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- qwerty/constants.md: master reference table, 100+ constants §1-§178 - qwerty/equalities.md: all major QWERTY equalities by theme - equations/blackroad-equations.md: all 19 BlackRoad equations - equations/consciousness.md: Psi_care, Phi_universal, CECE update rule - equations/quantum.md: qutrit, Weyl pair, density matrix, SVD - equations/universal.md: Three Tests, Euler-Lagrange, fine-structure - proofs/ternary-efficiency.md: ln(3)/3 > ln(2)/2 - proofs/self-reference.md: the QWERTY encoding is self-referential - proofs/pure-state.md: density matrix rank=1, SVD=SELF - figures/durer-square.md: magic square with 2000 substitution - figures/trinary-table.md: TAND TMUL TNEG TXOR truth tables - figures/qutrit-operators.md: Weyl X/Z, Gell-Mann matrices - figures/keyboard.md: QWERTY encoding layout - notebooks/README.md: page-by-page index of all 24 notebook pages Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
149 lines
3.2 KiB
Markdown
149 lines
3.2 KiB
Markdown
# Quantum Equations
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> Qutrits, Weyl operators, Gell-Mann matrices, density matrices.
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## Qutrit State Space (§172, §178)
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A **qutrit** is a three-level quantum system. Basis states: `{|0⟩, |1⟩, |2⟩}`.
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General state:
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```
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|Ψ⟩ = α|0⟩ + β|1⟩ + γ|?⟩ (Equation 5, page 16)
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```
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With concrete amplitudes from page 24:
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```
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|ψ⟩ = [ 0.4711 ]
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[ 0.7708 ]
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[ 0.8620 ]
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```
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QUTRIT = WEYL = PSI = 30 = 2×G_key.
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---
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## Weyl Pair (§172)
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The two fundamental qutrit operators, with ω = e^(2πi/3) (cube root of unity = root of x²+x+1, §166):
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**Shift operator X (clock):**
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```
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X|j⟩ = |j+1 mod 3⟩
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```
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Cycles through {|0⟩, |1⟩, |2⟩}.
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**Clock operator Z:**
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```
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Z|j⟩ = ωʲ|j⟩
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```
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Multiplies by powers of ω.
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Together: every 3×3 unitary can be written in terms of X^a Z^b.
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```
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CLOCK = BLOCH = HIERARCHY = COSMOS = 90
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SHIFT = SPIN = PAULI = OPERATOR = 55
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CLOCK + SHIFT = 90 + 55 = 145 = EVERYTHINGELSE
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QFT = Z (QWERTY=20) — she named the clock operator Z
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```
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---
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## Gell-Mann Matrices (§172, §178)
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The 8 generators of SU(3). For a qutrit, the density matrix is expressed:
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```
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ρ = I/3 + Σₖ rₖλₖ/2, k = 1..8
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```
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The Gell-Mann matrices λ₁...λ₈ are the quark color charge matrices.
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COLOR = TRINARY = LIGHT = 63. Quark color = ternary = light.
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```
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GELLMAN = INTEGRATION = 118 [her spelling]
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MANN = BIRTHDAY = 87
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```
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---
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## Density Matrix (§174, §178)
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For a pure state |ψ⟩:
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```
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ρ = |ψ⟩⟨ψ|
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```
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From page 24 (concrete computation):
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```
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ρ = [ 0.2219 0.3629 0.4062 ]
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[ 0.3629 0.5941 0.6639 ]
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[ 0.4062 0.6639 0.7401 ]
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```
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Properties:
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- Symmetric: ρ = ρᵀ (real state) → SYMMETRIC = UNIVERSAL = OCTONION = 112
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- Rank 1 (pure state)
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- One nonzero singular value: σ₁ ≈ 1.559
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```
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DENSITY = METHOD = 72 = reverse(27)
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PURE = 4! = 24 = B key
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SYMMETRIC = UNIVERSAL = 112
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```
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---
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## Time Evolution (§178)
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The Liouville–von Neumann equation: `dρ/dt = −i[H, ρ]/ℏ`
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From page 24:
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```
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ρ̇ = [ 0.0600+0j 0.0872−0.2680j 0.0753−0.2680j ]
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[ 0.0872+0.2680j −0.0400+0j 0.0560−0.2680j ]
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[ 0.0753+0.2680j 0.0560+0.2680j −0.0200+0j ]
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```
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**Tr(ρ̇) = 0.0600 − 0.0400 − 0.0200 = 0.** She is conserved.
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```
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EVOLUTION = EVERYTHING = ARITHMETIC = 108
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TRACE = QUBIT = SUM = UNIT = 45
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COMPLEX = 2×PAULI = 110
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```
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---
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## SVD Decomposition (§178)
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Singular Value Decomposition of ρ:
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- One nonzero singular value: σ₁ ≈ 1.559
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- All others: machine zero (~10⁻¹⁶)
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- This confirms ρ is a **pure state** (rank 1)
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```
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SVD = SELF = SPHERE = ZSH = 48 = 2×PURE
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PURE = 4! = 24
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VALUE = TRINARY = LIGHT = 63 ← the one surviving singular value = ternary = light
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SINGULAR = MAXWELL = 101 prime
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```
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**SVD = SELF.** She decomposed the density matrix and found herself.
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---
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## Entanglement Measure (§174)
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Von Neumann entropy of the reduced density matrix:
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```
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E_QC = −Tr(ρ_reduced · log ρ_reduced)
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ρ_reduced = Tr_chem(|Ψ_total⟩⟨Ψ_total|)
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```
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```
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ENTANGLEMENT = CONFINEMENT = 165 = 3×PAULI = 3×SPIN = 3×OPERATOR
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EIGENVALUE = PRESERVATION = ADVANTAGE = 117
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```
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Quantum entanglement = biological confinement. Same number.
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