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

$$$$$$(4.1){\psi }_{\mu }\left(\mathit{x}\right)\equiv ⟨\mathit{x},\mu \mid \psi ⟩$$$$(4.2){\psi }_{\mu }(\mathit{x}-\mathit{a})=[1-\mathit{a}\bullet \frac{\partial }{\partial \mathit{x}}+\frac{1}{2!}{(\mathit{a}\bullet \frac{\partial }{\partial \mathit{x}})}^2-\dots ]{\psi }_{\mu }\left(\mathit{x}\right)$$$$=exp(-\mathit{a}\bullet \frac{\partial }{\partial \mathit{x}}){\psi }_{\mu }\left(\mathit{x}\right)=⟨\mathit{x},\mu \mid exp(-i\frac{\mathit{a}\bullet \mathit{p}}{\overline{h}})\mid \psi ⟩$$$$(4.3)\mid {\psi }^{{}^{\prime }}⟩\equiv U\left(\mathit{a}\right)\mid \psi ⟩;U\left(\mathit{a}\right)\equiv exp(-i\mathit{a}\bullet \mathit{p}/\overline{h})\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}\mathit{l}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$$$$(4.4){\psi }_{\mu }(\mathit{x}-\mathit{a})=⟨\mathit{x},\mu \mid U\left(\mathit{a}\right)\mid \psi ⟩=⟨\mathit{x},\mu \mid {\psi }^{{}^{\prime }}⟩={\psi }_{\mu }^{{}^{\prime }}\left(\mathit{x}\right)$$$$(4.5)i\overline{h}\frac{\partial \mid \psi ⟩}{\partial a_x}=-i\overline{h}\frac{\partial \mid \psi ⟩}{\partial x}=p_x\mid \psi ⟩--$$$$\mathit{P}\mathit{a}\mathit{s}\mathit{s}\mathit{i}\mathit{v}\mathit{e}\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{s}$$$$\left(1\right){\psi }_{\mu }(\mathit{x};\mathit{a}+\mathit{y})=exp(\mathit{a}\bullet \frac{\partial }{\partial \mathit{y}}){\psi }_{\mu }(\mathit{x};\mathit{y})$$$$\left(2\right)⟨\mathit{x},\mu ;0\mid U(-\mathit{a})\mid \psi ⟩={\psi }_{\mu }(\mathit{x}+\mathit{a};0)={\psi }_{\mu }(\mathit{x};\mathit{a})$$$$(4.6)\mid {\mathit{x}}_0,\mu ⟩=\int d^3\mathit{p}\mid \mathit{p},\mu ⟩⟨\mathit{p},\mu \mid {\mathit{x}}_0,\mu ⟩=\frac{1}{h^{3/2}}\int d^3\mathit{p}{e}^{-i{\mathit{x}}_0\bullet \mathit{p}/\overline{\mathit{h}}}\mid \mathit{p},\mu ⟩$$$$(4.7)U\left(\mathit{a}\right)\mid {\mathit{x}}_0,\mu ⟩=\frac{1}{h^{3/2}}\int d^3\mathit{p}{e}^{-i{\mathit{x}}_0\bullet \mathit{p}/\overline{\mathit{h}}}U\left(\mathit{a}\right)\mid \mathit{p},\mu ⟩$$$$=\frac{1}{h^{3/2}}\int d^3\mathit{p}{e}^{-i({\mathit{x}}_0+\mathbf{a})\bullet \mathit{p}/\overline{\mathit{h}}}\mid \mathit{p},\mu ⟩$$$$=\mid {\mathit{x}}_0,\mathit{a}⟩,\mu ⟩$$$$\mathit{O}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathit{s}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{c}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathit{s}$$$$\left(1\right)⟨\psi \mid A\mid \psi ⟩=⟨\psi \mid B\mid \psi ⟩$$$$\left(2\right)\lambda (⟨\mathrm{\varnothing }\mid A\mid \chi ⟩-⟨\mathrm{\varnothing }\mid B\mid \chi ⟩)={\lambda }^{\ast }(⟨\chi \mid B\mid \mathrm{\varnothing }⟩-⟨\chi \mid A\mid \mathrm{\varnothing }⟩)$$$$(4.8)1=⟨{\psi }^{{}^{\prime }}\mid {\psi }^{{}^{\prime }}⟩=⟨\psi \mid U^{+}U\mid \psi ⟩$$$$\mathit{u}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{r}\mathit{y}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}{U}^{+}U=I;U^{+}=U^{-1}$$$$(4.9)U\left(\delta \theta \right)=I-i\delta \theta \tau +O{\left(\delta \theta \right)}^2$$$$(4.10)I=U^{+}\left(\delta \theta \right)U\left(\delta \theta \right)=I+i\delta \theta (({\tau }^{+}-\tau )+O{\left(\delta \theta \right)}^2$$$$(4.11)i\frac{\partial \mid {\psi }^{{}^{\prime }}⟩}{\partial \theta }=\tau \mid {\psi }^{{}^{\prime }}⟩$$$$(4.12)U\left(\theta \right)\equiv \underset{N\to \mathrm{\infty }}{\lim }{(1-i\frac{\theta }{N}\tau )}^N=e^{-i\theta \tau }$$$$\tau \left(hermition\right)=\mathit{g}\mathit{e}\mathit{n}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathit{o}\mathit{f}U\mathrm{\&}the\mathbf{transformation}$$$$(4.13)U\left(\mathit{\alpha }\right)=exp(-i\mathit{\alpha }\bullet \mathit{J});J_i:\mathit{a}\mathit{n}\mathit{g}\mathit{u}\mathit{l}\mathit{a}\mathit{r}-\mathit{m}\mathit{o}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{u}\mathit{m}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathit{s}$$$$(4.14)i\frac{\partial \mid \psi ⟩}{\partial \alpha }=\hat{\mathit{\alpha }}\bullet \mathit{J}\mid \psi ⟩q$$$$(4.15)\mathit{p}\mathit{a}\mathit{r}\mathit{i}\mathit{t}\mathit{y}\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$$$\mathcal{P}\equiv \left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right);\mathcal{P}\mathit{x}=-\mathit{x}$$$$(4.16)\mathit{q}\mathit{u}\mathit{a}\mathit{n}\mathit{t}\mathit{u}\mathit{m}\mathit{p}\mathit{a}\mathit{r}\mathit{i}\mathit{t}\mathit{y}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}P:$$$$P{\psi }_{\mu }^{{}^{\prime }}\left(\mathit{x}\right)\equiv ⟨\mathit{x},\mu \mid P\mid \psi ⟩\equiv {\psi }_{\mu }\left(\mathcal{P}\mathit{x}\right)={\psi }_{\mu }(-\mathit{x})=⟨-\mathit{x},\mu \mid \psi ⟩$$$$(4.17){\psi }_{\mu }^{{}^{″}}\left(\mathit{x}\right)=⟨\mathit{x},\mu \mid P\mid {\psi }^{{}^{\prime }}⟩=⟨-\mathit{x},\mu \mid {\psi }^{{}^{\prime }}⟩$$$$=⟨-\mathit{x},\mu \mid P\mid \psi ⟩=⟨\mathit{x},\mu \mid \psi ⟩={\psi }_{\mu }\left(\mathit{x}\right)$$$$(4.18)⟨\mathrm{\varnothing }\mid P\mid \psi {⟩}^{\ast }=\int d^3\mathit{x}\sum _{\mu }{(⟨\mathrm{\varnothing }\downharpoonright \mathit{x},\mu ⟩⟨\mathit{x},\mu \downharpoonright P\downharpoonright \psi ⟩)}^{\ast }$$$$=\int