⟨j,m|Ĵ²|j,m⟩ = ℏ²j(j+1)Ĥ = ℏω(â†â + ½)SWAP = CNOT₁₂ · CNOT₂₁ · CNOT₁₂∮ E⃗ · dA⃗ = Q/ε₀|α|² + |β|² = 1⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]σ² = σₓ² + σᵧ² + σᵤ²K(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}R̂ᵧ(θ) = e^{−iθσᵧ/2}⟨φ|ψ⟩⟨Â⟩ = Tr(ρ̂Â)â†|n⟩ = √(n+1) |n+1⟩C(ρ) = S(ρ_A) = S(ρ_B)⟨x|ψ⟩ = ψ(x)R̂ᵤ(θ) = e^{−iθσᵤ/2}Γᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tF⃗ = qE⃗ + qv⃗ × B⃗∄ Û : Û|ψ⟩|0⟩ = |ψ⟩|ψ⟩ ∀|ψ⟩Â_H(t) = Û†(t)Â_SÛ(t)iℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩[Â, B̂Ĉ] = [Â,B̂]Ĉ + B̂[Â,Ĉ]⟨j,m|Ĵ²|j,m⟩ = ℏ²j(j+1)Ĥ = ℏω(â†â + ½)SWAP = CNOT₁₂ · CNOT₂₁ · CNOT₁₂∮ E⃗ · dA⃗ = Q/ε₀|α|² + |β|² = 1⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]σ² = σₓ² + σᵧ² + σᵤ²K(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}R̂ᵧ(θ) = e^{−iθσᵧ/2}⟨φ|ψ⟩⟨Â⟩ = Tr(ρ̂Â)â†|n⟩ = √(n+1) |n+1⟩C(ρ) = S(ρ_A) = S(ρ_B)⟨x|ψ⟩ = ψ(x)R̂ᵤ(θ) = e^{−iθσᵤ/2}Γᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tF⃗ = qE⃗ + qv⃗ × B⃗∄ Û : Û|ψ⟩|0⟩ = |ψ⟩|ψ⟩ ∀|ψ⟩Â_H(t) = Û†(t)Â_SÛ(t)iℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩[Â, B̂Ĉ] = [Â,B̂]Ĉ + B̂[Â,Ĉ]
Ĵ±|j,m⟩ = ℏ√(j∓m)(j±m+1)|j,m±1⟩Tr(Ŵρ) < 0 ⟹ entangled∮ p dx = (n + ½)2πℏiℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩[x̂, p̂] = iℏ⟨n|â†â|n⟩ = nT̂ = |0⟩⟨0| + e^{iπ/4}|1⟩⟨1|ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|[Lₓ, Lᵧ] = iℏLᵤr⃗ = (sin θ cos φ, sin θ sin φ, cos θ)CNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef){ĉᵢ, ĉⱼ†} = δᵢⱼσₐσᵦ ≥ ½|⟨[Â,B̂]⟩|T̂ = p̂²/2mρ(t) = Σ Eₖ(t)ρ(0)Eₖ†(t)L̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩[Â, B̂Ĉ] = [Â,B̂]Ĉ + B̂[Â,Ĉ]â|n⟩ = √n |n−1⟩|ψ(t)⟩ = Û(t)|ψ(0)⟩[Ĥ, Â] = iℏ ∂Â/∂tĤ = ℏω(â†â + ½)Ĵ±|j,m⟩ = ℏ√(j∓m)(j±m+1)|j,m±1⟩Tr(Ŵρ) < 0 ⟹ entangled∮ p dx = (n + ½)2πℏiℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩[x̂, p̂] = iℏ⟨n|â†â|n⟩ = nT̂ = |0⟩⟨0| + e^{iπ/4}|1⟩⟨1|ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|[Lₓ, Lᵧ] = iℏLᵤr⃗ = (sin θ cos φ, sin θ sin φ, cos θ)CNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef){ĉᵢ, ĉⱼ†} = δᵢⱼσₐσᵦ ≥ ½|⟨[Â,B̂]⟩|T̂ = p̂²/2mρ(t) = Σ Eₖ(t)ρ(0)Eₖ†(t)L̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩[Â, B̂Ĉ] = [Â,B̂]Ĉ + B̂[Â,Ĉ]â|n⟩ = √n |n−1⟩|ψ(t)⟩ = Û(t)|ψ(0)⟩[Ĥ, Â] = iℏ ∂Â/∂tĤ = ℏω(â†â + ½)
