Kleinert · 제1장 기초

$$ \begin{align*} g^{11} & =\frac{1}{g}\left\{I_{\eta}+\left(I_{\xi}-I_{\eta}\right) \sin ^{2} \gamma\right\} I_{\zeta} \\ g^{21} & =\frac{1}{g} \sin \beta \sin \gamma \cos \gamma\left(I_{\xi}-I_{\eta}\right) I_{\zeta} \\ g^{31} & =\frac{1}{g}\left\{\cos \beta\left[-\sin ^{2} \gamma\left(I_{\xi}-I_{\eta}\right)-I_{\eta}\right]\right\} I_{\zeta} \\ g^{22} & =\frac{1}{g}\left\{\sin ^{2} \beta\left[I_{\xi}-\sin ^{2} \gamma\left(I_{\xi}-I_{\eta}\right)\right]\right\} I_{\zeta} \\ g^{32} & =\frac{1}{g}\left\{\sin \beta \cos \beta \sin \gamma \cos \gamma\left(I_{\eta}-I_{\xi}\right)\right\} I_{\zeta}, \\ g^{33} & =\frac{1}{g}\left\{\sin ^{2} \beta I_{\xi} I_{\eta}+\cos ^{2} \beta I_{\eta} I_{\zeta}+\cos ^{2} \beta \sin ^{2} \gamma\left(I_{\xi}-I_{\eta}\right) I_{\zeta}\right\} . \end{align*} $$

$$ \left.\delta \mathcal{A}\left[q_{i}\right]\right|_{q_{i}(t)=q_{i}^{\mathrm{cl}}(t)}=0 $$

$$ \sigma^{i} \sigma^{j}=\delta_{i j}+i \epsilon_{i j k} \sigma^{k} $$

$$ \alpha R e^{-R^{2} \cos 2 \alpha}+\frac{1}{R \sin 2 \alpha}\left[e^{-R^{2} \cos 2 \varphi}\right]_{\varphi=\alpha}^{\varphi=\pi / 4} $$

$$ \begin{align*} \bar{\Gamma}_{11}^{1}= & {\left[\cos \beta \cos \gamma \sin \gamma\left(I_{\eta}^{2}-I_{\eta} I_{\zeta}-I_{\xi}^{2}+I_{\xi} I_{\zeta}\right)\right] / I_{\xi} I_{\eta}, } \\ \bar{\Gamma}_{21}^{1}= & \left\{\operatorname { c o s } \beta \left[\sin ^{2} \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-\left(I_{\xi}-I_{\eta}\right) I_{\zeta}\right)\right.\right. \\ & \left.\left.+I_{\eta}\left(I_{\xi}+I_{\eta}-I_{\zeta}\right)\right]\right\} / 2 \sin \beta I_{\xi} I_{\eta}, \\ \bar{\Gamma}_{31}^{1}= & \left\{\cos \gamma \sin \gamma\left[I_{\eta}^{2}-I_{\xi}^{2}+\left(I_{\xi}-I_{\eta}\right) I_{\zeta}\right]\right\} / 2 I_{\xi} I_{\eta}, \\ \bar{\Gamma}_{22}^{1}= & 0, \\ \bar{\Gamma}_{32}^{1}= & {\left[\sin ^{2} \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-\left(I_{\xi}-I_{\eta}\right) I_{\zeta}\right)-I_{\eta}\left(I_{\xi}-I_{\eta}+I_{\zeta}\right)\right] / 2 \sin \beta I_{\xi} I_{\eta}, } \\ \bar{\Gamma}_{33}^{1}= & 0, \\ \bar{\Gamma}_{11}^{2}= & \left\{\cos \beta \sin \beta\left[\sin ^{2} \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)\right)-I_{\xi}\left(I_{\xi}-I_{\zeta}\right)\right]\right\} / I_{\xi} I_{\eta}, \\ \bar{\Gamma}_{21}^{2}= & \left\{\cos \beta \cos \gamma \sin _{\gamma}\left[I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)\right]\right\} / 2 I_{\xi} I_{\eta}, \\ \bar{\Gamma}_{31}^{2}= & \left\{\sin \beta\left[\sin ^{2} \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)\right)-I_{\xi}\left(I_{\xi}-I_{\eta}-I_{\zeta}\right)\right]\right\} / 2 I_{\xi} I_{\eta}, \\ \bar{\Gamma}_{22}^{2}= & 0, \\ \bar{\Gamma}_{32}^{2}= & {\left[\cos \gamma \sin \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)\right)\right] / 2 I_{\xi} I_{\eta}, } \\ \bar{\Gamma}_{33}^{2}= & 0, \\ \bar{\Gamma}_{11}^{3}= & \left\{\operatorname { c o s } \gamma \operatorname { s i n } \gamma \left[\sin ^{2} \beta\left(I_{\xi} I_{\eta}\left(I_{\xi}-I_{\eta}\right)-I_{\zeta}\left(I_{\xi}^{2}-I_{\eta}^{2}\right)+I_{\zeta}^{2}\left(I_{\xi}-I_{\eta}\right)\right)\right.\right. \\ & \left.\left.+\left(I_{\xi}^{2}-I_{\eta}^{2}\right) I_{\zeta}-I_{\zeta}^{2}\left(I_{\xi}-I_{\eta}\right)\right]\right\} / I_{\xi} I_{\eta} I_{\zeta}, \\ \bar{\Gamma}_{21}^{3}= & \left\{\operatorname { s i n } { } ^ { 2 } \beta \left[\sin ^{2} \gamma\left(2 I_{\xi} I_{\eta}\left(I_{\eta}-I_{\xi}\right)+I_{\zeta}\left(I_{\xi}^{2}-I_{\eta}^{2}\right)-I_{\zeta}^{2}\left(I_{\xi}-I_{\eta}\right)\right)\right.\right. \\ & \left.+I_{\xi} I_{\eta}\left(I_{\xi}-I_{\eta}\right)+I_{\eta} I_{\zeta}\left(I_{\eta}-I_{\zeta}\right)\right]-\sin ^{2} \gamma\left(\left(I_{\xi}^{2}-I_{\eta}^{2}\right) I_{\zeta}-I_{\zeta}^{2}\left(I_{\xi}-I_{\eta}\right)\right) \\ & \left.-I_{\eta} I_{\zeta}\left(I_{\xi}+I_{\eta}-I_{\zeta}\right)\right\} / 2 \sin \beta I_{\xi} I_{\eta} I_{\zeta}, \\ \bar{\Gamma}_{31}^{3}= & {\left[\cos \beta \cos \gamma \sin \gamma\left(I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)\right)\right] / 2 I_{\xi} I_{\eta}, } \\ \bar{\Gamma}_{22}^{3}= & \cos \gamma \sin \gamma^{3}\left(I_{\eta}-I_{\xi}\right) / I_{\zeta}, \\ \bar{\Gamma}_{32}^{3}= & \left\{\cos \beta\left[\sin { }^{2} \gamma\left(I_{\eta}^{2}-I_{\xi}^{2}+\left(I_{\xi}-I_{\eta}\right) I_{\zeta}\right)+I_{\eta}\left(I_{\xi}-I_{\eta}+I_{\zeta}\right)\right]\right\} / 2 \sin \beta I_{\eta} I_{\xi}, \\ \bar{\Gamma}_{33}^{3}= & 0 . \end{align*} $$

$$ \delta q_{i}\left(t_{a}\right)=\delta q_{i}\left(t_{b}\right)=0 . $$

$$ \hat{\mathbf{B}} \cdot \boldsymbol{\sigma} \delta \mathbf{B}(t) \cdot \boldsymbol{\sigma}=\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)+i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \boldsymbol{\sigma}, \quad \delta \mathbf{B}(t) \cdot \boldsymbol{\sigma} \hat{\mathbf{B}} \cdot \boldsymbol{\sigma}=\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)-i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \sigma $$

$$ \int_{\infty}^{\infty} d p e^{-i p^{2}}=e^{-i \pi / 4} \sqrt{\pi} $$

$$ \begin{align*} \bar{R}_{11}= & \left\{\operatorname { s i n } ^ { 2 } \beta \left[\sin ^{2} \gamma\left(I_{\eta}^{3}-I_{\xi}^{3}-\left(I_{\xi} I_{\eta}-I_{\zeta}^{2}\right)\left(I_{\xi}-I_{\eta}\right)\right)\right.\right. \\ & \left.\left.+\left(\left(I_{\xi}+I_{\zeta}\right)^{2}-I_{\eta}^{2}\right)\left(I_{\xi}-I_{\zeta}\right)\right]+I_{\zeta}^{3}-I_{\zeta}\left(I_{\xi}-I_{\eta}\right)^{2}\right\} / 2 I_{\xi} I_{\eta} I_{\zeta} \\ \bar{R}_{21}= & \left\{\sin \beta \sin \gamma \cos \gamma\left[I_{\eta}^{3}-I_{\xi}^{3}+\left(I_{\xi} I_{\eta}-I_{\zeta}^{2}\right)\left(I_{\eta}-I_{\xi}\right)\right]\right\} / 2 I_{\xi} I_{\eta} I_{\zeta} \\ \bar{R}_{31}= & -\left\{\cos \beta\left[\left(I_{\xi}-I_{\eta}\right)^{2}-I_{\zeta}^{2}\right]\right\} / 2 I_{\xi} I_{\eta} \\ \bar{R}_{22}= & \left\{\sin ^{2} \gamma\left[I_{\xi}^{3}-I_{\eta}^{3}+\left(I_{\xi} I_{\eta}-I_{\zeta}^{2}\right)\left(I_{\xi}-I_{\eta}\right)\right]+I_{\eta}^{3}-\left(I_{\xi}-I_{\zeta}\right)^{2} I_{\eta}\right\} / 2 I_{\xi} I_{\eta} I_{\zeta}, \\ \bar{R}_{32}= & 0, \\ \bar{R}_{33}= & -\left[\left(I_{\xi}-I_{\eta}\right)^{2}-I_{\zeta}^{2}\right] / 2 I_{\xi} I_{\eta} \end{align*} $$

