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\begin{align}
& \left\langle {{{\hat{F}}}_{S}} \right\rangle ={{\left\langle  \Psi  \right|}_{t}}{{{\hat{F}}}_{S}}{{\left| \Psi  \right\rangle }_{t}}={{\left\langle  \Psi  \right|}_{t}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{{\hat{F}}}_{S}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{\left| \Psi  \right\rangle }_{t}} \\
& {{\left\langle  \Psi  \right|}_{t}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}={{\left\langle  \Psi  \right|}_{W}} \\
& {{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{{\hat{F}}}_{S}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}={{{\hat{F}}}_{W}}(t) \\
& {{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{\left| \Psi  \right\rangle }_{0}}={{\left| \Psi  \right\rangle }_{W}} \\
& \left\langle {{{\hat{F}}}_{S}} \right\rangle ={{\left\langle  \Psi  \right|}_{W}}{{{\hat{F}}}_{W}}(t){{\left| \Psi  \right\rangle }_{W}} \\
\end{align}

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⟨ F ^ S ⟩ = ⟨ Ψ | t ⁢ F ^ S ⁢ | Ψ ⟩ t = ⟨ Ψ | t ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t ⁢ e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ F ^ S ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t ⁢ e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ | Ψ ⟩ t ⟨ Ψ | t ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t = ⟨ Ψ | W e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ F ^ S ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t = F ^ W ⁢ ( t ) e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ | Ψ ⟩ 0 = | Ψ ⟩ W ⟨ F ^ S ⟩ = ⟨ Ψ | W ⁢ F ^ W ⁢ ( t ) ⁢ | Ψ ⟩ W missing-subexpression delimited-⟨⟩ subscript ^ 𝐹 𝑆 subscript bra Ψ 𝑡 subscript ^ 𝐹 𝑆 subscript ket Ψ 𝑡 subscript bra Ψ 𝑡 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ^ 𝐹 𝑆 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ket Ψ 𝑡 missing-subexpression subscript bra Ψ 𝑡 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript bra Ψ 𝑊 missing-subexpression superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ^ 𝐹 𝑆 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ^ 𝐹 𝑊 𝑡 missing-subexpression superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ket Ψ 0 subscript ket Ψ 𝑊 missing-subexpression delimited-⟨⟩ subscript ^ 𝐹 𝑆 subscript bra Ψ 𝑊 subscript ^ 𝐹 𝑊 𝑡 subscript ket Ψ 𝑊 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\left\langle{{{\hat{% F}}}_{S}}\right\rangle={{\left\langle\Psi\right|}_{t}}{{{\hat{F}}}_{S}}{{\left% |\Psi\right\rangle}_{t}}={{\left\langle\Psi\right|}_{t}}{{e}^{-\frac{i}{\hbar}% {{{\hat{H}}}^{0}}t}}{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{{\hat{F}}}_{S}% }{{e}^{-\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^% {0}}t}}{{\left|\Psi\right\rangle}_{t}}\\ &\displaystyle{{\left\langle\Psi\right|}_{t}}{{e}^{-\frac{i}{\hbar}{{{\hat{H}}% }^{0}}t}}={{\left\langle\Psi\right|}_{W}}\\ &\displaystyle{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{{\hat{F}}}_{S}}{{e}^% {-\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}={{{\hat{F}}}_{W}}(t)\\ &\displaystyle{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{\left|\Psi\right% \rangle}_{0}}={{\left|\Psi\right\rangle}_{W}}\\ &\displaystyle\left\langle{{{\hat{F}}}_{S}}\right\rangle={{\left\langle\Psi% \right|}_{W}}{{{\hat{F}}}_{W}}(t){{\left|\Psi\right\rangle}_{W}}\\ \end{aligned}}}

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⟨F^S⟩=⟨Ψ|tF^S|Ψ⟩t=⟨Ψ|te−iℏH^0te+iℏH^0tF^Se−iℏH^0te+iℏH^0t|Ψ⟩t⟨Ψ|te−iℏH^0t=⟨Ψ|We+iℏH^0tF^Se−iℏH^0t=F^W(t)e+iℏH^0t|Ψ⟩0=|Ψ⟩W⟨F^S⟩=⟨Ψ|WF^W(t)|Ψ⟩W

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