∮ p dx = (n + ½)2πℏF⃗ = qE⃗ + qv⃗ × B⃗T₂ : ⟨σₓ⟩ → 0 dephasingTr(ÂB̂) = Tr(B̂Â)Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_fp̂ = i√(mℏω/2)(↠− â)[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖÂ_H(t) = Û†(t)Â_SÛ(t)K(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}Â|aₙ⟩ = aₙ|aₙ⟩∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tρ' = Σ MₖρMₖ†⟨x|ψ⟩ = ψ(x)⟨n|â†â|n⟩ = n{ĉᵢ, ĉⱼ†} = δᵢⱼEₙ⁽¹⁾ = ⟨n⁰|V̂|n⁰⟩|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0Tr(ρ̂²) ≤ 1Γᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)L̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩F(ρ,σ) = (Tr√(√ρ σ √ρ))²∮ p dx = (n + ½)2πℏF⃗ = qE⃗ + qv⃗ × B⃗T₂ : ⟨σₓ⟩ → 0 dephasingTr(ÂB̂) = Tr(B̂Â)Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_fp̂ = i√(mℏω/2)(↠− â)[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖÂ_H(t) = Û†(t)Â_SÛ(t)K(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}Â|aₙ⟩ = aₙ|aₙ⟩∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tρ' = Σ MₖρMₖ†⟨x|ψ⟩ = ψ(x)⟨n|â†â|n⟩ = n{ĉᵢ, ĉⱼ†} = δᵢⱼEₙ⁽¹⁾ = ⟨n⁰|V̂|n⁰⟩|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0Tr(ρ̂²) ≤ 1Γᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)L̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩F(ρ,σ) = (Tr√(√ρ σ √ρ))²
|ψ⟩ ⊗ |Φ⁺⟩ → classical bits → |ψ⟩∮ B⃗ · dl⃗ = μ₀Ia₀ = ℏ²/(mₑe²) ≈ 0.529 ÅTr(ρ̂) = 1⟨p|ψ⟩ = φ(p)E(|ψ⟩) = S(Tr_B(|ψ⟩⟨ψ|))∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0[x̂ᵢ, p̂ⱼ] = iℏδᵢⱼψₙₗₘ(r,θ,φ) = Rₙₗ(r)Yₗₘ(θ,φ)Tr(ÂB̂) = Tr(B̂Â)ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dp|ψ_out⟩ = Ûₙ···Û₂Û₁|ψ_in⟩L̂ᵤ |l,m⟩ = ℏm|l,m⟩W = (2π/ℏ)|Mfi|²ρ(E)R̂ᵤ(θ) = e^{−iθσᵤ/2}σₓ = (⁰₁ ¹₀)[Lₓ, Lᵧ] = iℏLᵤ|ψ(t)⟩ = Û(t)|ψ(0)⟩Ĥ = T̂ + V̂⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]Eₙ⁽²⁾ = Σ |⟨m⁰|V̂|n⁰⟩|²/(Eₙ⁰ − Eₘ⁰)ΔEΔt ≥ ℏ/2|ψ⟩ ⊗ |Φ⁺⟩ → classical bits → |ψ⟩∮ B⃗ · dl⃗ = μ₀Ia₀ = ℏ²/(mₑe²) ≈ 0.529 ÅTr(ρ̂) = 1⟨p|ψ⟩ = φ(p)E(|ψ⟩) = S(Tr_B(|ψ⟩⟨ψ|))∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0[x̂ᵢ, p̂ⱼ] = iℏδᵢⱼψₙₗₘ(r,θ,φ) = Rₙₗ(r)Yₗₘ(θ,φ)Tr(ÂB̂) = Tr(B̂Â)ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dp|ψ_out⟩ = Ûₙ···Û₂Û₁|ψ_in⟩L̂ᵤ |l,m⟩ = ℏm|l,m⟩W = (2π/ℏ)|Mfi|²ρ(E)R̂ᵤ(θ) = e^{−iθσᵤ/2}σₓ = (⁰₁ ¹₀)[Lₓ, Lᵧ] = iℏLᵤ|ψ(t)⟩ = Û(t)|ψ(0)⟩Ĥ = T̂ + V̂⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]Eₙ⁽²⁾ = Σ |⟨m⁰|V̂|n⁰⟩|²/(Eₙ⁰ − Eₘ⁰)ΔEΔt ≥ ℏ/2