$$ \begin{align*} \delta \mathcal{A}\left[q_{i}\right] & =\left\{\mathcal{A}\left[q_{i}+\delta q_{i}\right]-\mathcal{A}\left[q_{i}\right]\right\}_{\text {lin }} \\ & =\int_{t_{a}}^{t_{b}} d t\left\{L\left(q_{i}(t)+\delta q_{i}(t), \dot{q}_{i}(t)+\delta \dot{q}_{i}(t), t\right)-L\left(q_{i}(t), \dot{q}_{i}(t), t\right)\right\}_{\text {lin }} \\ & =\int_{t_{a}}^{t_{b}} d t\left\{\frac{\partial L}{\partial q_{i}} \delta q_{i}(t)+\frac{\partial L}{\partial \dot{q}_{i}} \delta \dot{q}_{i}(t)\right\} \\ & =\int_{t_{a}}^{t_{b}} d t\left\{\frac{\partial L}{\partial q_{i}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{q}_{i}}\right\} \delta q_{i}(t)+\left.\frac{\partial L}{\partial \dot{q}_{i}} \delta q_{i}(t)\right|_{t_{a}} ^{t_{b}} \end{align*} $$

$$ \begin{align*} \hat{\mathbf{B}} \cdot \boldsymbol{\sigma} \delta \mathbf{B}(t) \cdot \boldsymbol{\sigma} \hat{\mathbf{B}} \cdot \boldsymbol{\sigma} & =[\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)] \hat{\mathbf{B}} \cdot \boldsymbol{\sigma}+i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \boldsymbol{\sigma} \hat{\mathbf{B}} \cdot \boldsymbol{\sigma} \\ & =i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \hat{\mathbf{B}}+\{[\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)] \hat{\mathbf{B}}-[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \times \hat{\mathbf{B}}\} \cdot \boldsymbol{\sigma} \end{align*} $$

$$ \bar{R}=\left[2\left(I_{\xi} I_{\eta}+I_{\eta} I_{\zeta}+I_{\zeta} I_{\xi}\right)-I_{\xi}^{2}-I_{\eta}^{2}-I_{\zeta}^{2}\right] / 2 I_{\xi} I_{\eta} I_{\zeta} $$

$$ \frac{d}{d t} \frac{\partial L}{\partial \dot{q}_{i}}=\frac{\partial L}{\partial q_{i}} $$

$$ \begin{array}{r} \cos B\left(t_{b}-t\right) / 2 \cos B\left(t-t_{a}\right) / 2 \delta \mathbf{B}(t) \cdot \boldsymbol{\sigma} \\ +i \sin B\left(t_{b}-t\right) / 2 \cos B\left(t-t_{a}\right) / 2\{\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)+i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \boldsymbol{\sigma}\} \\ +i \cos B\left(t_{b}-t\right) / 2 \sin B\left(t-t_{a}\right) / 2\{\hat{\mathbf{B}} \cdot \delta \mathbf{B}(t)-i[\hat{\mathbf{B}} \times \delta \mathbf{B}(t)] \cdot \boldsymbol{\sigma}\} \\ +\sin B\left(t_{b}-t\right) / 2 \sin B\left(t-t_{a}\right) / 2 \delta \mathbf{B} \cdot \boldsymbol{\sigma} \end{array} $$

$$ H \equiv \frac{\partial L}{\partial \dot{q}_{i}} \dot{q}_{i}-L\left(q_{i}, \dot{q}_{i}, t\right) $$

$$ p_{i} \equiv \frac{\partial}{\partial \dot{q}_{i}} L\left(q_{i}, \dot{q}_{i}, t\right) $$

$$ \dot{q}_{i}=v_{i}\left(p_{i}, q_{i}, t\right) $$

$$ h_{i j}\left(q_{i}, \dot{q}_{i}, t\right) \equiv \frac{\partial^{2}}{\partial \dot{q}_{i} \partial \dot{q}_{j}} L\left(q_{i}, \dot{q}_{i}, t\right) $$

$$ H\left(p_{i}, q_{i}, t\right)=p_{i} v_{i}\left(p_{i}, q_{i}, t\right)-L\left(q_{i}, v_{i}\left(p_{i}, q_{i}, t\right), t\right) $$

$$ \mathcal{A}\left[p_{i}, q_{i}\right]=\int_{t_{a}}^{t_{b}} d t\left[p_{i}(t) \dot{q}_{i}(t)-H\left(p_{i}(t), q_{i}(t), t\right)\right] $$

$$ \begin{align*} q_{i}(t)=q_{i}^{\mathrm{cl}}(t)+\delta q_{i}(t), \quad \delta q_{i}\left(t_{a}\right)=\delta q_{i}\left(t_{b}\right)=0 \\ p_{i}(t)=p_{i}^{\mathrm{cl}}(t)+\delta p_{i}(t) \end{align*} $$

$$ \begin{gather*} \delta \mathcal{A}\left[p_{i}, q_{i}\right]=\int_{t_{a}}^{t_{b}} d t\left[\delta p_{i}(t) \dot{q}_{i}(t)+p_{i}(t) \delta \dot{q}_{i}(t)-\frac{\partial H}{\partial p_{i}} \delta p_{i}-\frac{\partial H}{\partial q_{i}} \delta q_{i}\right] \\ =\int_{t a}^{t_{b}} d t\left\{\left[\dot{q}_{i}(t)-\frac{\partial H}{\partial p_{i}}\right] \delta p_{i}-\left[\dot{p}_{i}(t)+\frac{\partial H}{\partial q_{i}}\right] \delta q_{i}\right\} \\ +\left.p_{i}(t) \delta q_{i}(t)\right|_{t_{a}} ^{t_{b}} \end{gather*} $$

$$ \begin{align*} \dot{p}_{i} & =-\frac{\partial H}{\partial q_{i}} \\ \dot{q}_{i} & =\frac{\partial H}{\partial p_{i}} \end{align*} $$

$$ \frac{d}{d t} O\left(p_{i}(t), q_{i}(t), t\right)=\frac{\partial O}{\partial p_{i}} \dot{p}_{i}+\frac{\partial O}{\partial q_{i}} \dot{q}_{i}+\frac{\partial O}{\partial t} $$

$$ \begin{align*} \frac{d O}{d t} & =\frac{\partial H}{\partial p_{i}} \frac{\partial O}{\partial q_{i}}-\frac{\partial O}{\partial p_{i}} \frac{\partial H}{\partial q_{i}}+\frac{\partial O}{\partial t} \\ & \equiv\{H, O\}+\frac{\partial O}{\partial t} \end{align*} $$

$$ \{A, B\} \equiv \frac{\partial A}{\partial p_{i}} \frac{\partial B}{\partial q_{i}}-\frac{\partial B}{\partial p_{i}} \frac{\partial A}{\partial q_{i}} $$

$$ \begin{align*} \{A, B\}=-\{B, A\} & \text { antisymmetry } \\ \{A,\{B, C\}\}+\{B,\{C, A\}\}+\{C,\{A, B\}\}=0 & \text { Jacobi identity. } \end{align*} $$

$$ \begin{align*} \frac{d}{d t} p_{i} & =\left\{H, p_{i}\right\}=\frac{\partial H}{\partial p_{j}} \frac{\partial p_{i}}{\partial q_{j}}-\frac{\partial p_{i}}{\partial p_{j}} \frac{\partial H}{\partial q_{j}}=-\frac{\partial H}{\partial q_{i}} \\ \frac{d}{d t} q_{i} & =\left\{H, q_{i}\right\}=\frac{\partial H}{\partial p_{j}} \frac{\partial q_{i}}{\partial q_{j}}-\frac{\partial q_{i}}{\partial p_{j}} \frac{\partial H}{\partial q_{j}}=\frac{\partial H}{\partial p_{i}} \end{align*} $$

$$ \begin{align*} & \left\{p_{i}, q_{j}\right\}=\delta_{i j} \\ & \left\{p_{i}, p_{j}\right\}=0 \\ & \left\{q_{i}, q_{j}\right\}=0 \end{align*} $$

$$ H=H\left(p_{i}, q_{i}\right) $$

$$ q_{i}=f_{i}\left(Q_{j}, t\right) $$

$$ Q_{i}=f_{i}^{-1}\left(q_{j}, t\right) . $$

$$ \operatorname{det}\left(\frac{\partial f_{i}}{\partial Q_{j}}\right) \neq 0 $$

