4-1-x-x-4-2-x-a-1-a-x-1-2-a-x-2-x-exp-a-x-x-x-exp-i-a-p-h-4-3-U-a-U-a-exp-ia-p-h-translat

Question Number 118633 by wsyip last updated on 18/Oct/20

(4.1) ψ_μ (x)≡⟨x,μ∣ψ⟩ (4.2) ψ_μ (x−a)=[1−a•(∂/∂x)+(1/(2!))(a•(∂/∂x))^2 −…]ψ_μ (x) =exp(−a•(∂/∂x))ψ_μ (x)=⟨x,μ∣exp(−i((a•p)/h^� ))∣ψ⟩ (4.3) ∣ψ^′ ⟩≡U(a)∣ψ⟩ ; U(a)≡exp(−ia•p/h^� ) translation operator (4.4) ψ_μ (x−a)=⟨x,μ∣U(a)∣ψ⟩=⟨x,μ∣ψ^′ ⟩=ψ_μ ^′ (x) (4.5) ih^� ((∂∣ψ⟩)/∂a_x )=−ih^� ((∂∣ψ⟩)/∂x)=p_x ∣ψ⟩−− Passive transformations (1) ψ_μ (x;a+y)=exp(a•(∂/∂y))ψ_μ (x;y) (2) ⟨x,μ;0∣U(−a)∣ψ⟩=ψ_μ (x+a;0)=ψ_μ (x;a) (4.6) ∣x_0 ,μ⟩=∫d^3 p∣p,μ⟩⟨p,μ∣x_0 ,μ⟩=(1/h^(3/2) )∫d^3 pe^(−ix_0 •p/h^� ) ∣p,μ⟩ (4.7) U(a)∣x_0 ,μ⟩=(1/h^(3/2) )∫d^3 pe^(−ix_0 •p/h^� ) U(a)∣p,μ⟩ =(1/h^(3/2) )∫d^3 pe^(−i(x_0 +a)•p/h^� ) ∣p,μ⟩ =∣x_0 ,a⟩,μ⟩ Operators from expectation values (1) ⟨ψ∣A∣ψ⟩=⟨ψ∣B∣ψ⟩ (2) λ(⟨∅∣A∣χ⟩−⟨∅∣B∣χ⟩)=λ^∗ (⟨χ∣B∣∅⟩−⟨χ∣A∣∅⟩) (4.8) 1=⟨ψ^′ ∣ψ^′ ⟩=⟨ψ∣U^+ U∣ψ⟩ unitiry operator U^+ U=I ; U^+ =U^(−1) (4.9) U(δθ)=I−iδθτ+O(δθ)^2 (4.10) I=U^+ (δθ)U(δθ)=I+iδθ((τ^+ −τ)+O(δθ)^2 (4.11) i((∂∣ψ^′ ⟩)/∂θ)=τ∣ψ^′ ⟩ (4.12) U(θ)≡lim_(N→∞) (1−i(θ/N)τ)^N =e^(−iθτ) τ (hermition) = generator of U & the transformation (4.13) U(𝛂)=exp(−i𝛂•J) ; J_i :angular-momentum operators (4.14) i((∂∣ψ⟩)/∂α)=𝛂^� •J∣ψ⟩q (4.15) parity transformation P≡ (((−1),0,0),(0,(−1),0),(0,0,(−1)) ) ; Px=−x (4.16) quantum parity operator P: P ψ_μ ^′ (x)≡⟨x,μ∣P∣ψ⟩≡ψ_μ (Px)=ψ_μ (−x)=⟨−x,μ∣ψ⟩ (4.17) ψ_μ ^(′′) (x)=⟨x,μ∣P∣ψ^′ ⟩=⟨−x,μ∣ψ^′ ⟩ =⟨−x,μ∣P∣ψ⟩=⟨x,μ∣ψ⟩=ψ_μ (x) (4.18) ⟨∅∣P∣ψ⟩^∗ =∫d^3 xΣ_μ (⟨∅⇂x,μ⟩⟨x,μ⇂P⇂ψ⟩)^∗ =∫d^3 xΣ_μ (⟨∅⇂x,μ⟩⟨−x,μ⇂ψ⟩)^∗ =∫d^3 xΣ_μ (⟨ψ⇂−x,μ⟩⟨x,μ⇂P^2 ⇂∅⟩) =∫d^3 xΣ_μ (⟨ψ⇂−x,μ⟩⟨−x,μ⇂P⇂∅⟩)=⟨ψ∣P∣∅⟩ (4.19) Mirror operators ⟨x,y∣M∣ψ⟩=⟨y,x∣ψ⟩ (4.20) U^+ (a)xU(a)=x+a (4.21) x+δa⋍(1+i((δa•p)/h^� ))x(1−i((δa•p)/h^� )) =x−(i/h^� )[x,δa•p]+O(δa)^2 (4.22) [x_i ,p_j ]=ih^� δ_(ij) (4.23) U^+ (a)xU(a)=U^+ (a)U(a)x+U^+ (a)[x,U(a)]=x+U^+ (a)[x,U(a)] (4.24) U^+ (a)xU(a)=x−(i/h^� )U^+ (a)[x,a•p]U(a)=x+a Rotations in ordinary space R^T =R^(−1) ; det(R)=+1 ; R(𝛂)𝛂^� =𝛂^� TrR(𝛂)=1+2cos∣𝛂∣ ; v^′ =v+𝛂×v (4.25) R(𝛂)⟨ψ∣x∣ψ⟩=⟨ψ^′ ∣x∣ψ^′ ⟩=⟨ψ∣U^+ (𝛂)xU(𝛂)∣ψ⟩ (4.26) R(𝛂)x=U^+ (𝛂)xU(𝛂) (4.27) x+δ𝛂×x⋍(1+iδ𝛂•J)x(1−iδ𝛂•J) =x+i[δ𝛂•J,x]+O(δ𝛂)^2 (4.28) (δ𝛂×x)_i =Σ_(ij) ε_(ijk) δα_j x_k (4.29) Σ_(ij) ε_(ijk) δα_j x_k =iΣ_j δα_j [J_j ,x_i ] (4.30) [J_i ,x_j ]=iΣ_k ε_(ijk) x_k (4.31) [J_i ,v_j ]=iΣ_k ε_(ijk) v_k (4.32) [J_i ,p_j ]=iΣ_k ε_(ijk) p_k (4.33) [J_i ,J_j ]=iΣ_k ε_(ijk) J_k (4.34) ⟨ψ^′ ∣S∣ψ^′ ⟩=⟨ψ∣U^+ (𝛂)SU(𝛂)∣ψ⟩=⟨ψ∣S∣ψ⟩ (4.35) S⋍(1+iδ𝛂•J)S(1−iδ𝛂•J) =S+iδ𝛂•[J,S]+O(δ𝛂)^2 (4.36) [J,S]=0 (4.37) [J,J^2 ]=0 (4.38) The parity operator: x→Px=−x −⟨ψ∣x∣ψ⟩=P⟨ψ∣x∣ψ⟩=⟨ψ^′ ∣x∣ψ^′ ⟩=⟨ψ∣P^+ xP∣ψ⟩ (4.39) {x,P}≡xP+Px=0 (4.40) {v,P}≡vP+Pv=0 (4.41) v⇂𝛚^′ ⟩=v(P⇂𝛚⟩)=−Pv⇂𝛚⟩=−𝛚P⇂𝛚⟩=−𝛚⇂𝛚^′ ⟩ (4.42) −⟨±∣v∣±⟩=P⟨±∣v∣±⟩=⟨±∣P^+ vP∣±⟩=(±)^2 ⟨±∣v∣±⟩ (4.43a) ⟨x∣PV∣ψ⟩=⟨−x∣V∣ψ⟩=V(−x)⟨−x∣ψ⟩=V(x)⟨−x∣ψ⟩ (4.43b) ⟨x∣VP∣ψ⟩=V(x)⟨x∣P∣ψ⟩=V(x)⟨−x∣ψ⟩ (4.44) p^2 P=Σ_k p_k p_k P=−Σ_k p_k Pp_k =Σ_k Pp_k p_k =Pp^2 ⇒[p^2 ,P]=0 (4.45) {P,[v_i ,J_j ]}=iΣ_k ε_(ijk) {P,v_k }=0 (4.46) 0={P,[v_i ,J_j ]}=[{P,v_i },J_j ]−{[P,J_j ],v_i }=−{[P,J_j ],v_i } (4.47) [P,J_j ]=λP (4.48) ⟨ψ^′ ∣J∣ψ^′ ⟩=⟨ψ∣P^+ JP∣ψ⟩=⟨ψ∣J∣ψ⟩ (4.48) ⟨ψ∣M^+ xM∣ψ⟩=⟨ψ∣y∣ψ⟩ Mirror operators (4.50) M^+ xM=y ⇒ xM=My (4.51) ∣ψ,t⟩=e^(−iHt/h^� ) ∣ψ,0⟩ (4.52) U(t)=e^(−iHt/h^� ) time-evolution operator (4.53) U(θ)U(t)∣ψ⟩=U(t)U(θ)∣ψ⟩ (4.54a) ⟨x∣VU(𝛂)∣ψ⟩=V(x)⟨x∣U(𝛂)∣ψ⟩=V(x)⟨R(𝛂)x∣ψ⟩ (4.54b) ⟨x∣U(𝛂)V∣ψ⟩=⟨R(𝛂)x∣V∣ψ⟩=V(R(𝛂)x)⟨R(𝛂)x∣ψ⟩ (4.55) H=Σ_(i=1) ^n (p_i ^2 /(2m_i ))+Σ_(i<j) V(x_i −x_j ) (4.56) ∣ψ,t⟩=U(t)∣ψ,0⟩ (4.57) Q_t ^∼ ≡U^+ (t)QU(t) (4.58) ⟨Q⟩_t =⟨ψ,t∣Q∣ψ,t⟩=⟨ψ,0∣U^+ (t)QU(t)∣ψ,0⟩=⟨ψ,0∣Q_t ^∼ ∣ψ,0⟩ (4.59) ⟨∅,t⇂ψ,t⟩=⟨∅,0⇂ψ,0⟩ ; ∣∅,t⟩≡U(t)∣∅,0⟩ (4.60) (dQ_t ^∼ /dt)=(dU^+ /dt)QU+U^+ Q(dU/dt) (4.61) (dU/dt)=−((iH)/h^� )U⇒(dU^+ /dt)=((iH)/h^� )U^+ (4.62) ih^� (dQ_t ^∽ /dt)=−HU^+ QU+U^+ QUH=[Q_t ^∽ ,H] (4.63) exp(−i𝛂•J)≡R(𝛂) (4.64) I=R^T (𝛂)R(𝛂)=exp(−i𝛂•J)^T exp(−i𝛂•J) =exp(−i𝛂•J^T )exp(−i𝛂•J) (4.65) 0=−in•J^T exp(−iθn•J^T )exp(−iθn•J) +exp(−iθn•J^T )exp(−iθn•J)(−in•J) −in•{J^T +J} (4.66) {R^T (𝛂)R(𝛃)R(𝛂)}𝛃^′ =R^T (𝛂)R(𝛃)𝛃=R^T (𝛂)𝛃^′ (4.67) R^T (𝛂)R(𝛃)R(𝛂)=R(𝛃^′ )=R(R(−𝛂)𝛃) (4.68) (1+i𝛂•J)(1+i𝛃•J)(1−i𝛂•J)⋍1−i(𝛃−𝛂×𝛃)•J (4.69) α_i β_j [J_i ,J_j ]=iα_i β_j Σ_k ε_(ijk) J_k (4.70) [J_i ,J_j ]=iΣ_k ε_(ijk) J_k (4.71) Prob(at x⇂ψ)=Σ_μ ∣⟨x,μ⇂ψ⟩∣^2 (4.72) R(∅)= (((cos ∅),(−sin ∅),0),((sin ∅),(cos ∅),0),(0,0,1) ) (4.73) J_z ^′ =≡M•J_z •M^+ (4.74) S_x =(1/( (√2))) ((0,1,0),(1,0,1),(0,1,0) ) ; S_y =(1/( (√2))) ((0,(−i),0),(i,0,(−i)),(0,i,0) ) ; S_z = ((1,0,0),(0,0,0),(0,0,(−1)) ) (4.75) ⟨x∣p⟩=e^(ip•x/h^� ) (4.76) [{A,B},C]={A,[B,C]}+{[A,C],B} (4.77) G≡(1/2)(1−P) (4.78) S⟨ψ∣x∣ψ⟩=⟨ψ∣S^+ xS∣ψ⟩ (4.79) S_(ij) =δ_(ij) −2n_i n_j (4.80) V(x)=f(R)+λxy ; R=(√(x^2 +y^2 ))