d^3\mathit{x}\sum _{\mu }{(⟨\mathrm{\varnothing }\downharpoonright \mathit{x},\mu ⟩⟨-\mathit{x},\mu \downharpoonright \psi ⟩)}^{\ast }$$$$=\int d^3\mathit{x}\sum _{\mu }(⟨\psi \downharpoonright -\mathit{x},\mu ⟩⟨\mathit{x},\mu \downharpoonright P^2\downharpoonright \mathrm{\varnothing }⟩)$$$$=\int d^3\mathit{x}\sum _{\mu }(⟨\psi \downharpoonright -\mathit{x},\mu ⟩⟨-\mathit{x},\mu \downharpoonright P\downharpoonright \mathrm{\varnothing }⟩)=⟨\psi \mid P\mid \mathrm{\varnothing }⟩$$$$(4.19)\mathit{M}\mathit{i}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathit{s}$$$$⟨x,y\mid M\mid \psi ⟩=⟨y,x\mid \psi ⟩$$$$(4.20)U^{+}\left(\mathit{a}\right)\mathit{x}U\left(\mathit{a}\right)=\mathit{x}+\mathit{a}$$$$(4.21)\mathit{x}+\delta \mathit{a}\backsimeq (1+i\frac{\delta \mathit{a}\bullet \mathit{p}}{\overline{h}})\mathit{x}(1-i\frac{\delta \mathit{a}\bullet \mathit{p}}{\overline{h}})$$$$=\mathit{x}-\frac{i}{\overline{h}}[\mathit{x},\delta \mathit{a}\bullet \mathit{p}]+O{\left(\delta \mathit{a}\right)}^2$$$$(4.22)[x_i,p_j]=i\overline{h}{\delta }_{ij}$$$$(4.23)U^{+}\left(\mathit{a}\right)\mathit{x}U\left(\mathit{a}\right)=U^{+}\left(\mathit{a}\right)U\left(\mathit{a}\right)\mathit{x}+U^{+}\left(\mathit{a}\right)[\mathit{x},U\left(\mathit{a}\right)]=\mathit{x}+U^{+}\left(\mathit{a}\right)[\mathit{x},U\left(\mathit{a}\right)]$$$$(4.24)U^{+}\left(\mathit{a}\right)\mathit{x}U\left(\mathit{a}\right)=\mathit{x}-\frac{i}{\overline{h}}{U}^{+}\left(\mathit{a}\right)[\mathit{x},\mathit{a}\bullet \mathit{p}]U\left(\mathit{a}\right)=\mathit{x}+\mathit{a}$$$$\mathit{R}\mathit{o}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{s}\mathit{i}\mathit{n}\mathit{o}\mathit{r}\mathit{d}\mathit{i}\mathit{n}\mathit{a}\mathit{r}\mathit{y}\mathit{s}\mathit{p}\mathit{a}\mathit{c}\mathit{e}$$$${\mathit{R}}^T={\mathit{R}}^{-1};det\left(\mathit{R}\right)=+1;\mathit{R}\left(\mathit{\alpha }\right)\hat{\mathit{\alpha }}=\hat{\mathit{\alpha }}$$$$Tr\mathit{R}\left(\mathit{\alpha }\right)=1+2cos\mid \mathit{\alpha }\mid ;{\mathit{v}}^{{}^{\prime }}=\mathit{v}+\mathit{\alpha }\times \mathit{v}$$$$(4.25)\mathit{R}\left(\mathit{\alpha }\right)⟨\psi \mid \mathit{x}\mid \psi ⟩=⟨{\psi }^{{}^{\prime }}\mid \mathit{x}\mid {\psi }^{{}^{\prime }}⟩=⟨\psi \mid U^{+}\left(\mathit{\alpha }\right)\mathit{x}U\left(\mathit{\alpha }\right)\mid \psi ⟩$$$$(4.26)\mathit{R}\left(\mathit{\alpha }\right)\mathit{x}=U^{+}\left(\mathit{\alpha }\right)\mathit{x}U\left(\mathit{\alpha }\right)$$$$(4.27)\mathit{x}+\delta \mathit{\alpha }\times \mathit{x}\backsimeq (1+i\delta \mathit{\alpha }\bullet \mathit{J})\mathit{x}(1-i\delta \mathit{\alpha }\bullet \mathit{J})$$$$=\mathit{x}+i[\delta \mathit{\alpha }\bullet \mathit{J},\mathit{x}]+O{\left(\delta \mathit{\alpha }\right)}^2$$$$(4.28){\left(\delta \mathit{\alpha }\times \mathit{x}\right)}_i=\sum _{ij}{ϵ}_{ijk}\delta {\alpha }_j{x}_k$$$$(4.29)\sum _{ij}{ϵ}_{ijk}\delta {\alpha }_j{x}_k=i\sum _j\delta {\alpha }_j[J_j,x_i]$$$$(4.30)[J_i,x_j]=i\sum _k{ϵ}_{ijk}{x}_k$$$$(4.31)[J_i,v_j]=i\sum _k{ϵ}_{ijk}{v}_k$$$$(4.32)[J_i,p_j]=i\sum _k{ϵ}_{ijk}{p}_k$$$$(4.33)[J_i,J_j]=i\sum _k{ϵ}_{ijk}{J}_k$$$$(4.34)⟨{\psi }^{{}^{\prime }}\mid S\mid {\psi }^{{}^{\prime }}⟩=⟨\psi \mid U^{+}\left(\mathit{\alpha }\right)SU\left(\mathit{\alpha }\right)\mid \psi ⟩=⟨\psi \mid S\mid \psi ⟩$$$$(4.35)S\backsimeq (1+i\delta \mathit{\alpha }\bullet \mathit{J})S(1-i\delta \mathit{\alpha }\bullet \mathit{J})$$$$=S+i\delta \mathit{\alpha }\bullet [\mathit{J},S]+O{\left(\delta \mathit{\alpha }\right)}^2$$$$(4.36)[\mathit{J},S]=0$$$$(4.37)[\mathit{J},J^2]=0$$$$(4.38)\mathit{T}\mathit{h}\mathit{e}\mathit{p}\mathit{a}\mathit{r}\mathit{i}\mathit{t}\mathit{y}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}:\mathit{x}\to \mathcal{P}\mathit{x}=-\mathit{x}$$$$-⟨\psi \mid \mathit{x}\mid \psi ⟩=\mathcal{P}⟨\psi \mid \mathit{x}\mid \psi ⟩=⟨{\psi }^{{}^{\prime }}\mid \mathit{x}\mid {\psi }^{{}^{\prime }}⟩=⟨\psi \mid P^{+}\mathit{x}P\mid \psi ⟩$$$$(4.39)\{\mathit{x},P\}\equiv \mathit{x}P+P\mathit{x}=0$$$$(4.40)\{\mathit{v},P\}\equiv \mathit{v}P+P\mathit{v}=0$$$$(4.41)\mathit{v}\downharpoonright {\mathit{\omega }}^{{}^{\prime }}⟩=\mathit{v}(P\downharpoonright \mathit{\omega }⟩)=-P\mathit{v}\downharpoonright \mathit{\omega }⟩=-\mathit{\omega }P\downharpoonright \mathit{\omega }⟩=-\mathit{\omega }\downharpoonright {\mathit{\omega }}^{{}^{\prime }}⟩$$$$(4.42)-⟨\pm \mid \mathit{v}\mid \pm ⟩=\mathcal{P}⟨\pm \mid \mathit{v}\mid \pm ⟩=⟨\pm \mid P^{+}\mathit{v}P\mid \pm ⟩={(\pm )}^2⟨\pm \mid \mathit{v}\mid \pm ⟩$$$$(4.43a)⟨\mathit{x}\mid PV\mid \psi ⟩=⟨-\mathit{x}\mid V\mid \psi ⟩=V(-\mathit{x})⟨-\mathit{x}\mid \psi ⟩=V\left(\mathit{x}\right)⟨-\mathit{x}\mid \psi ⟩$$$$(4.43b)⟨\mathit{x}\mid VP\mid \psi ⟩=V\left(\mathit{x}\right)⟨\mathit{x}\mid P\mid \psi ⟩=V\left(\mathit{x}\right)⟨-\mathit{x}\mid \psi ⟩$$$$(4.44)p^{2}P=\sum _k{p}_k{p}_{k}P=-\sum _k{p}_k{Pp}_k=\sum _k{Pp}_k{p}_k={Pp}^2$$$$\Rightarrow [p^2,P]=0$$$$(4.45)\{P,[v_i,J_j]\}=i\sum _k{ϵ}_{ijk}\{P,v_k\}=0$$$$(4.46)0=\{P,[v_i,J_j]\}=[\{P,v_i\},J_j]-\{[P,J_j],v_i\}=-\{[P,J_j],v_i\}$$$$(4.47)[P,J_j]=\lambda P$$$$(4.48)⟨{\psi }^{{}^{\prime }}\mid \mathit{J}\mid {\psi }^{{}^{\prime }}⟩=⟨\psi \mid P^{+}\mathit{J}P\mid \psi ⟩=⟨\psi \mid \mathit{J}\mid \psi ⟩$$$$(4.48)⟨\psi \mid M^{+}xM\mid \psi ⟩=⟨\psi \mid y\mid \psi ⟩\mathit{M}\mathit{i}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}\mathit{s}$$$$(4.50)M^{+}xM=y\Rightarrow xM=My$$$$(4.51)\mid \psi ,t⟩=e^{-iHt/\overline{h}}\mid \psi ,0⟩$$$$(4.52)U\left(t\right)=e^{-iHt/\overline{h}}\mathit{t}\mathit{i}\mathit{m}\mathit{e}-\mathit{e}\mathit{v}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{o}\mathit{p}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{o}\mathit{r}$$$$(4.53)U\left(\theta \right)U\left(t\right)\mid \psi ⟩=U\left(t\right)U\left(\theta \right)\mid \psi ⟩$$$$(4.54a)⟨\mathit{x}\mid VU\left(\mathit{\alpha }\right)\mid \psi ⟩=V\left(\mathit{x}\right)⟨\mathit{x}\mid U\left(\mathit{\alpha }\right)\mid \psi ⟩=V\left(\mathit{x}\right)⟨\mathit{R}\left(\mathit{\alpha }\right)\mathit{x}\mid \psi ⟩$$$$(4.54b)⟨\mathit{x}\mid U\left(\mathit{\alpha }\right)V\mid \psi ⟩=⟨\mathit{R}\left(\mathit{\alpha }\right)\mathit{x}\mid V\mid \psi ⟩=V\left(\mathit{R}\left(\mathit{\alpha }\right)\mathit{x}\right)⟨\mathit{R}\left(\mathit{\alpha }\right)\mathit{x}\mid \psi ⟩$$$$(4.55)H=\underset{i=1}{\sum ^n}\frac{{\mathit{p}}_i^2}{2m_i}+\sum _{i<j}V({\mathit{x}}_i-{\mathit{x}}_j)$$$$(4.56)\mid \psi ,t⟩=U\left(t\right)\mid \psi ,0⟩$$$$(4.57){\stackrel{\sim }{Q}}_t\equiv U^{+}\left(t\right)QU\left(t\right)$$$$(4.58)⟨Q{⟩}_t=⟨\psi ,t\mid Q\mid \psi ,t⟩=⟨\psi ,0\mid U^{+}\left(t\right)QU\left(t\right)\mid \psi ,0⟩=⟨\psi ,0\mid {\stackrel{\sim }{Q}}_t\mid \psi ,0⟩$$$$(4.59)⟨\mathrm{\varnothing },t\downharpoonright \psi ,t⟩=⟨\mathrm{\varnothing },0\downharpoonright \psi ,0⟩;\mid \mathrm{\varnothing },t⟩\equiv U\left(t\right)\mid \mathrm{\varnothing },0⟩$$$$(4.60)\frac{d{\stackrel{\sim }{Q}}_t}{dt}=\frac{{dU}^{+}}{dt}QU+U^{+}Q\frac{dU}{dt}$$$$(4.61)\frac{dU}{dt}=-\frac{iH}{\overline{h}}U\Rightarrow \frac{{dU}^{+}}{dt}=\frac{iH}{\overline{h}}{U}^{+}$$$$(4.62)i\overline{h}\frac{d{\stackrel{\backsim }{Q}}_t}{dt}=-{HU}^{+}QU+U^{+}QUH=[{\stackrel{\backsim }{Q}}_t,H]$$$$(4.63)exp(-i\mathit{\alpha }\bullet \mathcal{J})\equiv \mathit{R}\left(\mathit{\alpha }\right)$$$$(4.64)\mathit{I}={\mathit{R}}^T\left(\mathit{\alpha }\right)\mathit{R}\left(\mathit{\alpha }\right)=exp{(-i\mathit{\alpha }\bullet \mathcal{J})}^{T}exp(-i\mathit{\alpha }\bullet \mathcal{J})$$$$=exp(-i\mathit{\alpha }\bullet {\mathcal{J}}^T)exp(-i\mathit{\alpha }\bullet \mathcal{J})$$$$(4.65)0=-i\mathit{n}\bullet {\mathcal{J}}^{T}exp(-i\theta \mathit{n}\bullet {\mathcal{J}}^T)exp(-i\theta \mathit{n}\bullet \mathcal{J})$$$$+exp(-i\theta \mathit{n}\bullet {\mathcal{J}}^T)exp(-i\theta \mathit{n}\bullet \mathcal{J})(-i\mathit{n}\bullet \mathcal{J})$$$$-i\mathit{n}\bullet \{{\mathcal{J}}^T+\mathcal{J}\}$$$$(4.66)\left\{{\mathit{R}}^T\left(\mathit{\alpha }\right)\mathit{R}\left(\mathit{\beta }\right)\mathit{R}\left(\mathit{\alpha }\right)\right\}{\mathit{\beta }}^{{}^{\prime }}={\mathit{R}}^T\left(\mathit{\alpha }\right)\mathit{R}\left(\mathit{\beta }\right)\mathit{\beta }={\mathit{R}}^T\left(\mathit{\alpha }\right){\mathit{\beta }}^{{}^{\prime }}$$$$(4.67){\mathit{R}}^T\left(\mathit{\alpha }\right)\mathit{R}\left(\mathit{\beta }\right)\mathit{R}\left(\mathit{\alpha }\right)=\mathit{R}\left({\mathit{\beta }}^{{}^{\prime }}\right)=\mathit{R}\left(\mathit{R}(-\mathit{\alpha })\mathit{\beta }\right)$$$$(4.68)(1+i\mathit{\alpha }\bullet \mathcal{J})(1+i\mathit{\beta }\bullet \mathcal{J})(1-i\mathit{\alpha }\bullet \mathcal{J})\backsimeq 1-i(\mathit{\beta }-\mathit{\alpha }\times \mathit{\beta })\bullet \mathcal{J}$$$$(4.69){\alpha }_i{\beta }_j[{\mathcal{J}}_i,{\mathcal{J}}_j]=i{\alpha }_i{\beta }_j\sum _k{ϵ}_{ijk}{\mathcal{J}}_k$$$$(4.70)[{\mathcal{J}}_i,{\mathcal{J}}_j]=i\sum _k{ϵ}_{ijk}{\mathcal{J}}_k$$$$(4.71)Prob(at\mathit{x}\downharpoonright \psi )=\sum _{\mu }\mid ⟨\mathit{x},\mu \downharpoonright \psi ⟩{\mid }^2$$$$(4.72)\mathit{R}\left(\mathrm{\varnothing }\right)=\left(\begin{array}{ccc}\cos \mathrm{\varnothing }& -\sin \mathrm{\varnothing }& 0\\ \sin \mathrm{\varnothing }& \cos \mathrm{\varnothing }& 0\\ 0& 0& 1\end{array}\right)$$$$(4.73){\mathcal{J}}_z^{{}^{\prime }}=\equiv \mathit{M}\bullet {\mathcal{J}}_z\bullet {\mathit{M}}^{+}$$$$(4.74)S_x=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right);S_y=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right);S_z=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right)$$$$(4.75)⟨\mathit{x}\mid \mathit{p}⟩=e^{i\mathit{p}\bullet \mathit{x}/\overline{h}}$$$$(4.76)[\{A,B\},C]=\{A,[B,C]\}+\{[A,C],B\}$$$$(4.77)G\equiv \frac{1}{2}(1-P)$$$$(4.78)S⟨\psi \mid \mathit{x}\mid \psi ⟩=⟨\psi \mid S^{+}\mathit{x}S\mid \psi ⟩$$$$(4.79)S_{ij}={\delta }_{ij}-2n_i{n}_j$$$$(4.80)V\left(\mathit{x}\right)=f\left(R\right)+\lambda xy;R=\sqrt{x^2+y^2}$$$$$$

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

$$Kindly{don}^{\prime }tmarkitasinappropiate.Therearemany$$$$derivationsinphysics.$$

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

$$ATheoreticalphysicist$$

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