$$ L^{\prime}\left(Q_{j}, \dot{Q}_{j}, t\right) \equiv L\left(f_{i}\left(Q_{j}, t\right), \dot{f}_{i}\left(Q_{j}, t\right), t\right) $$

$$ \begin{align*} \mathcal{A} & =\int_{t_{a}}^{t_{b}} d t L^{\prime}\left(Q_{j}(t), \dot{Q}_{j}(t), t\right) \\ & =\int_{t_{a}}^{t_{b}} d t L\left(f_{i}\left(Q_{j}(t), t\right), \dot{f}_{i}\left(Q_{j}(t), t\right), t\right) \end{align*} $$

$$ \frac{d}{d t} \frac{\partial L^{\prime}}{\partial \dot{Q}_{j}}-\frac{\partial L^{\prime}}{\partial Q_{j}}=0 $$

$$ \begin{align*} \delta \mathcal{A} & =\int_{t_{a}}^{t_{b}} d t\left(\frac{\partial L}{\partial q_{i}} \delta f_{i}+\frac{\partial L}{\partial \dot{q}_{i}} \delta \dot{f}_{i}\right) \\ & =\int_{t_{a}}^{t_{b}} d t\left(\frac{\partial L}{\partial q_{i}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{q}_{i}}\right) \delta f_{i}+\left.\frac{\partial L}{\partial \dot{q}_{i}} \delta f_{i}\right|_{t_{a}} ^{t_{b}} \end{align*} $$

$$ \dot{q}_{i}=\dot{f}_{i}\left(Q_{j}, t\right)=\frac{\partial f_{i}}{\partial Q_{j}} \dot{Q}_{j}+\frac{\partial f_{i}}{\partial t} $$

$$ \begin{align*} p_{i} & =p_{i}\left(P_{j}, Q_{j}, t\right) \\ q_{i} & =q_{i}\left(P_{j}, Q_{j}, t\right) \end{align*} $$

$$ \begin{align*} P_{j} & =P_{j}\left(p_{i}, q_{i}, t\right) \\ Q_{j} & =Q_{j}\left(p_{i}, q_{i}, t\right) \end{align*} $$

$$ \begin{align*} \int_{t_{a}}^{t_{b}} d t\left[p_{i} \dot{q}_{i}-H\left(p_{i}, q_{i}, t\right)\right]=\int_{t_{a}}^{t_{b}} d t[ & \left.P_{j} \dot{Q}_{j}-H^{\prime}\left(P_{j}, Q_{j}, t\right)\right] \\ & +\left.F\left(P_{j}, Q_{j}, t\right)\right|_{t_{a}} ^{t_{b}} \end{align*} $$

$$ \begin{align*} \dot{P}_{i} & =-\frac{\partial H^{\prime}}{\partial Q_{i}} \\ \dot{Q}_{i} & =\frac{\partial H^{\prime}}{\partial P_{i}} \end{align*} $$

$$ \int_{t_{a}}^{t_{b}} d t\left\{p_{i}\left(\frac{\partial q_{i}}{\partial P_{j}} \dot{P}_{j}+\frac{\partial q_{i}}{\partial Q_{j}} \dot{Q}_{j}+\frac{\partial q_{i}}{\partial t}\right)-H\left(p_{i}\left(P_{j}, Q_{j}, t\right), q_{i}\left(P_{j}, Q_{j}, t\right), t\right)\right\} $$

$$ \begin{align*} \int_{t_{a}}^{t_{b}}\left\{\left(P_{j}\right.\right. & \left.-p_{i} \frac{\partial q_{i}}{\partial Q_{j}}\right) d Q_{j}-p_{i} \frac{\partial q_{i}}{\partial P_{j}} d P_{j} \\ & \left.-\left(H^{\prime}+p_{i} \frac{\partial q_{i}}{\partial t}-H\right) d t\right\}=-\left.F\left(P_{j}, Q_{j}, t\right)\right|_{t_{a}} ^{t_{b}} \end{align*} $$

$$ \begin{align*} \frac{\partial p_{i}}{\partial P_{k}} \frac{\partial q_{i}}{\partial Q_{l}}-\frac{\partial q_{i}}{\partial P_{k}} \frac{\partial p_{i}}{\partial Q_{l}} & =\delta_{k l} \\ \frac{\partial p_{i}}{\partial P_{k}} \frac{\partial q_{i}}{\partial P_{l}}-\frac{\partial q_{i}}{\partial P_{k}} \frac{\partial p_{i}}{\partial P_{l}} & =0 \\ \frac{\partial p_{i}}{\partial Q_{k}} \frac{\partial q_{i}}{\partial Q_{l}}-\frac{\partial q_{i}}{\partial Q_{k}} \frac{\partial p_{i}}{\partial Q_{l}} & =0 \end{align*} $$

$$ \begin{align*} \frac{\partial p_{i}}{\partial t} \frac{\partial q_{i}}{\partial P_{l}}-\frac{\partial q_{i}}{\partial t} \frac{\partial p_{i}}{\partial P_{l}} & =\frac{\partial\left(H^{\prime}-H\right)}{\partial P_{l}} \\ \frac{\partial p_{i}}{\partial t} \frac{\partial q_{i}}{\partial Q_{l}}-\frac{\partial q_{i}}{\partial t} \frac{\partial p_{i}}{\partial Q_{l}} & =\frac{\partial\left(H^{\prime}-H\right)}{\partial Q_{l}} \end{align*} $$

$$ \begin{align*} \left(P_{k}, Q_{l}\right) & =\delta_{k l} \\ \left(P_{k}, P_{l}\right) & =0 \\ \left(Q_{k}, Q_{l}\right) & =0 \end{align*} $$

$$ J=\left(\begin{array}{ll} \partial P_{i} / \partial p_{j} & \partial P_{i} / \partial q_{j} \\ \partial Q_{i} / \partial p_{j} & \partial Q_{i} / \partial q_{j} \end{array}\right) $$

$$ J^{-1}=\left(\begin{array}{cc} \partial p_{i} / \partial P_{j} & \partial p_{i} / \partial Q_{j} \\ \partial q_{i} / \partial P_{j} & \partial q_{i} / \partial Q_{j} \end{array}\right) $$

$$ E=\left(\begin{array}{rl} 0 & \delta_{i j} \\ -\delta_{i j} & 0 \end{array}\right) $$

$$ \begin{align*} \left\{P_{k}, Q_{l}\right\} & =\delta_{k l} \\ \left\{P_{k}, P_{l}\right\} & =0 \\ \left\{Q_{k}, Q_{l}\right\} & =0 \end{align*} $$

$$ \mathcal{L} \equiv\left(\begin{array}{rc} -\left(Q_{i}, P_{j}\right) & -\left(Q_{i}, Q_{j}\right) \\ \left(P_{i}, P_{j}\right) & \left(P_{i}, Q_{j}\right) \end{array}\right) $$

$$ \mathcal{P} \equiv\left(\begin{array}{cc} \left\{P_{i}, Q_{j}\right\} & -\left\{P_{i}, P_{j}\right\} \\ \left\{Q_{i}, Q_{j}\right\} & -\left\{Q_{i}, P_{j}\right\} \end{array}\right) $$

$$ p_{i} \dot{q}_{i}-P_{j} \dot{Q}_{j}=\frac{d}{d t} G\left(P_{j}, Q_{j}, t\right) $$

$$ \prod_{i} \int\left[d p_{i} d q_{i}\right]=\prod_{j} \int\left[d P_{j} d Q_{j}\right] $$

$$ \{A, B\}^{\prime} \equiv \frac{\partial A}{\partial P_{j}} \frac{\partial B}{\partial Q_{j}}-\frac{\partial B}{\partial P_{j}} \frac{\partial A}{\partial Q_{j}} $$

$$ \frac{d O}{d t}=\left\{H^{\prime}, O\right\}^{\prime}+\frac{\partial O}{\partial t} $$

$$ \begin{align*} & \left\{P_{i}, Q_{j}\right\}^{\prime}=\delta_{i j} \\ & \left\{P_{i}, P_{j}\right\}^{\prime}=0 \\ & \left\{Q_{i}, Q_{j}\right\}^{\prime}=0 \end{align*} $$

$$ F=F\left(q_{i}, Q_{j}, t\right) $$

$$ \int_{t_{a}}^{t_{b}} d t\left[p_{i} \dot{q}_{i}-H\left(p_{i}, q_{i}, t\right)\right]=\int_{t_{a}}^{t_{b}} d t\left[P_{j} \dot{Q}_{j}-H^{\prime}\left(P_{j}, Q_{j}, t\right)+\frac{d}{d t} F\left(q_{i}, Q_{j}, t\right)\right] $$

$$ \begin{align*} & \int_{t_{a}}^{t_{b}} d t\left\{p_{i} \dot{q}_{i}+\dot{P}_{j} Q_{j}-\left[H\left(p_{i}, q_{i}, t\right)-H^{\prime}\left(P_{j}, Q_{j}, t\right)\right]\right\} \\ & \quad=\int_{t_{a}}^{t_{b}} d t\left\{\frac{\partial F}{\partial q_{i}}\left(q_{i}, P_{j}, t\right) \dot{q}_{i}+\frac{\partial F}{\partial P_{j}}\left(q_{i}, P_{j}, t\right) \dot{P}_{j}+\frac{\partial F}{\partial t}\left(q_{i}, P_{j}, t\right)\right\} \end{align*} $$