(4.1) ψ_{μ} (x) \equiv ⟨ x, μ ∣ ψ ⟩

(4.2) ψ_{μ} (x - a) = [1 - a ∙ \frac{\partial}{\partial x} + \frac{1}{2!} {(a ∙ \frac{\partial}{\partial x})}^{2} - \dots] ψ_{μ} (x)

= e x p (- a ∙ \frac{\partial}{\partial x}) ψ_{μ} (x) = ⟨ x, μ ∣ e x p (- i \frac{a ∙ p}{\bar{h}}) ∣ ψ ⟩

(4.3) ∣ ψ^{^{'}} ⟩ \equiv U (a) ∣ ψ ⟩; U (a) \equiv e x p (- i a ∙ p / \bar{h}) t r a n s l a t i o n o p e r a t o r

(4.4) ψ_{μ} (x - a) = ⟨ x, μ ∣ U (a) ∣ ψ ⟩ = ⟨ x, μ ∣ ψ^{^{'}} ⟩ = ψ_{μ}^{^{'}} (x)

(4.5) i \bar{h} \frac{\partial ∣ ψ ⟩}{\partial a_{x}} = - i \bar{h} \frac{\partial ∣ ψ ⟩}{\partial x} = p_{x} ∣ ψ ⟩ - -

P a s s i v e t r a n s f o r m a t i o n s

(1) ψ_{μ} (x; a + y) = e x p (a ∙ \frac{\partial}{\partial y}) ψ_{μ} (x; y)

(2) ⟨ x, μ; 0 ∣ U (- a) ∣ ψ ⟩ = ψ_{μ} (x + a; 0) = ψ_{μ} (x; a)

(4.6) ∣ x_{0}, μ ⟩ = \int d^{3} p ∣ p, μ ⟩ ⟨ p, μ ∣ x_{0}, μ ⟩ = \frac{1}{h^{3 / 2}} \int d^{3} p e^{- i x_{0} ∙ p / \bar{h}} ∣ p, μ ⟩

(4.7) U (a) ∣ x_{0}, μ ⟩ = \frac{1}{h^{3 / 2}} \int d^{3} p e^{- i x_{0} ∙ p / \bar{h}} U (a) ∣ p, μ ⟩

= \frac{1}{h^{3 / 2}} \int d^{3} p e^{- i (x_{0} + a) ∙ p / \bar{h}} ∣ p, μ ⟩

=∣ x_{0}, a ⟩, μ ⟩

O p e r a t o r s f r o m e x p e c t a t i o n v a l u e s

(1) ⟨ ψ ∣ A ∣ ψ ⟩ = ⟨ ψ ∣ B ∣ ψ ⟩

(2) λ (⟨ \emptyset ∣ A ∣ χ ⟩ - ⟨ \emptyset ∣ B ∣ χ ⟩) = λ^{*} (⟨ χ ∣ B ∣ \emptyset ⟩ - ⟨ χ ∣ A ∣ \emptyset ⟩)

(4.8) 1 = ⟨ ψ^{^{'}} ∣ ψ^{^{'}} ⟩ = ⟨ ψ ∣ U^{+} U ∣ ψ ⟩

u n i t i r y o p e r a t o r U^{+} U = I; U^{+} = U^{- 1}

(4.9) U (δ θ) = I - i δ θ τ + O {(δ θ)}^{2}

(4.10) I = U^{+} (δ θ) U (δ θ) = I + i δ θ ((τ^{+} - τ) + O {(δ θ)}^{2}

(4.11) i \frac{\partial ∣ ψ^{^{'}} ⟩}{\partial θ} = τ ∣ ψ^{^{'}} ⟩

(4.12) U (θ) \equiv \lim_{N \to \infty} {(1 - i \frac{θ}{N} τ)}^{N} = e^{- i θ τ}

τ (h e r m i t i o n) = g e n e r a t o r o f U & t h e transformation

(4.13) U (α) = e x p (- i α ∙ J); J_{i} : a n g u l a r - m o m e n t u m o p e r a t o r s

(4.14) i \frac{\partial ∣ ψ ⟩}{\partial α} = \hat{α} ∙ J ∣ ψ ⟩ q

(4.15) p a r i t y t r a n s f o r m a t i o n

P \equiv (\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}); P x = - x

(4.16) q u a n t u m p a r i t y o p e r a t o r P :

