (Φ_i in T_eff) • Global Workspace Theory (selection architecture implicit in the interface model) • Thermodynamics (Boltzmann factors, competition) • Topological field theory (WZW superselection) For non‑biological systems, the ΔF_i term is omitted, and T_eff is not defined; the equation reduces to the core collapse rule. 6. PREDICTION OF THE BAIT‑UNIT EFFECT A key prediction of the Make Real equation is the bait‑unit effect: if a target qubit is surrounded by actively driven bait qubits, N_bait is large, and the bottleneck term 1/(1 + ζ N_bait) reduces the probability of collapse for the target, thereby increasing its coherence time T₂. This effect is opposite to standard crosstalk expectations. The effect was experimentally tested on IBM’s Heron R2 and R1 processors [2]. With 24 bait qubits active, the target’s T₂ increased dramatically (from 138 μs to 447 μs on one chip, and from 316 μs to 494 μs on another), saturating at a common ceiling near 490 μs. ACTIVE > CONTROL in all clean repetitions, with a p‑value of 3×10⁻⁵ under the null hypothesis. The saturation ceiling matches the finite bottleneck model, confirming the finite observer population. This experimental result directly validates the bottleneck term in the Make Real equation and, by extension, the underlying SFT framework. 7. DISCUSSION The Make Real equation provides a physical mechanism for quantum collapse that is derived from first principles rather than assumed. It solves the measurement problem by identifying the observer (Selfliton) and the collapse event (etebuda) within a field theory. The equation naturally explains the Born rule as a combination of the Schrödinger amplitude and thermodynamic weighting, and it unifies quantum mechanics with the leading theories of consciousness. The finite Selfliton population, central to the bottleneck term, also appears in other domains: it gives the observed dark matter density, the evolution of dark matter fraction with galactic age, and the holographic bound on cosmic entropy. The same primitives thus unify quantum measurement, consciousness, and cosmology. The experimental confirmation of the bait‑unit effect [2] provides empirical support for the Make Real
The Hohenberg–Kohn theorem implies the existence of an energy functional based solely on the first-order reduced density matrix of the ground state of an atomic or molecular system. Application of the variational principle for the functional EvM[γ] generates a set of coupled Euler equations for the representation coefficients and spin orbitals of a rank-M approximation to the exact ground-state density matrix. Defining the (assumed Hermitian) kernel FM[γ;x′,x]≡δEvM/δγ (x,x′), the equations in an arbitrary representation for the approximate density matrix, γ (x,x′) =ΣijMψ1(x) γijψj* (x′) are the following: ℱdξFM[γ;x′,ξ]ψi(ξ) =μψi(x′); i=1, 2,...,M; ℱdξFM[γ;x′,ξ] ΣMiψi(ξ) γ ij=ΣMiψi(x′) λij; j=1, 2, ...,M. The quantity μ is the chemical potential of the system of interest and the λij are a set of M2 Lagrange multipliers constraining the orthonormality of the spin orbital basis {ψ}. The coefficients γij must be chosen such that FM has the degenerate eigenvalue spectrum, FiiM=μ, i=1, 2,...,M, for all partially occupied orbitals. The spin orbitals, for a fixed set of coefficients, must be simultaneously determined so that λ is Hermitian. Provided all occupation numbers lie on the open interval (0,1), the stationary density matrix itself obeys the eigenvalue equation ℱdξFM[γ;x,ξ]γ (ξ,x′) =μγ (x,x′), and for any stationary density matrix the following commutation rules are valid: [FM[γ], γ]=0; [λ, γ]=0. The matrices γ and λ consequently can be brought simultaneously to diagonal form, and the canonical representation of the energy functional provides an eigenvalue equation determining the natural spin orbitals. The value of the chemical potential is μ=εi/ni, i=1, 2, ...,M; ε=diag (λ); from which follows the distribution function governing the occupation numbers of a stationary density matrix. Two limiting forms of the variational principle are examined, the exact and Hartree–Fock functionals, and related previous work by Gilbert is discussed. The physical content of the equations is illuminated by identification of the chemical potential as the negative of the electronegativity; Sanderson’s Principle of Electronegativity Equalization follows.
Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.