$$ \begin{align*} p_{i} & =\frac{\partial}{\partial q_{i}} F\left(q_{i}, P_{j}, t\right) \\ Q_{j} & =\frac{\partial}{\partial P_{j}} F\left(q_{i}, P_{j}, t\right) \end{align*} $$

$$ H^{\prime}\left(P_{j}, Q_{j}, t\right)=H\left(p_{i}, q_{i}, t\right)+\frac{\partial}{\partial t} F\left(q_{i}, P_{j}, t\right) $$

$$ \frac{\partial}{\partial t} F\left(q_{i}, P_{j}, t\right)=-H\left(p_{i}, q_{i}, t\right) $$

$$ \partial_{t} F\left(q_{i}, P_{j}, t\right)=-H\left(\partial_{q_{i}} F\left(q_{i}, P_{j}, t\right), q_{i}, t\right) $$

$$ \delta \mathcal{A}\left[p_{i}, q_{i}\right]=p_{i}\left(t_{b}\right) \delta q_{i}\left(t_{b}\right)-p_{i}\left(t_{a}\right) \delta q_{i}\left(t_{a}\right) $$

$$ p_{i}=\frac{\partial}{\partial q_{i}} A\left(q_{i}, t\right) $$

$$ \frac{d}{d t} A\left(q_{i}(t), t\right)=p_{i}(t) \dot{q}_{i}(t)-H\left(p_{i}(t), q_{i}(t), t\right) $$

$$ \partial_{t} A\left(q_{i}, t\right)=-H\left(p_{i}, q_{i}, t\right) $$

$$ \partial_{t} A\left(q_{i}, t\right)=-H\left(\partial_{q_{i}} A\left(q_{i}, t\right), q_{i}, t\right) $$

$$ \mathcal{A}=-M c^{2} \int d \tau L(q, \dot{q})=-M c^{2} \int d \tau \sqrt{g_{\mu \nu} \dot{q}^{\mu}(\tau) \dot{q}^{\nu}(\tau)} $$

$$ \frac{d}{d t}\left[\frac{1}{L(q, \dot{q})} g_{\mu \nu} \dot{q}^{\nu}\right]=\frac{1}{2 L(q, \dot{q})}\left(\partial_{\mu} g_{\kappa \lambda}\right) \dot{q}^{\kappa} \dot{q}^{\lambda} $$

$$ \frac{d}{d t}\left(g_{\mu \nu} \dot{q}^{\nu}\right)=\frac{1}{2}\left(\partial_{\mu} g_{\kappa \lambda}\right) \dot{q}^{\kappa} \dot{q}^{\lambda} $$

$$ g_{\mu \nu} \ddot{q}^{\nu}=\left(\frac{1}{2} \partial_{\mu} g_{\kappa \lambda}-\partial_{\lambda} g_{\mu \kappa}\right) \dot{q}^{\kappa} \dot{q}^{\lambda} $$

$$ \bar{\Gamma}_{\lambda \nu \mu} \equiv \frac{1}{2}\left(\partial_{\lambda} g_{\nu \mu}+\partial_{\nu} g_{\lambda \mu}-\partial_{\mu} g_{\lambda \nu}\right) $$

$$ \bar{\Gamma}_{\kappa \nu}^{\mu} \equiv g^{\mu \sigma} \bar{\Gamma}_{\kappa \nu \sigma} . $$

$$ \ddot{q}^{\mu}+\bar{\Gamma}_{\kappa \lambda}^{\mu} \dot{q}^{\kappa} \dot{q}^{\lambda}=0 . $$

$$ \mathbf{p}=\left(p^{1}, p^{2}, \ldots, p^{D}\right) $$

$$ \mathbf{x}=\left(x^{1}, x^{2}, \ldots, x^{D}\right) $$

$$ \Psi_{\mathbf{p}}(\mathbf{x}, t)=e^{i \mathbf{k x}-i \omega t} $$

$$ \frac{d N}{d t} \propto\left|\Psi_{1}+\Psi_{2}\right|^{2} \approx\left|e^{i k\left(R+\frac{1}{2} d \sin \varphi\right)}+e^{i k\left(R-\frac{1}{2} d \sin \varphi\right)}\right|^{2} \frac{1}{R^{2}} $$

$$ \mathbf{p}=\hbar \mathbf{k}, $$

$$ \hbar \equiv \frac{h}{2 \pi}=1.0545919(80) \times 10^{-27} \mathrm{erg} \mathrm{sec} $$

$$ E=\hbar \omega $$

$$ \Psi_{\mathbf{p}}(\mathbf{x}, t)=\mathcal{N} e^{i\left(\mathbf{p} \mathbf{x}-E_{\mathbf{p}} t\right) / \hbar} $$

$$ \mathbf{j}(\mathbf{x}, t) \equiv-i \frac{\hbar}{2 m} \psi^{*}(\mathbf{x}, t) \stackrel{\leftrightarrow}{\nabla} \psi(\mathbf{x}, t) $$

$$ \begin{align*} \psi^{*}(\mathbf{x}, t) \overleftrightarrow{\nabla} \psi(\mathbf{x}, t) & \equiv \psi^{*}(\mathbf{x}, t) \vec{\nabla} \psi(\mathbf{x}, t)-\psi^{*}(\mathbf{x}, t) \overleftarrow{\nabla} \psi(\mathbf{x}, t) \\ & \equiv \psi^{*}(\mathbf{x}, t) \nabla \psi(\mathbf{x}, t)-\left[\nabla \psi^{*}(\mathbf{x}, t)\right] \psi(\mathbf{x}, t) \end{align*} $$

$$ \Psi(\mathbf{x}, t)=\int \frac{d^{3} p}{(2 \pi \hbar)^{3}} f(\mathbf{p}) e^{i\left(\mathbf{p} \mathbf{x}-E_{\mathbf{p}} t\right) / \hbar} $$

$$ f(\mathbf{p})=\int d^{3} x e^{-i \mathbf{p} \mathbf{x} / \hbar} \Psi(\mathbf{x}, 0) $$

$$ \Delta \mathrm{x} \Delta \mathrm{p} \sim \hbar $$

$$ \overline{\mathbf{v}}=\partial E_{\overline{\mathbf{p}}} / \partial \overline{\mathbf{p}} $$

$$ H(\mathbf{p})=E_{\mathbf{p}}=\frac{\mathbf{p}^{2}}{2 M} $$

$$ \int \frac{d^{3} p}{(2 \pi \hbar)^{3}} f(\mathbf{p})\left[H(\mathbf{p})-E_{\mathbf{p}}\right] e^{i\left(\mathbf{p} \mathbf{x}-E_{\mathbf{p}} t\right) / \hbar}=0 $$

$$ \begin{align*} & \hat{\mathbf{p}}=-i \hbar \nabla, \\ & \hat{E}=i \hbar \partial_{t} \end{align*} $$

$$ \left.\left[H(-i \hbar \nabla)-i \hbar \partial_{t}\right)\right] \Psi(\mathbf{x}, t)=0 . $$

$$ \left(\hat{H}-i \hbar \partial_{t}\right) \Psi(\mathbf{x}, t)=0 $$

$$ \hat{H} \equiv H(-i \hbar \nabla, \mathbf{x}, t) . $$

$$ \left[\hat{p}_{i}, x_{j}\right]=-i \hbar, \quad[\hat{E}, t]=0=i \hbar $$

$$ \hat{H}(\hat{\mathbf{p}}, \mathbf{x}) \Psi_{E_{n}}(\mathbf{x})=E_{n} \Psi_{E_{n}}(\mathbf{x}) $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{\mathbf{p}^{2}}{2 M}-\frac{e^{2}}{r}, $$

$$ \int d^{3} x|\Psi(\mathbf{x}, t)|^{2}=1 $$

$$ \int d^{3} x\left[\hat{H} \Psi_{2}(\mathbf{x}, t)\right]^{*} \Psi_{1}(\mathbf{x}, t)=\int d^{3} x \Psi_{2}^{*}(\mathbf{x}, t) \hat{H} \Psi_{1}(\mathbf{x}, t) $$

$$ \int d^{3} x \Psi_{2}^{*}(\mathbf{x}, t) \hat{H}^{\dagger} \Psi_{1}(\mathbf{x}, t) \equiv \int d^{3} x\left[\hat{H} \Psi_{2}(\mathbf{x}, t)\right]^{*} \Psi_{1}(\mathbf{x}, t) $$

$$ \hat{H}=\hat{H}^{\dagger} $$

$$ \begin{align*} & i \hbar \frac{d}{d t} \int d^{3} x \Psi_{2}^{*}(\mathbf{x}, t) \Psi_{1}(\mathbf{x}, t) \\ & \quad=\int d^{3} x \Psi_{2}^{*}(\mathbf{x}, t) \hat{H} \Psi_{1}(\mathbf{x}, t)-\int d^{3} x\left[\hat{H} \Psi_{2}(\mathbf{x}, t)\right]^{*} \Psi_{1}(\mathbf{x}, t)=0 \end{align*} $$