P ψ_{μ}^{^{'}} (x) \equiv ⟨ x, μ ∣ P ∣ ψ ⟩ \equiv ψ_{μ} (P x) = ψ_{μ} (- x) = ⟨ - x, μ ∣ ψ ⟩

(4.17) ψ_{μ}^{^{″}} (x) = ⟨ x, μ ∣ P ∣ ψ^{^{'}} ⟩ = ⟨ - x, μ ∣ ψ^{^{'}} ⟩

= ⟨ - x, μ ∣ P ∣ ψ ⟩ = ⟨ x, μ ∣ ψ ⟩ = ψ_{μ} (x)

(4.18) ⟨ \emptyset ∣ P ∣ ψ ⟩^{*} = \int d^{3} x \sum_{μ} {(⟨ \emptyset ⇂ x, μ ⟩ ⟨ x, μ ⇂ P ⇂ ψ ⟩)}^{*}

= \int d^{3} x \sum_{μ} {(⟨ \emptyset ⇂ x, μ ⟩ ⟨ - x, μ ⇂ ψ ⟩)}^{*}

= \int d^{3} x \sum_{μ} (⟨ ψ ⇂ - x, μ ⟩ ⟨ x, μ ⇂ P^{2} ⇂ \emptyset ⟩)

= \int d^{3} x \sum_{μ} (⟨ ψ ⇂ - x, μ ⟩ ⟨ - x, μ ⇂ P ⇂ \emptyset ⟩) = ⟨ ψ ∣ P ∣ \emptyset ⟩

(4.19) M i r r o r o p e r a t o r s

⟨ x, y ∣ M ∣ ψ ⟩ = ⟨ y, x ∣ ψ ⟩

(4.20) U^{+} (a) x U (a) = x + a

(4.21) x + δ a ⋍ (1 + i \frac{δ a ∙ p}{\bar{h}}) x (1 - i \frac{δ a ∙ p}{\bar{h}})

= x - \frac{i}{\bar{h}} [x, δ a ∙ p] + O {(δ a)}^{2}

(4.22) [x_{i}, p_{j}] = i \bar{h} δ_{i j}

(4.23) U^{+} (a) x U (a) = U^{+} (a) U (a) x + U^{+} (a) [x, U (a)] = x + U^{+} (a) [x, U (a)]

(4.24) U^{+} (a) x U (a) = x - \frac{i}{\bar{h}} U^{+} (a) [x, a ∙ p] U (a) = x + a

R o t a t i o n s i n o r d i n a r y s p a c e

R^{T} = R^{- 1}; d e t (R) = + 1; R (α) \hat{α} = \hat{α}

T r R (α) = 1 + 2 c o s ∣ α ∣; v^{^{'}} = v + α \times v

(4.25) R (α) ⟨ ψ ∣ x ∣ ψ ⟩ = ⟨ ψ^{^{'}} ∣ x ∣ ψ^{^{'}} ⟩ = ⟨ ψ ∣ U^{+} (α) x U (α) ∣ ψ ⟩

(4.26) R (α) x = U^{+} (α) x U (α)

(4.27) x + δ α \times x ⋍ (1 + i δ α ∙ J) x (1 - i δ α ∙ J)

= x + i [δ α ∙ J, x] + O {(δ α)}^{2}

(4.28) {(δ α \times x)}_{i} = \sum_{i j} ϵ_{i j k} δ α_{j} x_{k}

(4.29) \sum_{i j} ϵ_{i j k} δ α_{j} x_{k} = i \sum_{j} δ α_{j} [J_{j}, x_{i}]

(4.30) [J_{i}, x_{j}] = i \sum_{k} ϵ_{i j k} x_{k}

(4.31) [J_{i}, v_{j}] = i \sum_{k} ϵ_{i j k} v_{k}

(4.32) [J_{i}, p_{j}] = i \sum_{k} ϵ_{i j k} p_{k}

(4.33) [J_{i}, J_{j}] = i \sum_{k} ϵ_{i j k} J_{k}

(4.34) ⟨ ψ^{^{'}} ∣ S ∣ ψ^{^{'}} ⟩ = ⟨ ψ ∣ U^{+} (α) S U (α) ∣ ψ ⟩ = ⟨ ψ ∣ S ∣ ψ ⟩

