[proofplan] The commutation of the two unitary groups says that the time evolution $e^{-itH_{\mathrm{op}}}$ preserves the whole functional calculus of the observable $A$. In particular, each spectral projection $E_A(B)$ commutes with $e^{-itH_{\mathrm{op}}}$. Once this projection commutation is known, the spectral probability of $A$ in the evolved state is computed by moving $e^{-itH_{\mathrm{op}}}$ across $E_A(B)$ and cancelling it by unitarity. The expectation identity follows because equality of all spectral probabilities gives equality of the associated scalar spectral measures. [/proofplan] [step:Promote commutation of the unitary group to commutation of spectral projections] Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. Let $\mathcal L(H)$ denote the Banach algebra of bounded linear operators from $H$ to $H$. For each $s\in\mathbb R$, write $U_A(s):H\to H$ for the unitary operator $e^{-isA}$. For each $t\in\mathbb R$, write $U_H(t):H\to H$ for the unitary operator $e^{-itH_{\mathrm{op}}}$. Fix $t\in\mathbb R$, and define the bounded unitary operator $W:H\to H$ by $W=U_H(t)$. Since $W$ is unitary, $W\in\mathcal L(H)$. By hypothesis, \begin{align*} U_A(s)W=WU_A(s) \end{align*} for every $s\in\mathbb R$. We use the bounded [Borel functional calculus](/theorems/2696) commutation theorem for [self-adjoint operators](/page/Self-Adjoint%20Operators): if $A$ is self-adjoint on $H$, $E_A$ is its spectral measure, and a bounded operator $W\in\mathcal L(H)$ satisfies $U_A(s)W=WU_A(s)$ for every $s\in\mathbb R$, then $W$ commutes with the full bounded Borel functional calculus of $A$. In particular, taking the bounded Borel function $\mathbb{1}_B:\mathbb R\to\mathbb R$ gives $E_A(B)W=WE_A(B)$ for every Borel set $B\in\mathcal B(\mathbb R)$. The hypotheses of this theorem hold here because $A$ is self-adjoint, $E_A$ is the spectral measure of $A$, $W=U_H(t)$ is unitary and therefore bounded, and the displayed commutation relation holds for every $s\in\mathbb R$. Let $B\in\mathcal B(\mathbb R)$ be a Borel set. Applying the spectral commutation theorem to this $B$ gives \begin{align*} E_A(B)W=WE_A(B). \end{align*} Since $W=U_H(t)$, this is \begin{align*} E_A(B)U_H(t)=U_H(t)E_A(B). \end{align*} [guided] Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$, and let $\mathcal L(H)$ denote the Banach algebra of bounded linear operators from $H$ to $H$. For each $s\in\mathbb R$, let $U_A(s):H\to H$ denote $e^{-isA}$, and for each $t\in\mathbb R$, let $U_H(t):H\to H$ denote $e^{-itH_{\mathrm{op}}}$. Fix a time $t\in\mathbb R$. The operator carrying the state forward by time $t$ is the map $W:H\to H$ defined by $W=U_H(t)$. Since $U_H(t)$ is unitary, $W$ is bounded and belongs to $\mathcal L(H)$. The assumption of the theorem says that this fixed bounded operator commutes with every member of the unitary group generated by $A$: \begin{align*} U_A(s)W=WU_A(s) \end{align*} for every $s\in\mathbb R$. The key input is the bounded Borel functional calculus commutation theorem for self-adjoint operators. It says that if $A$ is self-adjoint, $E_A$ is its spectral measure, and a bounded operator $W\in\mathcal L(H)$ commutes with $U_A(s)=e^{-isA}$ for every $s\in\mathbb R$, then $W$ commutes with every operator $f(A)$ obtained from the bounded Borel functional calculus of $A$. For a Borel set $B\in\mathcal B(\mathbb R)$, the indicator function $\mathbb{1}_B:\mathbb R\to\mathbb R$ is