$$ H(\mathbf{p}, \mathbf{x}, t)=T(\mathbf{p}, t)+V(\mathbf{x}, t) $$

$$ \mathbf{j}(\mathbf{x}, t) \equiv-i \frac{\hbar}{2 m} \psi(\mathbf{x}, t) \overleftrightarrow{\nabla} \psi(\mathbf{x}, t) $$

$$ \rho(\mathbf{x}, t)=\psi^{*}(\mathbf{x}, t) \psi(\mathbf{x}, t) $$

$$ \partial_{t} \rho(\mathbf{x}, t)=-\boldsymbol{\nabla} \cdot \mathbf{j}(\mathbf{x}, t) $$

$$ \int_{V} d^{3} x \partial_{t} \rho(\mathbf{x}, t)=-\int_{V} d^{3} x \boldsymbol{\nabla} \cdot \mathbf{j}(\mathbf{x}, t)=-\int_{S} d \mathbf{S} \cdot \mathbf{j}(\mathbf{x}, t) $$

$$ H\left(\mathbf{p}_{\nu}, \mathbf{x}_{\nu}, t\right)=\sum_{\nu=1}^{N} \frac{\mathbf{p}_{\nu}^{2}}{2 M_{\nu}}+V\left(\mathbf{x}_{\nu}, t\right) $$

$$ \left\{-\sum_{\nu=1}^{N}\left[\frac{\hbar^{2}}{2 M_{\nu}} \partial_{\mathbf{x}_{\nu}}^{2}+V\left(\mathbf{x}_{\nu}, t\right)\right]\right\} \Psi\left(\mathbf{x}_{\nu}, t\right)=i \hbar \partial_{t} \Psi\left(\mathbf{x}_{\nu}, t\right) $$

$$ \Psi(\mathbf{x}, t) \equiv \Psi_{\mathbf{x}}(t) $$

$$ |\mathbf{v}|^{2}=\sum_{i} v_{i}^{*} v_{i} $$

$$ |\Psi|^{2}=\int d^{3} x \Psi_{\mathbf{x}}^{*}(t) \Psi_{\mathbf{x}}(t)=\int d^{3} x \Psi^{*}(\mathbf{x}, t) \Psi(\mathbf{x}, t) $$

$$ v_{i}=\sum_{a} b_{i}^{a} v_{a} $$

$$ v_{a} \equiv \sum_{i} b_{i}^{a *} v_{i} $$

$$ \sum_{i} b_{i}^{a *} b_{i}^{a^{\prime}}=\delta^{a a^{\prime}} $$

$$ \sum_{a} b_{i}^{a *} b_{j}^{a}=\delta^{i j} $$

$$ \mathbf{x}_{\mathbf{n}}=\left(n_{1}, n_{2}, n_{3}\right) \epsilon, \quad n_{1,2,3}=0, \pm 1, \pm 2, \ldots $$

$$ h^{\mathbf{n}}(\mathbf{x})=\left\{\begin{array}{lc} 1 / \sqrt{\epsilon^{3}} & \left|x_{i}-x_{\mathbf{n} i}\right| \leq \epsilon / 2, \quad i=1,2,3 \\ 0 & \text { otherwise. } \end{array}\right. $$

$$ \int d^{3} x h^{\mathbf{n}}(\mathbf{x})^{*} h^{\mathbf{n}^{\prime}}(\mathbf{x})=\delta^{\mathbf{n n}} $$

$$ \Psi(\mathbf{x}, t)=\sum_{\mathbf{n}} h^{\mathbf{n}}(\mathbf{x}) \Psi_{\mathbf{n}}(t) $$

$$ \Psi_{\mathbf{n}}(t)=\int d^{3} x h^{\mathbf{n}}(\mathbf{x})^{*} \Psi(\mathbf{x}, t) \approx \sqrt{\epsilon^{3}} \Psi\left(\mathbf{x}_{\mathbf{n}}, t\right) $$

$$ \int d^{3} x f^{a}(\mathbf{x})^{*} f^{a^{\prime}}(\mathbf{x})=\delta^{a a^{\prime}} $$

$$ \Psi(\mathbf{x}, t)=\sum_{a} f^{a}(\mathbf{x}) \Psi_{a}(t) $$

$$ \Psi_{a}(t)=\int d^{3} x f^{a}(\mathbf{x})^{*} \Psi(\mathbf{x}, t) $$

$$ \int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} \tilde{f}^{b^{\prime}}(\mathbf{x})=\delta^{b b^{\prime}}, \quad \sum_{b} \tilde{f}^{b}(\mathbf{x}) \tilde{f}^{b}\left(\mathbf{x}^{\prime}\right)^{*}=\delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \Psi(\mathbf{x}, t)=\sum_{b} \tilde{f}^{b}(\mathbf{x}) \tilde{\Psi}_{b}(t) $$

$$ \tilde{\Psi}_{b}(t)=\int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} \Psi(\mathbf{x}, t) $$

$$ \tilde{\Psi}_{b}(t)=\sum_{a}\left[\int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} f^{a}(\mathbf{x})\right] \Psi_{a}(t) $$

$$ \langle\tilde{b} \mid a\rangle \equiv \int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} f^{a}(\mathbf{x}) $$

$$ \begin{align*} \Psi_{a}(t) & =\langle a \mid \Psi(t)\rangle, \\ \tilde{\Psi}_{b}(t) & =\langle\tilde{b} \mid \Psi(t)\rangle . \end{align*} $$

$$ \langle\tilde{b} \mid \Psi(t)\rangle=\sum_{a}\langle\tilde{b} \mid a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \sum_{a}|a\rangle\langle a|=1 $$

$$ \langle\tilde{b} \mid \Psi(t)\rangle=\langle\tilde{b}| 1|\Psi(t)\rangle=\sum_{a}\langle\tilde{b} \mid a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \begin{align*} & \left\langle a \mid a^{\prime}\right\rangle=\int d^{3} x f^{a}(\mathbf{x})^{*} f^{a^{\prime}}(\mathbf{x})=\delta^{a a^{\prime}} \\ & \left\langle\tilde{b} \mid \tilde{b}^{\prime}\right\rangle=\int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} \tilde{f}^{b^{\prime}}(\mathbf{x})=\delta^{b b^{\prime}} \end{align*} $$

$$ \left\langle\mathbf{x}_{\mathbf{n}} \mid \mathbf{x}_{\mathbf{n}^{\prime}}\right\rangle \equiv \int d^{3} x h^{\mathbf{n}}(\mathbf{x})^{*} h^{\mathbf{n}^{\prime}}(\mathbf{x})=\delta_{\mathbf{n n}^{\prime}} $$

$$ \Psi_{\mathbf{n}}(t) \equiv\left\langle\mathbf{x}_{\mathbf{n}} \mid \Psi(t)\right\rangle \approx \sqrt{\epsilon^{3}} \Psi\left(\mathbf{x}_{\mathbf{n}}, t\right) $$

$$ \Psi_{\mathbf{n}}(t)=\left\langle\mathbf{x}_{\mathbf{n}} \mid \Psi(t)\right\rangle=\sum_{a}\left\langle\mathbf{x}_{\mathbf{n}} \mid a\right\rangle\langle a \mid \Psi(t)\rangle . $$

$$ \langle a \mid \Psi(t)\rangle=\sum_{\mathbf{n}}\left\langle a \mid \mathbf{x}_{\mathbf{n}}\right\rangle\left\langle\mathbf{x}_{\mathbf{n}} \mid \Psi(t)\right\rangle . $$

$$ \int d^{3} x h^{\mathbf{n}}(\mathbf{x})^{*}\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \sum_{n}\left|x_{n}\right\rangle\left\langle x_{n}\right| \approx 1 . $$

$$ \langle\mathbf{x} \mid \Psi(t)\rangle \approx \frac{1}{\sqrt{\epsilon^{3}}}\left\langle\mathbf{x}_{\mathbf{n}} \mid \Psi(t)\right\rangle $$

$$ \langle\mathbf{x} \mid \Psi(t)\rangle \equiv \Psi(\mathbf{x}, t) $$

$$ \begin{align*} \langle a \mid \Psi(t)\rangle & \approx \sum_{\mathbf{n}}\left\langle a \mid \mathbf{x}_{\mathbf{n}}\right\rangle\left\langle\mathbf{x}_{\mathbf{n}} \mid \Psi(t)\right\rangle \\ & \left.\approx \sum_{\mathbf{n}} \epsilon^{3}\langle a \mid \mathbf{x}\rangle\langle\mathbf{x} \mid \Psi(t)\rangle\right|_{\mathbf{x}=\mathbf{x}^{\mathbf{n}}} \end{align*} $$

$$ \langle a \mid \Psi(t)\rangle=\int d^{3} x\langle a \mid \mathbf{x}\rangle\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \int d^{3} x|\mathbf{x}\rangle\langle\mathbf{x}|=1 $$

$$ \langle a \mid \Psi(t)\rangle=\int d^{3} x\langle a \mid \mathbf{x}\rangle\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \sum_{a}|a\rangle\langle a|=1 $$

$$ |\Psi(t)\rangle=\sum_{a}|a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \langle b \mid \Psi(t)\rangle=\sum_{a}\langle b \mid a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \int d^{3} x|\mathbf{x}\rangle\langle\mathbf{x}|=1 $$