(4.35) S ⋍ (1 + i δ α ∙ J) S (1 - i δ α ∙ J)

= S + i δ α ∙ [J, S] + O {(δ α)}^{2}

(4.36) [J, S] = 0

(4.37) [J, J^{2}] = 0

(4.38) T h e p a r i t y o p e r a t o r : x \to P x = - x

- ⟨ ψ ∣ x ∣ ψ ⟩ = P ⟨ ψ ∣ x ∣ ψ ⟩ = ⟨ ψ^{^{'}} ∣ x ∣ ψ^{^{'}} ⟩ = ⟨ ψ ∣ P^{+} x P ∣ ψ ⟩

(4.39) {x, P} \equiv x P + P x = 0

(4.40) {v, P} \equiv v P + P v = 0

(4.41) v ⇂ ω^{^{'}} ⟩ = v (P ⇂ ω ⟩) = - P v ⇂ ω ⟩ = - ω P ⇂ ω ⟩ = - ω ⇂ ω^{^{'}} ⟩

(4.42) - ⟨ \pm ∣ v ∣ \pm ⟩ = P ⟨ \pm ∣ v ∣ \pm ⟩ = ⟨ \pm ∣ P^{+} v P ∣ \pm ⟩ = {(\pm)}^{2} ⟨ \pm ∣ v ∣ \pm ⟩

(4.43 a) ⟨ x ∣ P V ∣ ψ ⟩ = ⟨ - x ∣ V ∣ ψ ⟩ = V (- x) ⟨ - x ∣ ψ ⟩ = V (x) ⟨ - x ∣ ψ ⟩

(4.43 b) ⟨ x ∣ V P ∣ ψ ⟩ = V (x) ⟨ x ∣ P ∣ ψ ⟩ = V (x) ⟨ - x ∣ ψ ⟩

(4.44) p^{2} P = \sum_{k} p_{k} p_{k} P = - \sum_{k} p_{k} {P p}_{k} = \sum_{k} {P p}_{k} p_{k} = {P p}^{2}

\Rightarrow [p^{2}, P] = 0

(4.45) {P, [v_{i}, J_{j}]} = i \sum_{k} ϵ_{i j k} {P, v_{k}} = 0

(4.46) 0 = {P, [v_{i}, J_{j}]} = [{P, v_{i}}, J_{j}] - {[P, J_{j}], v_{i}} = - {[P, J_{j}], v_{i}}

(4.47) [P, J_{j}] = λ P

(4.48) ⟨ ψ^{^{'}} ∣ J ∣ ψ^{^{'}} ⟩ = ⟨ ψ ∣ P^{+} J P ∣ ψ ⟩ = ⟨ ψ ∣ J ∣ ψ ⟩

(4.48) ⟨ ψ ∣ M^{+} x M ∣ ψ ⟩ = ⟨ ψ ∣ y ∣ ψ ⟩ M i r r o r o p e r a t o r s

(4.50) M^{+} x M = y \Rightarrow x M = M y

(4.51) ∣ ψ, t ⟩ = e^{- i H t / \bar{h}} ∣ ψ, 0 ⟩

(4.52) U (t) = e^{- i H t / \bar{h}} t i m e - e v o l u t i o n o p e r a t o r

(4.53) U (θ) U (t) ∣ ψ ⟩ = U (t) U (θ) ∣ ψ ⟩

(4.54 a) ⟨ x ∣ V U (α) ∣ ψ ⟩ = V (x) ⟨ x ∣ U (α) ∣ ψ ⟩ = V (x) ⟨ R (α) x ∣ ψ ⟩

(4.54 b) ⟨ x ∣ U (α) V ∣ ψ ⟩ = ⟨ R (α) x ∣ V ∣ ψ ⟩ = V (R (α) x) ⟨ R (α) x ∣ ψ ⟩

(4.55) H = \underset{i = 1}{\sum^{n}} \frac{p_{i}^{2}}{2 m_{i}} + \sum_{i < j} V (x_{i} - x_{j})

(4.56) ∣ ψ, t ⟩ = U (t) ∣ ψ, 0 ⟩

(4.57) {\tilde{Q}}_{t} \equiv U^{+} (t) Q U (t)