bounded and Borel, and the functional calculus gives $\mathbb{1}_B(A)=E_A(B)$. Therefore $W$ commutes with every spectral projection of $A$: \begin{align*} E_A(B)W=WE_A(B) \end{align*} for every Borel set $B\in\mathcal B(\mathbb R)$. We verify the hypotheses before applying it. The operator $A$ is self-adjoint by the theorem statement. The family $E_A$ is the spectral measure associated to $A$ by the spectral theorem. The operator $W=U_H(t)$ is unitary, hence bounded. Finally, the displayed relation $U_A(s)W=WU_A(s)$ holds for every $s\in\mathbb R$ by the assumed commutation of the two unitary groups. Therefore, for the chosen Borel set $B\in\mathcal B(\mathbb R)$, \begin{align*} E_A(B)W=WE_A(B). \end{align*} Since $W=U_H(t)$, this is exactly \begin{align*} E_A(B)U_H(t)=U_H(t)E_A(B). \end{align*} This is the analytic form of the Noether principle here: the symmetry generated by $H_{\mathrm{op}}$ preserves every spectral subspace of the conserved observable $A$. [/guided] [/step] [step:Use projection commutation and unitarity to preserve spectral probabilities] Let $B\in\mathcal B(\mathbb R)$, let $t\in\mathbb R$, and let $\psi\in H$ be a unit vector. Since $U_H(t)$ is unitary, its Hilbert-space adjoint is $U_H(t)^*=U_H(-t)$, and \begin{align*} (U_H(t)x,U_H(t)y)_H=(x,y)_H \end{align*} for all $x,y\in H$. Using the commutation relation from the previous step, we compute \begin{align*} (E_A(B)U_H(t)\psi,U_H(t)\psi)_H=(U_H(t)E_A(B)\psi,U_H(t)\psi)_H. \end{align*} Applying unitarity of $U_H(t)$ to the [inner product](/page/Inner%20Product) gives \begin{align*} (U_H(t)E_A(B)\psi,U_H(t)\psi)_H=(E_A(B)\psi,\psi)_H. \end{align*} Therefore \begin{align*} (E_A(B)U_H(t)\psi,U_H(t)\psi)_H=(E_A(B)\psi,\psi)_H. \end{align*} [/step] [step:Identify the scalar spectral measures of the original and evolved states] For the fixed unit vector $\psi\in H$, define the scalar spectral measure \begin{align*} \mu_\psi:\mathcal B(\mathbb R)\to[0,1],\qquad B\mapsto (E_A(B)\psi,\psi)_H. \end{align*} For the evolved vector $U_H(t)\psi$, define \begin{align*} \mu_{t,\psi}:\mathcal B(\mathbb R)\to[0,1],\qquad B\mapsto (E_A(B)U_H(t)\psi,U_H(t)\psi)_H. \end{align*} The preceding step proves that \begin{align*} \mu_{t,\psi}(B)=\mu_\psi(B) \end{align*} for every $B\in\mathcal B(\mathbb R)$. Hence \begin{align*} \mu_{t,\psi}=\mu_\psi \end{align*} as Borel probability measures on $\mathbb R$. [/step] [step:Integrate the identity of scalar spectral measures to conserve the expectation] Assume that \begin{align*} \int_{\mathbb R}|\lambda|\,d\mu_\psi(\lambda)<\infty. \end{align*} Since $\mu_{t,\psi}=\mu_\psi$, the same absolute integrability holds for the evolved state: \begin{align*} \int_{\mathbb R}|\lambda|\,d\mu_{t,\psi}(\lambda)=\int_{\mathbb R}|\lambda|\,d\mu_\psi(\lambda)<\infty. \end{align*} Therefore the function $\lambda\mapsto \lambda$ is integrable with respect to both scalar spectral measures. Equality of the measures gives equality of their integrals: \begin{align*} \int_{\mathbb R}\lambda\,d\mu_{t,\psi}(\lambda)=\int_{\mathbb R}\lambda\,d\mu_\psi(\lambda). \end{align*} Substituting the definitions of $\mu_{t,\psi}$ and $\mu_\psi$ yields \begin{align*} \int_{\mathbb R}\lambda\,d(E_A(\lambda)U_H(t)\psi,U_H(t)\psi)_H=\int_{\mathbb R}\lambda\,d(E_A(\lambda)\psi,\psi)_H. \end{align*} This is the asserted conservation of the expectation of $A$ under the time evolution generated by $H_{\mathrm{op}}$. [/step]