$$ |\Psi(t)\rangle=\int d^{3} x|\mathbf{x}\rangle\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ |\mathbf{x}\rangle \approx \frac{1}{\sqrt{\epsilon^{3}}}\left|\mathbf{x}_{\mathbf{n}}\right\rangle $$

$$ \begin{align*} & \langle a \mid \tilde{b}\rangle=\int d^{3} x f^{a}(\mathbf{x})^{*} \tilde{f}^{b}(\mathbf{x}) \\ & \langle\tilde{b} \mid a\rangle=\int d^{3} x \tilde{f}^{b}(\mathbf{x})^{*} f^{a}(\mathbf{x}) \end{align*} $$

$$ \langle\tilde{b} \mid a\rangle \equiv\langle a \mid \tilde{b}\rangle^{*} $$

$$ |\Psi(t)\rangle=\sum_{a}|a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \langle\Psi(t)|=\sum_{a}\langle\Psi(t) \mid a\rangle\langle a|, $$

$$ \left\langle\mathbf{x} \mid \mathbf{x}^{\prime}\right\rangle \approx \frac{1}{\epsilon^{3}}\left\langle\mathbf{x}_{\mathbf{n}} \mid \mathbf{x}_{\mathbf{n}^{\prime}}\right\rangle=\frac{1}{\epsilon^{3}} \delta_{\mathbf{n n}^{\prime}} $$

$$ \epsilon^{3} \sum_{\mathbf{n}^{\prime}} \frac{1}{\epsilon^{3}} \delta_{\mathbf{n n}^{\prime}}=1 $$

$$ \left\langle\mathbf{x} \mid \mathbf{x}^{\prime}\right\rangle \equiv \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=0 \quad \text { for } \quad \mathbf{x} \neq \mathbf{x}^{\prime} $$

$$ \int d^{3} x^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=1 $$

$$ \delta^{(3)}\left(a\left(\mathbf{x}-\mathbf{x}^{\prime}\right)\right)=\frac{1}{|a|} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \delta(f(x))=\sum_{i} \frac{1}{\left|f^{\prime}\left(x_{i}\right)\right|} \delta\left(x-x_{i}\right) $$

$$ \delta[f ; \mathbf{x}] \equiv \int d^{3} x \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) f\left(\mathbf{x}^{\prime}\right)=f(\mathbf{x}) $$

$$ \hat{H}|\Psi(t)\rangle \equiv H(\hat{\mathbf{p}}, \hat{\mathbf{x}}, t)|\Psi(t)\rangle=i \hbar \partial_{t}|\Psi(t)\rangle $$

$$ \begin{align*} \langle\mathbf{x}| \hat{\mathbf{p}} & \equiv-i \hbar \nabla\langle\mathbf{x}| \\ \langle\mathbf{x}| \hat{\mathbf{x}} & \equiv \mathbf{x}\langle\mathbf{x}| \end{align*} $$

$$ \begin{align*} \langle\mathbf{x}| \hat{\mathbf{p}}\left|\mathbf{x}^{\prime}\right\rangle & =-i \hbar \nabla\left\langle\mathbf{x} \mid \mathbf{x}^{\prime}\right\rangle=-i \hbar \nabla \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \\ \langle\mathbf{x}| \hat{\mathbf{x}}\left|\mathbf{x}^{\prime}\right\rangle & =\mathbf{x}\left\langle\mathbf{x} \mid \mathbf{x}^{\prime}\right\rangle=\mathbf{x} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \end{align*} $$

$$ \begin{align*} \langle\mathbf{x}| H(\hat{\mathbf{p}}, \hat{\mathbf{x}}, t)|\Psi(t)\rangle & =H(-i \hbar \nabla, \mathbf{x}, t)\langle\mathbf{x} \mid \Psi(t)\rangle \\ & =i \hbar \partial_{t}\langle\mathbf{x} \mid \Psi(t)\rangle \end{align*} $$

$$ \begin{align*} \langle a| \hat{\mathbf{p}}\left|a^{\prime}\right\rangle & =\left\langle a^{\prime}\right| \hat{\mathbf{p}}|a\rangle^{*} \\ \langle a| \hat{\mathbf{x}}\left|a^{\prime}\right\rangle & =\left\langle a^{\prime}\right| \hat{\mathbf{x}}|a\rangle^{*} \end{align*} $$

$$ \langle a| \hat{H}\left|a^{\prime}\right\rangle=\left\langle a^{\prime}\right| \hat{H}|a\rangle^{*} $$

$$ \hat{O}(t) \equiv O(\hat{\mathbf{p}}, \hat{\mathbf{x}}, t) $$

$$ \langle a| \hat{O}^{\dagger}(t)\left|a^{\prime}\right\rangle \equiv\left\langle a^{\prime}\right| \hat{O}(t)|a\rangle^{*} $$

$$ \begin{align*} & \hat{\mathbf{p}}=\hat{\mathbf{p}}^{\dagger} \\ & \hat{\mathbf{x}}=\hat{\mathbf{x}}^{\dagger} \\ & \hat{H}=\hat{H}^{\dagger} \end{align*} $$

$$ \hat{H}\left|E_{n}\right\rangle=E_{n}\left|E_{n}\right\rangle $$

$$ \hat{\mathbf{p}}|\mathbf{p}\rangle=\mathbf{p}|\mathbf{p}\rangle $$

$$ \langle\mathbf{x}| \hat{\mathbf{p}}|\mathbf{p}\rangle=-i \hbar \partial_{\mathbf{x}}\langle\mathbf{x} \mid \mathbf{p}\rangle=\mathbf{p}\langle\mathbf{x} \mid \mathbf{p}\rangle $$

$$ \langle\mathbf{x} \mid \mathbf{p}\rangle \propto e^{i \mathbf{p x} / \hbar} $$

$$ \mathbf{p}^{\mathbf{m}}=\frac{2 \pi \hbar}{L}\left(m_{1}, m_{2}, m_{3}\right), \quad m_{i}=0, \pm 1, \pm 2, \ldots $$

$$ \left\langle\mathbf{x} \mid \mathbf{p}^{\mathbf{m}}\right\rangle=\frac{1}{\sqrt{L^{3}}} \exp \left(i \mathbf{p}^{\mathbf{m}} \mathbf{x} / \hbar\right) $$

$$ \int d^{3} x\left|\left\langle\mathbf{x} \mid \mathbf{p}^{\mathbf{m}}\right\rangle\right|^{2}=1 $$

$$ \sum_{\mathbf{m}}\left|\mathbf{p}^{\mathbf{m}}\right\rangle\left\langle\mathbf{p}^{\mathbf{m}}\right|=1 $$

$$ \Psi(\mathbf{x}, t)=\langle\mathbf{x} \mid \Psi(t)\rangle=\sum_{\mathbf{m}}\left\langle\mathbf{x} \mid \mathbf{p}^{\mathbf{m}}\right\rangle\left\langle\mathbf{p}^{\mathbf{m}} \mid \Psi(t)\right\rangle $$

$$ \sum_{\mathbf{m}} \approx \int \frac{d^{3} p L^{3}}{(2 \pi \hbar)^{3}} $$

$$ |\mathbf{p}\rangle \approx \sqrt{L^{3}}\left|\mathbf{p}^{\mathbf{m}}\right\rangle $$

$$ \left\langle\mathbf{p} \mid \mathbf{p}^{\prime}\right\rangle=(2 \pi \hbar)^{3} \delta^{(3)}\left(\mathbf{p}-\mathbf{p}^{\prime}\right) $$

$$ \int \frac{d^{3} p}{(2 \pi \hbar)^{3}}|\mathbf{p}\rangle\langle\mathbf{p}|=1 $$

$$ \Psi(\mathbf{x}, t)=\int \frac{d^{3} p}{(2 \pi \hbar)^{3}}\langle\mathbf{x} \mid \mathbf{p}\rangle\langle\mathbf{p} \mid \Psi(t)\rangle $$

$$ \langle\mathbf{x} \mid \mathbf{p}\rangle=e^{i \mathbf{p x} / \hbar} $$

$$ \langle\mathbf{p} \mid \Psi(t)\rangle=f(\mathbf{p}) e^{-i E \mathbf{p} t / \hbar} $$

$$ \begin{align*} \langle\mathbf{p} \mid \Psi(t)\rangle & =\int d^{3} x\langle\mathbf{p} \mid \mathbf{x}\rangle\langle\mathbf{x} \mid \Psi(t)\rangle \\ & =\int d^{3} x e^{-i \mathbf{p} \mathbf{x} / \hbar} \Psi(\mathbf{x}, t) \end{align*} $$

$$ \int d^{3} x|\mathbf{x}\rangle\langle\mathbf{x}|=1 $$

$$ \begin{align*} \left\langle\mathbf{p} \mid \mathbf{p}^{\prime}\right\rangle & =\int d^{3} x\langle\mathbf{p} \mid \mathbf{x}\rangle\left\langle\mathbf{x} \mid \mathbf{p}^{\prime}\right\rangle \\ & =\int d^{3} x e^{-i\left(\mathbf{p}-\mathbf{p}^{\prime}\right) \mathbf{x} / \hbar}=(2 \pi \hbar)^{3} \delta^{(3)}\left(\mathbf{p}-\mathbf{p}^{\prime}\right) \end{align*} $$