(4.58) ⟨ Q ⟩_{t} = ⟨ ψ, t ∣ Q ∣ ψ, t ⟩ = ⟨ ψ, 0 ∣ U^{+} (t) Q U (t) ∣ ψ, 0 ⟩ = ⟨ ψ, 0 ∣ {\tilde{Q}}_{t} ∣ ψ, 0 ⟩

(4.59) ⟨ \emptyset, t ⇂ ψ, t ⟩ = ⟨ \emptyset, 0 ⇂ ψ, 0 ⟩; ∣ \emptyset, t ⟩ \equiv U (t) ∣ \emptyset, 0 ⟩

(4.60) \frac{d {\tilde{Q}}_{t}}{d t} = \frac{{d U}^{+}}{d t} Q U + U^{+} Q \frac{d U}{d t}

(4.61) \frac{d U}{d t} = - \frac{i H}{\bar{h}} U \Rightarrow \frac{{d U}^{+}}{d t} = \frac{i H}{\bar{h}} U^{+}

(4.62) i \bar{h} \frac{d {\overset{∽}{Q}}_{t}}{d t} = - {H U}^{+} Q U + U^{+} Q U H = [{\overset{∽}{Q}}_{t}, H]

(4.63) e x p (- i α ∙ J) \equiv R (α)

(4.64) I = R^{T} (α) R (α) = e x p {(- i α ∙ J)}^{T} e x p (- i α ∙ J)

= e x p (- i α ∙ J^{T}) e x p (- i α ∙ J)

(4.65) 0 = - i n ∙ J^{T} e x p (- i θ n ∙ J^{T}) e x p (- i θ n ∙ J)

+ e x p (- i θ n ∙ J^{T}) e x p (- i θ n ∙ J) (- i n ∙ J)

- i n ∙ {J^{T} + J}

(4.66) {R^{T} (α) R (β) R (α)} β^{^{'}} = R^{T} (α) R (β) β = R^{T} (α) β^{^{'}}

(4.67) R^{T} (α) R (β) R (α) = R (β^{^{'}}) = R (R (- α) β)

(4.68) (1 + i α ∙ J) (1 + i β ∙ J) (1 - i α ∙ J) ⋍ 1 - i (β - α \times β) ∙ J

(4.69) α_{i} β_{j} [J_{i}, J_{j}] = i α_{i} β_{j} \sum_{k} ϵ_{i j k} J_{k}

(4.70) [J_{i}, J_{j}] = i \sum_{k} ϵ_{i j k} J_{k}

(4.71) P r o b (a t x ⇂ ψ) = \sum_{μ} ∣ ⟨ x, μ ⇂ ψ ⟩ ∣^{2}

(4.72) R (\emptyset) = (\begin{matrix} \cos \emptyset & - \sin \emptyset & 0 \\ \sin \emptyset & \cos \emptyset & 0 \\ 0 & 0 & 1 \end{matrix})

(4.73) J_{z}^{^{'}} =\equiv M ∙ J_{z} ∙ M^{+}

(4.74) S_{x} = \frac{1}{\sqrt{2}} (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}); S_{y} = \frac{1}{\sqrt{2}} (\begin{matrix} 0 & - i & 0 \\ i & 0 & - i \\ 0 & i & 0 \end{matrix}); S_{z} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix})

(4.75) ⟨ x ∣ p ⟩ = e^{i p ∙ x / \bar{h}}

(4.76) [{A, B}, C] = {A, [B, C]} + {[A, C], B}

(4.77) G \equiv \frac{1}{2} (1 - P)

(4.78) S ⟨ ψ ∣ x ∣ ψ ⟩ = ⟨ ψ ∣ S^{+} x S ∣ ψ ⟩

(4.79) S_{i j} = δ_{i j} - 2 n_{i} n_{j}

(4.80) V (x) = f (R) + λ x y; R = \sqrt{x^{2} + y^{2}}

Commented by Dwaipayan Shikari last updated on 18/Oct/20

Kindly don′t mark it as inappropiate. There are many derivations in physics.

K i n d l y {d o n}^{'} t m a r k i t a s i n a p p r o p i a t e . T h e r e a r e m a n y

d e r i v a t i o n s i n p h y s i c s .

Commented by Dwaipayan Shikari last updated on 18/Oct/20

A T h e o r e t i c a l p h y s i c i s t

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