概率密度 ρ ( t , r ) = | ψ ( t , r ) | 2 {\displaystyle \rho (t,\mathbf {r} )=|\psi (t,\mathbf {r} )|^{2}} r {\displaystyle \mathbf {r} }
∂ ρ ∂ t = ψ ∗ ∂ ψ ∂ t + ψ ∂ ψ ∗ ∂ t . {\displaystyle {\frac {\partial \rho }{\partial t}}=\psi ^{*}{\frac {\partial \psi }{\partial t}}+\psi {\frac {\partial \psi ^{*}}{\partial t}}.}
i ℏ ∂ ψ ∂ t = 1 2 m ( ℏ i ∂ ∂ r − A ) ⋅ ( ℏ i ∂ ∂ r − A ) ψ + V ψ , {\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}={\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi +V\psi ,} ℏ i ∂ ψ ∗ ∂ t = 1 2 m ( i ℏ ∂ ∂ r − A ) ⋅ ( i ℏ ∂ ∂ r − A ) ψ ∗ + V ψ ∗ , {\displaystyle {\hbar \over i}{\partial \psi ^{*} \over \partial t}={\frac {1}{2m}}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}+V\psi ^{*},}
∂ ρ ∂ t = − i ℏ ψ ∗ [ 1 2 m ( ℏ i ∂ ∂ r − A ) ⋅ ( ℏ i ∂ ∂ r − A ) ψ + V ψ ] {\displaystyle {\frac {\partial \rho }{\partial t}}=-{\frac {i}{\hbar }}\psi ^{*}\left[{\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi +V\psi \right]} + i ℏ ψ [ 1 2 m ( i ℏ ∂ ∂ r − A ) ⋅ ( i ℏ ∂ ∂ r − A ) ψ ∗ + V ψ ∗ ] . {\displaystyle +{\frac {i}{\hbar }}\psi \left[{\frac {1}{2m}}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}+V\psi ^{*}\right].} V {\displaystyle V}
∂ ρ ∂ t = − i 2 m ℏ [ ψ ∗ ( i ℏ ∂ ∂ r + A ) ⋅ ( i ℏ ∂ ∂ r + A ) ψ − ψ ( i ℏ ∂ ∂ r − A ) ⋅ ( i ℏ ∂ ∂ r − A ) ψ ∗ ] {\displaystyle {\frac {\partial \rho }{\partial t}}=-{\frac {i}{2m\hbar }}\left[\psi ^{*}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\psi -\psi \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}\right]} = ⋯ = − ℏ 2 m i ( ∂ 2 ψ ∂ r 2 ψ ∗ − ψ ∂ 2 ψ ∗ ∂ r 2 ) + 1 m ( ψ ψ ∗ ∂ ∂ r ⋅ A + A ∂ ψ ∂ r ψ ∗ + A ψ ∂ ψ ∗ ∂ r ) . {\displaystyle =\dots =-{\frac {\hbar }{2mi}}\left({\frac {\partial ^{2}\psi }{\partial \mathbf {r} ^{2}}}\psi ^{*}-\psi {\frac {\partial ^{2}\psi ^{*}}{\partial \mathbf {r} ^{2}}}\right)+{\frac {1}{m}}\left(\psi \psi ^{*}{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {A} +\mathbf {A} {\frac {\partial \psi }{\partial \mathbf {r} }}\psi ^{*}+\mathbf {A} \psi {\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\right).} j = ℏ 2 m i ( ψ ∗ ∂ ψ ∂ r − ∂ ψ ∗ ∂ r ψ ) − 1 m A ψ ∗ ψ {\displaystyle \mathbf {j} ={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi }{\partial \mathbf {r} }}-{\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\psi \right)-{\frac {1}{m}}\mathbf {A} \psi ^{*}\psi }
∂ ∂ r ⋅ j = ℏ 2 m i ( ∂ 2 ψ ∂ r 2 ψ ∗ − ψ ∂ 2 ψ ∗ ∂ r 2 ) − 1 m ( ψ ψ ∗ ∂ ∂ r ⋅ A + A ∂ ψ ∂ r ψ ∗ + A ψ ∂ ψ ∗ ∂ r ) . {\displaystyle {\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {j} ={\frac {\hbar }{2mi}}\left({\frac {\partial ^{2}\psi }{\partial \mathbf {r} ^{2}}}\psi ^{*}-\psi {\frac {\partial ^{2}\psi ^{*}}{\partial \mathbf {r} ^{2}}}\right)-{\frac {1}{m}}\left(\psi \psi ^{*}{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {A} +\mathbf {A} {\frac {\partial \psi }{\partial \mathbf {r} }}\psi ^{*}+\mathbf {A} \psi {\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\right).} 结果
∂ ρ ∂ t = − ∂ ∂ r ⋅ j . {\displaystyle {\frac {\partial \rho }{\partial t}}=-{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {j} .}
在空间区域 R {\displaystyle R} ∂ R : {\displaystyle \partial R:}
∂ ∂ t ∫ R ρ d 3 r = − ∫ R ∂ ∂ r ⋅ j d 3 r . {\displaystyle {\partial \over \partial t}\int _{R}\rho \,d^{3}r=-\int _{R}{\partial \over \partial \mathbf {r} }\cdot \mathbf {j} \,d^{3}r.} 根据 高斯定律 ， j {\displaystyle \mathbf {j} } ∂ R {\displaystyle \partial R} j {\displaystyle \mathbf {j} } R : {\displaystyle R:} 散度 的积分：
∮ ∂ R j ⋅ d Σ = ∫ R ∂ ∂ r ⋅ j d 3 r . {\displaystyle \oint _{\partial R}\mathbf {j} \cdot d\Sigma =\int _{R}{\partial \over \partial \mathbf {r} }\cdot \mathbf {j} \,d^{3}r.} 因此，我们有
∂ ∂ t ∫ R ρ d 3 r = − ∮ ∂ R j ⋅ d Σ . {\displaystyle {\partial \over \partial t}\int _{R}\rho \,d^{3}r=-\oint _{\partial R}\mathbf {j} \cdot d\Sigma .} 如果 ρ {\displaystyle \rho } j {\displaystyle \mathbf {j} } R {\displaystyle R} R {\displaystyle R} 连续性方程 。
ρ {\displaystyle \rho } j , {\displaystyle \mathbf {j} ,} ψ {\displaystyle \psi } ∫ R ρ d 3 r = ∫ R | ψ | 2 d 3 r {\displaystyle \int _{R}\rho \,d^{3}r=\int _{R}|\psi |^{2}\,d^{3}r} R {\displaystyle R} R {\displaystyle R} R {\displaystyle R} ψ {\displaystyle \psi } n {\displaystyle n} R {\displaystyle R}
![{\displaystyle {\frac {\partial \rho }{\partial t}}=-{\frac {i}{\hbar }}\psi ^{*}\left[{\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi +V\psi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23fdabbde0cdbdb661a59e5da9d9e6383413680)
![{\displaystyle +{\frac {i}{\hbar }}\psi \left[{\frac {1}{2m}}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}+V\psi ^{*}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98727a8a7448922f0b5f337839182910d485ccd8)
![{\displaystyle {\frac {\partial \rho }{\partial t}}=-{\frac {i}{2m\hbar }}\left[\psi ^{*}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\psi -\psi \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb63f2ee8fad5069e8ef2da029a0eeb8e925ce34)