$$ \int d x|x\rangle\langle x|=1 $$

$$ \sum_{n=-N}^{N}\left|x_{n}\right\rangle\left\langle x_{n}\right| $$

$$ \sum_{n=-N}^{N}\left\langle p \mid x_{n}\right\rangle\left\langle x_{n} \mid p^{\prime}\right\rangle=\sum_{n=-N}^{N}\left\langle p \mid x_{n}\right\rangle\left\langle x_{n} \mid p^{\prime}\right\rangle=\sum_{n=-N}^{N} e^{i\left(p-p^{\prime}\right) n a / \hbar} $$

$$ \sum_{n=-\infty}^{\infty} e^{2 \pi i \mu n}=\sum_{m=-\infty}^{\infty} \delta(\mu-m) $$

$$ \sum_{n=-\infty}^{\infty}\left\langle p \mid x_{n}\right\rangle\left\langle x_{n} \mid p^{\prime}\right\rangle=\sum_{m=-\infty}^{\infty} \delta\left(\frac{\left(p-p^{\prime}\right) a}{2 \pi \hbar}-m\right)=\sum_{m=-\infty}^{\infty} \frac{2 \pi \hbar}{a} \delta\left(p-p^{\prime}-\frac{2 \pi \hbar m}{a}\right) . $$

$$ \begin{align*} \sum_{n=-N}^{N} e^{2 \pi i \mu n} & =1+\left(e^{2 \pi i \mu}+e^{2 \cdot 2 \pi i \mu}+\ldots+e^{N \cdot 2 \pi i \mu}+\text { c.c. }\right) \\ & =-1+\left(\frac{1-e^{2 \pi i \mu(N+1)}}{1-e^{2 \pi i \mu}}+\text { c.c. }\right) \\ & =1+\frac{e^{2 \pi i \mu}-e^{2 \pi i \mu(N+1)}}{1-e^{2 \pi i \mu}}+\text { c.c. }=\frac{\sin \pi \mu(2 N+1)}{\sin \pi \mu} \end{align*} $$

$$ \sum_{n=-N}^{N}\left\langle p \mid x_{n}\right\rangle\left\langle x_{n} \mid p^{\prime}\right\rangle=\frac{\sin \left(p-p^{\prime}\right) a(2 N+1) / 2 \hbar}{\sin \left(p-p^{\prime}\right) a / 2 \hbar} . $$

$$ \int_{-n / 2 N}^{n / 2 N} d \mu \frac{\sin \pi \mu(2 N+1)}{\sin \pi \mu}=\int_{-n / 2 N}^{n / 2 N} d \mu \frac{\sin 2 \pi \mu N \cos \pi \mu+\cos 2 \pi \mu N \sin \pi \mu}{\sin \pi \mu} $$

$$ \begin{align*} \int_{-n / 2 N}^{n / 2 N} d \mu & \frac{\sin \pi \mu(2 N+1)}{\sin \pi \mu} \xrightarrow{N \rightarrow \infty} \int_{-n / 2 N}^{n / 2 N} d \mu \frac{\sin 2 \pi \mu N}{\pi \mu}+\int_{-n / 2 N}^{n / 2 N} d \mu \cos 2 \pi \mu N \\ & \xrightarrow{N \rightarrow \infty} \frac{1}{\pi} \int_{-\pi n}^{\pi n} d x \frac{\sin x}{x}+\frac{1}{2 \pi N} \int_{-\pi n}^{\pi n} d x \cos x \xrightarrow{N \rightarrow \infty} 1 \end{align*} $$

$$ \int_{-\infty}^{\infty} d x \frac{\sin x}{x}=\pi $$

$$ \sum_{m=-\infty}^{\infty} f(m) $$

$$ \sum_{m=-\infty}^{\infty} f(m)=\int_{-\infty}^{\infty} d \mu \sum_{n=-\infty}^{\infty} e^{2 \pi i \mu n} f(\mu) $$

$$ \hat{A} \equiv A(\hat{\mathbf{p}}, \hat{\mathbf{x}}) $$

$$ \hat{A}|a\rangle=a|a\rangle $$

$$ \sum_{a}|a\rangle\langle a|=1 $$

$$ |\Psi(t)\rangle=\sum_{a}|a\rangle\langle a \mid \Psi(t)\rangle $$

$$ \langle a \mid \Psi(t)\rangle $$

$$ \Psi(\mathbf{x}, t)=\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \langle\Psi(t)| \hat{A}|\Psi(t)\rangle \equiv \int d^{3} x\langle\Psi(t) \mid \mathbf{x}\rangle A(-i \hbar \nabla, \mathbf{x})\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \Delta \mathrm{x} \Delta \mathrm{p} \sim \hbar $$

$$ \begin{align*} & {\left[\hat{p}_{i}, \hat{x}_{j}\right]=-i \hbar \delta_{i j}} \\ & {\left[\hat{x}_{i}, \hat{x}_{j}\right]=0} \\ & {\left[\hat{p}_{i}, \hat{p}_{j}\right]=0} \end{align*} $$

$$ \hat{A}|a\rangle=a|a\rangle $$

$$ |\Psi(t)\rangle=\sum_{a}|a\rangle\langle a \mid \Psi(t)\rangle, $$

$$ \hat{B}|a\rangle=b_{a}|a\rangle, $$

$$ \hat{B} \hat{A}|a\rangle=b_{a} a|a\rangle=a b_{a}|a\rangle=\hat{A} \hat{B}|a\rangle, $$

$$ [\hat{A}, \hat{B}]=0 $$

$$ \hat{\rho}(t) \equiv|\Psi(t)\rangle\langle\Psi(t)|, $$

$$ \rho\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; t\right)=\left\langle\mathbf{x}_{1} \mid \Psi(t)\right\rangle\left\langle\Psi(t) \mid \mathbf{x}_{2}\right\rangle $$

$$ \langle\Psi(t)| f(\mathbf{x}, \hat{\mathbf{p}})|\Psi(t)\rangle=\operatorname{tr}[f(\mathbf{x}, \hat{\mathbf{p}}) \hat{\rho}(t)]=\int d^{3} x\langle\Psi(t) \mid \mathbf{x}\rangle f(\mathbf{x},-i \hbar \nabla)\langle\mathbf{x} \mid \Psi(t)\rangle $$

$$ \hat{\rho}(t) \equiv \sum_{n, m}\left|E_{n}\right\rangle \rho_{n m}(t)\left\langle E_{m}\right|=\sum_{n, m}\left|E_{n}\right\rangle\left\langle E_{n} \mid \Psi(t)\right\rangle\left\langle\Psi(t) \mid E_{m}\right\rangle\left\langle E_{m}\right| $$

$$ W(\mathbf{X}, \mathbf{p} ; t) \equiv \int \frac{d^{3} \Delta x}{(2 \pi \hbar)^{3}} e^{i \mathbf{p} \Delta \mathbf{x} / \hbar} \rho(\mathbf{X}+\Delta \mathbf{x} / 2, \mathbf{X}-\Delta \mathbf{x} / 2 ; t) $$

$$ \left(\partial_{t}+\mathbf{v} \cdot \boldsymbol{\nabla}_{\mathbf{X}}\right) W(\mathbf{X}, \mathbf{p} ; t)=W_{t}(\mathbf{X}, \mathbf{p} ; t), \quad \mathbf{v} \equiv \frac{\mathbf{p}}{M} $$

$$ W_{t}(\mathbf{X}, \mathbf{p} ; t) \equiv \frac{2}{\hbar} \int \frac{d^{3} q}{(2 \pi \hbar)^{3}} W(\mathbf{X}, \mathbf{p}-\mathbf{q} ; t) \int d^{3} \Delta \mathbf{x} V(\mathbf{X}-\Delta \mathbf{x} / 2) e^{i \mathbf{q} \Delta \mathbf{x} / \hbar} $$

$$ \left(\partial_{t}+\mathbf{v} \cdot \boldsymbol{\nabla}_{\mathbf{X}}\right) W(\mathbf{X}, \mathbf{p} ; t)=-F(\mathbf{X}) \boldsymbol{\nabla}_{\mathbf{p}} W(\mathbf{X}, \mathbf{p} ; t), \quad \mathbf{v} \equiv \frac{\mathbf{p}}{M} $$

$$ H\left(\hat{\mathbf{p}}_{\nu}, \hat{\mathbf{x}}_{\nu}, t\right)|\Psi(t)\rangle=i \hbar \partial_{t}|\Psi(t)\rangle $$

$$ \begin{align*} & \left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N} \mid \mathbf{x}_{1}^{\prime}, \ldots, \mathbf{x}_{N}^{\prime}\right\rangle=\delta^{(3)}\left(\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime}\right) \cdots \delta^{(3)}\left(\mathbf{x}_{N}-\mathbf{x}_{N}^{\prime}\right) \\ & \int d^{3} x_{1} \cdots d^{3} x_{N}\left|\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right\rangle\left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right|=1 \end{align*} $$

$$ \begin{align*} \left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right| \hat{\mathbf{p}}_{\nu} & =-i \hbar \partial_{\mathbf{x}_{\nu}}\left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right| \\ \left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right| \hat{\mathbf{x}}_{\nu} & =\mathbf{x}_{\nu}\left\langle\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\right| \end{align*} $$