iℏ ∂ψ/∂t = ĤψEₙ = ℏω(n + ½)dÂ_H/dt = (i/ℏ)[Ĥ, Â_H]W = (2π/ℏ)|Mfi|²ρ(E)σᵤ = (¹₀ ⁰₋₁)Ŝ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)∮ E⃗ · dA⃗ = Q/ε₀|↑⟩ = (¹₀), |↓⟩ = (⁰₁)R̂ₓ(θ) = e^{−iθσₓ/2}⟨n|â†â|n⟩ = n|n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩⟨Â⟩ = Tr(ρ̂Â)ΔxΔp ≥ ℏ/2φ(p) = (2πℏ)^{−1/2} ∫ψ(x)e^{−ipx/ℏ}dx[Lₓ, Lᵧ] = iℏLᵤJ = L + S|α|² + |β|² = 1n̂ = â†âCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)−(ℏ²/2m)∇²ψ + Vψ = EψŜ² |s,mₛ⟩ = ℏ²s(s+1)|s,mₛ⟩iℏ ∂ψ/∂t = ĤψEₙ = ℏω(n + ½)dÂ_H/dt = (i/ℏ)[Ĥ, Â_H]W = (2π/ℏ)|Mfi|²ρ(E)σᵤ = (¹₀ ⁰₋₁)Ŝ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)∮ E⃗ · dA⃗ = Q/ε₀|↑⟩ = (¹₀), |↓⟩ = (⁰₁)R̂ₓ(θ) = e^{−iθσₓ/2}⟨n|â†â|n⟩ = n|n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩⟨Â⟩ = Tr(ρ̂Â)ΔxΔp ≥ ℏ/2φ(p) = (2πℏ)^{−1/2} ∫ψ(x)e^{−ipx/ℏ}dx[Lₓ, Lᵧ] = iℏLᵤJ = L + S|α|² + |β|² = 1n̂ = â†âCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)−(ℏ²/2m)∇²ψ + Vψ = EψŜ² |s,mₛ⟩ = ℏ²s(s+1)|s,mₛ⟩
r⃗ = (sin θ cos φ, sin θ sin φ, cos θ)∇ × E⃗ = −∂B⃗/∂t|ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩⟨Â⟩ = Σ aₙ P(aₙ)ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|dσ/dΩ = |f(θ)|²Ĥψ = EψS = −kᵦ Tr(ρ̂ ln ρ̂)χ = α|↑⟩ + β|↓⟩⟨φ|ψ⟩σᵤ = (¹₀ ⁰₋₁)E(|ψ⟩) = S(Tr_B(|ψ⟩⟨ψ|))[σᵢ, σⱼ] = 2iεᵢⱼₖσₖ⟨m|n⟩ = δₘₙΔEΔt ≥ ℏ/2⟨x_f|e^{−iĤT/ℏ}|x_i⟩ = ∫𝒟x e^{iS[x]/ℏ}Tr(ÂB̂) = Tr(B̂Â)L̂ᵤ |l,m⟩ = ℏm|l,m⟩â|n⟩ = √n |n−1⟩Ĥ = T̂ + V̂Tr(Ŵρ) < 0 ⟹ entangled|0_L⟩ = (|000⟩ + |111⟩)/√2r⃗ = (sin θ cos φ, sin θ sin φ, cos θ)∇ × E⃗ = −∂B⃗/∂t|ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩⟨Â⟩ = Σ aₙ P(aₙ)ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|dσ/dΩ = |f(θ)|²Ĥψ = EψS = −kᵦ Tr(ρ̂ ln ρ̂)χ = α|↑⟩ + β|↓⟩⟨φ|ψ⟩σᵤ = (¹₀ ⁰₋₁)E(|ψ⟩) = S(Tr_B(|ψ⟩⟨ψ|))[σᵢ, σⱼ] = 2iεᵢⱼₖσₖ⟨m|n⟩ = δₘₙΔEΔt ≥ ℏ/2⟨x_f|e^{−iĤT/ℏ}|x_i⟩ = ∫𝒟x e^{iS[x]/ℏ}Tr(ÂB̂) = Tr(B̂Â)L̂ᵤ |l,m⟩ = ℏm|l,m⟩â|n⟩ = √n |n−1⟩Ĥ = T̂ + V̂Tr(Ŵρ) < 0 ⟹ entangled|0_L⟩ = (|000⟩ + |111⟩)/√2
