算符
量子力學 > 量子算符表格

在這表格裏，算符的表現空間是位置空間。假若表現空間是其它種空間，則表示出的方程式會不一樣。在英文字母上方的尖角號表示整個符號代表的是個量子算符，不是單位向量。
| 算符名稱 | 直角坐標系分量表示 | 向量表示 |
| --- | --- | --- |
| 位置算符 |  {\begin{aligned}{\hat {x}}=x\\{\hat {y}}=y\\{\hat {z}}=z\end{aligned}} |  \mathbf {\hat {r}} =\mathbf {r}  |
| 動量算符 | 一般狀況   {\begin{aligned}{\hat {p}}_{x}&=-i\hbar {\frac {\partial }{\partial x}}\\{\hat {p}}_{y}&=-i\hbar {\frac {\partial }{\partial y}}\\{\hat {p}}_{z}&=-i\hbar {\frac {\partial }{\partial z}}\end{aligned}} | 一般狀況   \mathbf {\hat {p}} =-i\hbar \nabla  |
| 動量算符 | 電磁場   {\begin{aligned}{\hat {p}}_{x}=-i\hbar {\frac {\partial }{\partial x}}-qA_{x}\\{\hat {p}}_{y}=-i\hbar {\frac {\partial }{\partial y}}-qA_{y}\\{\hat {p}}_{z}=-i\hbar {\frac {\partial }{\partial z}}-qA_{z}\end{aligned}} | 電磁場（ \mathbf {A} 是磁向量勢）   \mathbf {\hat {p}} =-i\hbar \nabla -q\mathbf {A}  |
| 動能算符 | 平移運動   {\begin{aligned}{\hat {T}}_{x}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\\{\hat {T}}_{y}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}\\{\hat {T}}_{z}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}\\\end{aligned}} | 平移運動   {\begin{aligned}{\hat {T}}&={\hat {T}}_{x}+{\hat {T}}_{y}+{\hat {T}}_{z}\\&={\frac {-\hbar ^{2}}{2m}}\nabla ^{2}\\\end{aligned}} |
| 動能算符 | 電磁場   {\begin{aligned}{\hat {T}}_{x}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial x}}-qA_{x}\right)^{2}\\{\hat {T}}_{y}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial y}}-qA_{y}\right)^{2}\\{\hat {T}}_{z}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial z}}-qA_{z}\right)^{2}\end{aligned}} | 電磁場（ \mathbf {A} 是磁向量勢）   {\begin{aligned}{\hat {T}}&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}\\&={\frac {1}{2m}}(-i\hbar \nabla -q\mathbf {A} )\cdot (-i\hbar \nabla -q\mathbf {A} )\\&={\frac {1}{2m}}(-i\hbar \nabla -q\mathbf {A} )^{2}\end{aligned}} |
| 動能算符 | 旋轉運動（ I是轉動慣量）   {\begin{aligned}{\hat {T}}_{xx}&={\frac {{\hat {J}}_{x}^{2}}{2I_{xx}}}\\{\hat {T}}_{yy}&={\frac {{\hat {J}}_{y}^{2}}{2I_{yy}}}\\{\hat {T}}_{zz}&={\frac {{\hat {J}}_{z}^{2}}{2I_{zz}}}\\\end{aligned}} | 旋轉運動    {\hat {T}}={\frac {\mathbf {\hat {J}} \cdot \mathbf {\hat {J}} }{2I}} |
| 勢能算符 | N/A |  {\hat {V}}=V\left(\mathbf {r} ,t\right) |
| 總能量算符 | N/A | 含時位勢：   {\hat {E}}=i\hbar {\frac {\partial }{\partial t}} 不含時位勢：   {\hat {E}}=E |
| 哈密頓算符 | N/A |  {\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V\\&={\frac {{\hat {p}}^{2}}{2m}}+V\\\end{aligned}} |
| 角動量算符 |  {\begin{aligned}{\hat {L}}_{x}&=-i\hbar \left(y{\partial  \over \partial z}-z{\partial  \over \partial y}\right)\\{\hat {L}}_{y}&=-i\hbar \left(z{\partial  \over \partial x}-x{\partial  \over \partial z}\right)\\{\hat {L}}_{z}&=-i\hbar \left(x{\partial  \over \partial y}-y{\partial  \over \partial x}\right)\end{aligned}} |  \mathbf {\hat {L}} =-i\hbar \mathbf {r} \times \nabla  |
| 自旋算符 |  {\begin{aligned}{\hat {S}}_{x}={\hbar  \over 2}\sigma _{x}\\{\hat {S}}_{y}={\hbar  \over 2}\sigma _{y}\\{\hat {S}}_{z}={\hbar  \over 2}\sigma _{z}\end{aligned}} 其中，  \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}  \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}  \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}} 是自旋1/2粒子的包立矩陣。 |  \mathbf {\hat {S}} ={\hbar  \over 2}{\boldsymbol {\sigma }} 其中，向量 {\boldsymbol {\sigma }}的分量是包立矩陣。 |
| 總角動量算符 |  {\begin{aligned}{\hat {J}}_{x}&={\hat {L}}_{x}+{\hat {S}}_{x}\\{\hat {J}}_{y}&={\hat {L}}_{y}+{\hat {S}}_{y}\\{\hat {J}}_{z}&={\hat {L}}_{z}+{\hat {S}}_{z}\end{aligned}} |  {\begin{aligned}\mathbf {\hat {J}} &=\mathbf {\hat {L}} +\mathbf {\hat {S}} \\&=-i\hbar \mathbf {r} \times \nabla +{\frac {\hbar }{2}}{\boldsymbol {\sigma }}\end{aligned}} |
| 躍遷矩（電） （transition moment） |  {\begin{aligned}{\hat {d}}_{x}&=qx\\{\hat {d}}_{y}&=qy\\{\hat {d}}_{z}&=qz\end{aligned}} |  \mathbf {\hat {d}} =q\mathbf {r}  |