$$ \left|\Psi\left(t_{b}\right)\right\rangle=e^{-i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}\left|\Psi\left(t_{a}\right)\right\rangle $$

$$ \hat{U}\left(t_{b}, t_{a}\right)=e^{-i\left(t_{b}-t_{a}\right) \hat{H} / \hbar} $$

$$ i \hbar \partial_{t_{b}} \hat{U}\left(t_{b}, t_{a}\right)=\hat{H} \hat{U}\left(t_{b}, t_{a}\right) $$

$$ \hat{U}^{-1}\left(t_{b}, t_{a}\right) \equiv e^{i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}=\hat{U}\left(t_{a}, t_{b}\right) $$

$$ \hat{U}^{\dagger}=\hat{U}^{-1} . $$

$$ \begin{align*} \hat{U}^{\dagger}\left(t_{b}, t_{a}\right) & =e^{i\left(t_{b}-t_{a}\right) \hat{H}^{\dagger} / \hbar} \\ & =e^{i\left(t_{b}-t_{a}\right) \hat{H} / \hbar}=\hat{U}^{-1}\left(t_{b}, t_{a}\right) . \end{align*} $$

$$ \begin{align*} \left|\Psi\left(t_{a}+\epsilon\right)\right\rangle & \approx\left(1-\frac{i}{\hbar} \int_{t_{a}}^{t_{a}+\epsilon} d t \hat{H}(t)\right)\left|\Psi\left(t_{a}\right)\right\rangle \\ \left|\Psi\left(t_{a}+2 \epsilon\right)\right\rangle & \approx\left(1-\frac{i}{\hbar} \int_{t_{a}+\epsilon}^{t_{a}+2 \epsilon} d t \hat{H}(t)\right)\left|\Psi\left(t_{a}+\epsilon\right)\right\rangle \\ & \vdots \\ \left|\Psi\left(t_{a}+(N+1) \epsilon\right)\right\rangle & \approx\left(1-\frac{i}{\hbar} \int_{t_{a}+N \epsilon}^{t_{a}+(N+1) \epsilon} d t \hat{H}(t)\right)\left|\Psi\left(t_{a}+N \epsilon\right)\right\rangle \end{align*} $$

$$ \hat{U}\left(t_{b}, t_{a}\right) \approx\left(1-\frac{i}{\hbar} \int_{t_{N}}^{t_{b}} d t_{N+1}^{\prime} \hat{H}\left(t_{N+1}^{\prime}\right)\right) \times \cdots \times\left(1-\frac{i}{\hbar} \int_{t_{a}}^{t_{1}} d t_{1}^{\prime} \hat{H}\left(t_{1}^{\prime}\right)\right) $$

$$ \begin{align*} \hat{U}\left(t_{b}, t_{a}\right)= & 1-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t_{1}^{\prime} \hat{H}\left(t_{1}^{\prime}\right)+\left(\frac{-i}{\hbar}\right)^{2} \int_{t_{a}}^{t_{b}} d t_{2}^{\prime} \int_{t_{a}}^{t_{2}^{\prime}} d t_{1}^{\prime} \hat{H}\left(t_{2}^{\prime}\right) \hat{H}\left(t_{1}^{\prime}\right) \\ & +\left(\frac{-i}{\hbar}\right)^{3} \int_{t_{a}}^{t_{b}} d t_{3}^{\prime} \int_{t_{a}}^{t_{3}^{\prime}} d t_{2}^{\prime} \int_{t_{a}}^{t_{2}^{\prime}} d t_{1}^{\prime} \hat{H}\left(t_{3}^{\prime}\right) \hat{H}\left(t_{2}^{\prime}\right) \hat{H}\left(t_{1}^{\prime}\right)+\ldots \end{align*} $$

$$ \hat{O}_{n}\left(t_{n}\right) \cdots \hat{O}_{1}\left(t_{1}\right) $$

$$ \hat{T}\left(\hat{O}_{n}\left(t_{n}\right) \cdots \hat{O}_{1}\left(t_{1}\right)\right) \equiv \hat{O}_{i_{n}}\left(t_{i_{n}}\right) \cdots \hat{O}_{i_{1}}\left(t_{i_{1}}\right), $$

$$ t_{i_{n}}>t_{i_{n-1}}>\ldots>t_{i_{1}} $$

$$ \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{2}} d t_{1} \hat{H}\left(t_{2}\right) \hat{H}\left(t_{1}\right) $$

$$ \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{2}}^{t_{b}} d t_{1} \hat{H}\left(t_{2}\right) \hat{H}\left(t_{1}\right) $$

$$ \hat{T} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{2}}^{t_{b}} d t_{1} \hat{H}\left(t_{2}\right) \hat{H}\left(t_{1}\right) $$

$$ \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{2}}^{t_{b}} d t_{1} \hat{H}\left(t_{1}\right) \hat{H}\left(t_{2}\right) $$

$$ \int_{t_{a}}^{t_{b}} d t_{1} \int_{t_{a}}^{t_{1}} d t_{2} \hat{H}\left(t_{1}\right) \hat{H}\left(t_{2}\right) $$

$$ \frac{1}{2} \hat{T} \int_{t_{a}}^{t_{b}} d t_{2} \int_{t_{a}}^{t_{b}} d t_{1} \hat{H}\left(t_{2}\right) \hat{H}\left(t_{1}\right) $$

$$ \frac{1}{2} \hat{T}\left(\int_{t_{a}}^{t_{b}} d t \hat{H}(t)\right)^{2} $$

$$ \begin{align*} & \frac{1}{n!} \hat{T} \int_{t_{a}}^{t_{b}} d t_{n} \int_{t_{a}}^{t_{b}} d t_{n-1} \cdots \int_{t_{a}}^{t_{b}} d t_{1} \hat{H}\left(t_{n}\right) \hat{H}\left(t_{n-1}\right) \cdots \hat{H}\left(t_{1}\right) \\ & \quad=\frac{1}{n!} \hat{T}\left[\int_{t_{a}}^{t_{b}} d t \hat{H}(t)\right]^{n} \end{align*} $$

$$ \begin{align*} \hat{U}\left(t_{b}, t_{a}\right)=1 & -\frac{i}{\hbar} \hat{T} \int_{t_{a}}^{t_{b}} d t \hat{H}(t)+\frac{1}{2!}\left(\frac{-i}{\hbar}\right)^{2} \hat{T}\left(\int_{t_{a}}^{t_{b}} d t \hat{H}(t)\right)^{2} \\ & +\ldots+\frac{1}{n!}\left(\frac{-i}{\hbar}\right)^{n} \hat{T}\left(\int_{t_{a}}^{t_{b}} d t \hat{H}(t)\right)^{n}+\ldots \end{align*} $$

$$ \hat{U}\left(t_{b}, t_{a}\right)=\hat{T} \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \hat{H}(t)\right\} $$

$$ \begin{align*} \delta \hat{U}\left(t_{b}, t_{a}\right) & =-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t^{\prime} \hat{T} \exp \left\{-\frac{i}{\hbar} \int_{t^{\prime}}^{t_{b}} d t \hat{H}(t)\right\} \delta \hat{H}\left(t^{\prime}\right) \hat{T} \exp \left\{-\frac{i}{\hbar} \int_{t_{a}}^{t^{\prime}} d t \hat{H}(t)\right\} \\ & =-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t^{\prime} \hat{U}\left(t_{b}, t^{\prime}\right) \delta \hat{H}\left(t^{\prime}\right) \hat{U}\left(t^{\prime}, t_{a}\right) \end{align*} $$

$$ \hat{U}\left(t_{b}, t_{a}\right)=\hat{U}\left(t_{b}, t^{\prime}\right) \hat{U}\left(t^{\prime}, t_{a}\right), \quad t^{\prime} \in\left(t_{a}, t_{b}\right) . $$

$$ \begin{align*} & \hat{T} \exp \left(-\frac{i}{\hbar} \int_{t^{\prime}}^{t_{b}} \hat{H}(t) d t\right) \hat{T} \exp \left(-\frac{i}{\hbar} \int_{t_{a}}^{t^{\prime}} \hat{H}(t) d t\right) \\ & \quad=\hat{T}\left[\exp \left(-\frac{i}{\hbar} \int_{t^{\prime}}^{t_{b}} \hat{H}(t) d t\right) \exp \left(-\frac{i}{\hbar} \int_{t_{a}}^{t^{\prime}} \hat{H}(t) d t\right)\right] \\ & \quad=\hat{T} \exp \left(-\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} \hat{H}(t) d t\right) \end{align*} $$

$$ \hat{U}\left(t_{b}, t_{a}\right) \equiv \hat{U}\left(t_{a}, t_{b}\right)^{-1} $$

$$ \left|\Psi\left(t_{a}\right)\right\rangle=\hat{U}\left(t_{a}, t_{b}\right)\left|\Psi\left(t_{b}\right)\right\rangle $$

$$ \left|\Psi\left(t_{b}\right)\right\rangle=\hat{U}\left(t_{a}, t_{b}\right)^{-1}\left|\Psi\left(t_{a}\right)\right\rangle, \quad t_{b}