|Ψ⁻⟩ = (|01⟩ − |10⟩)/√2Tr(ÂB̂) = Tr(B̂Â)⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tT̂ = p̂²/2mD(ρ‖σ) = Tr(ρ log ρ − ρ log σ)|ψ(t)⟩ = Û(t)|ψ(0)⟩Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_f[Â, B̂] = ÂB̂ − B̂Â[x̂, p̂] = iℏ|n⟩ = (â†)ⁿ/√(n!) |0⟩J = L + SS = −kᵦ Tr(ρ̂ ln ρ̂)P(aₙ) = |⟨aₙ|ψ⟩|²∮ E⃗ · dA⃗ = Q/ε₀Ĥ = T̂ + V̂ΔxΔp ≥ ℏ/2∇ × E⃗ = −∂B⃗/∂tf(θ) = −(m/2πℏ²)⟨k'|V̂|k⟩ΔEΔt ≥ ℏ/2Ĥψ = Eψ|α|² + |β|² = 1|Ψ⁻⟩ = (|01⟩ − |10⟩)/√2Tr(ÂB̂) = Tr(B̂Â)⟨r⟩ = a₀n²[1 + ½(1 − l(l+1)/n²)]∇ × B⃗ = μ₀J⃗ + μ₀ε₀∂E⃗/∂tT̂ = p̂²/2mD(ρ‖σ) = Tr(ρ log ρ − ρ log σ)|ψ(t)⟩ = Û(t)|ψ(0)⟩Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_f[Â, B̂] = ÂB̂ − B̂Â[x̂, p̂] = iℏ|n⟩ = (â†)ⁿ/√(n!) |0⟩J = L + SS = −kᵦ Tr(ρ̂ ln ρ̂)P(aₙ) = |⟨aₙ|ψ⟩|²∮ E⃗ · dA⃗ = Q/ε₀Ĥ = T̂ + V̂ΔxΔp ≥ ℏ/2∇ × E⃗ = −∂B⃗/∂tf(θ) = −(m/2πℏ²)⟨k'|V̂|k⟩ΔEΔt ≥ ℏ/2Ĥψ = Eψ|α|² + |β|² = 1
|ψ⟩ ⊗ |Φ⁺⟩ → classical bits → |ψ⟩δ(x−x') = (1/2π)∫e^{ik(x−x')}dkK(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_fΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0∂ρ̂/∂t = −(i/ℏ)[Ĥ, ρ̂]⟨Â⟩ = Σ aₙ P(aₙ)⟨x|ψ⟩ = ψ(x)χ = α|↑⟩ + β|↓⟩a₀ = ℏ²/(mₑe²) ≈ 0.529 ÅŜ² |s,mₛ⟩ = ℏ²s(s+1)|s,mₛ⟩O(√N) queriesL̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩−(ℏ²/2m)∇²ψ + Vψ = EψŜᵤ |s,mₛ⟩ = ℏmₛ|s,mₛ⟩[Ĥ, Â] = iℏ ∂Â/∂tQFT|j⟩ = (1/√N) Σ e^{2πijk/N}|k⟩σᵧ = (⁰₋ᵢ ⁱ₀)[x̂ᵢ, p̂ⱼ] = iℏδᵢⱼCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓJ = L + S|ψ⟩ ⊗ |Φ⁺⟩ → classical bits → |ψ⟩δ(x−x') = (1/2π)∫e^{ik(x−x')}dkK(b,a) = ∫𝒟x(t) e^{iS[x]/ℏ}Û_G = Ĥ(2|0⟩⟨0| − 𝟙)Ĥ · Ô_fΓᵢ→f = (2π/ℏ)|⟨f|V̂|i⟩|²ρ(Ef)∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0∂ρ̂/∂t = −(i/ℏ)[Ĥ, ρ̂]⟨Â⟩ = Σ aₙ P(aₙ)⟨x|ψ⟩ = ψ(x)χ = α|↑⟩ + β|↓⟩a₀ = ℏ²/(mₑe²) ≈ 0.529 ÅŜ² |s,mₛ⟩ = ℏ²s(s+1)|s,mₛ⟩O(√N) queriesL̂² |l,m⟩ = ℏ²l(l+1)|l,m⟩−(ℏ²/2m)∇²ψ + Vψ = EψŜᵤ |s,mₛ⟩ = ℏmₛ|s,mₛ⟩[Ĥ, Â] = iℏ ∂Â/∂tQFT|j⟩ = (1/√N) Σ e^{2πijk/N}|k⟩σᵧ = (⁰₋ᵢ ⁱ₀)[x̂ᵢ, p̂ⱼ] = iℏδᵢⱼCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓJ = L + S
T̂ = |0⟩⟨0| + e^{iπ/4}|1⟩⟨1||n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0R̂ᵤ(θ) = e^{−iθσᵤ/2}|0_L⟩ = (|000⟩ + |111⟩)/√2γₙ = i ∮ ⟨n|∇_R|n⟩ · dR⃗[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓĤ = T̂ + V̂∮ E⃗ · dA⃗ = Q/ε₀|Φ⁺⟩ = (|00⟩ + |11⟩)/√2Eₙ = ℏω(n + ½)φ(p) = (2πℏ)^{−1/2} ∫ψ(x)e^{−ipx/ℏ}dxTr(ρ̂²) ≤ 1iℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩Ĥ = (1/√2)(|0⟩⟨0| + |0⟩⟨1| + |1⟩⟨0| − |1⟩⟨1|)P̂ = |ψ⟩⟨ψ|r⃗ = (sin θ cos φ, sin θ sin φ, cos θ)Ŝ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dpĴ±|j,m⟩ = ℏ√(j∓m)(j±m+1)|j,m±1⟩QFT|j⟩ = (1/√N) Σ e^{2πijk/N}|k⟩T̂ = |0⟩⟨0| + e^{iπ/4}|1⟩⟨1||n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩∂²ψ/∂x² + (2m/ℏ²)(E−V)ψ = 0R̂ᵤ(θ) = e^{−iθσᵤ/2}|0_L⟩ = (|000⟩ + |111⟩)/√2γₙ = i ∮ ⟨n|∇_R|n⟩ · dR⃗[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖCNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓĤ = T̂ + V̂∮ E⃗ · dA⃗ = Q/ε₀|Φ⁺⟩ = (|00⟩ + |11⟩)/√2Eₙ = ℏω(n + ½)φ(p) = (2πℏ)^{−1/2} ∫ψ(x)e^{−ipx/ℏ}dxTr(ρ̂²) ≤ 1iℏ ∂/∂t |ψ⟩ = Ĥ|ψ⟩Ĥ = (1/√2)(|0⟩⟨0| + |0⟩⟨1| + |1⟩⟨0| − |1⟩⟨1|)P̂ = |ψ⟩⟨ψ|r⃗ = (sin θ cos φ, sin θ sin φ, cos θ)Ŝ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dpĴ±|j,m⟩ = ℏ√(j∓m)(j±m+1)|j,m±1⟩QFT|j⟩ = (1/√N) Σ e^{2πijk/N}|k⟩
L̂± = L̂ₓ ± iL̂ᵧdσ/dΩ = |f(θ)|²[Ĥ, Â] = iℏ ∂Â/∂tEₙ = −13.6 eV / n²|Ψ⁻⟩ = (|01⟩ − |10⟩)/√2|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖZ = ∫𝒟φ e^{iS[φ]/ℏ}ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dpŜ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)|α|² + |β|² = 1R̂ₓ(θ) = e^{−iθσₓ/2}E₀ ≤ ⟨ψ_trial|Ĥ|ψ_trial⟩/⟨ψ_trial|ψ_trial⟩|ψ_out⟩ = Ûₙ···Û₂Û₁|ψ_in⟩⟨x|p⟩ = e^{ipx/ℏ}/√(2πℏ)|n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩⟨Â⟩ = Tr(ρ̂Â)iℏ ∂ψ/∂t = Ĥψ⟨φ|ψ⟩|n⟩ = (â†)ⁿ/√(n!) |0⟩ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|CNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓL̂± = L̂ₓ ± iL̂ᵧdσ/dΩ = |f(θ)|²[Ĥ, Â] = iℏ ∂Â/∂tEₙ = −13.6 eV / n²|Ψ⁻⟩ = (|01⟩ − |10⟩)/√2|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2[Jᵢ, Jⱼ] = iℏεᵢⱼₖJₖZ = ∫𝒟φ e^{iS[φ]/ℏ}ψ(x) = (2πℏ)^{−1/2} ∫φ(p)e^{ipx/ℏ}dpŜ·n̂ = (ℏ/2)(cos θ σᵤ + sin θ cos φ σₓ + sin θ sin φ σᵧ)|α|² + |β|² = 1R̂ₓ(θ) = e^{−iθσₓ/2}E₀ ≤ ⟨ψ_trial|Ĥ|ψ_trial⟩/⟨ψ_trial|ψ_trial⟩|ψ_out⟩ = Ûₙ···Û₂Û₁|ψ_in⟩⟨x|p⟩ = e^{ipx/ℏ}/√(2πℏ)|n⟩ = |n⁰⟩ + Σ ⟨m⁰|V̂|n⁰⟩/(Eₙ⁰−Eₘ⁰)|m⁰⟩⟨Â⟩ = Tr(ρ̂Â)iℏ ∂ψ/∂t = Ĥψ⟨φ|ψ⟩|n⟩ = (â†)ⁿ/√(n!) |0⟩ρ̂ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|CNOT = |0⟩⟨0|⊗𝟙 + |1⟩⟨1|⊗σₓ
