ψ ( x ) → e i α ( x ) ψ ( x ) {\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x)}
∂ ρ ∂ t = ∂ ∂ t ( ψ ( x ) ∗ ψ ( x ) ) = ψ ( x ) ∗ H ^ ψ i ℏ − ψ ( x ) ( H ^ ψ ) ∗ i ℏ {\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {\partial }{\partial t}}(\psi (x)^{*}\psi (x))=\psi (x)^{*}{\frac {{\hat {H}}\psi }{i\hbar }}-\psi (x){\frac {({\hat {H}}\psi )^{*}}{i\hbar }}}
= ψ ( x ) ∗ T ^ ψ i ℏ − ψ ( x ) ( T ^ ψ ) ∗ i ℏ − ψ ( x ) ∗ U ^ ψ i ℏ + ψ ( x ) ( U ^ ψ ) ∗ i ℏ {\displaystyle =\psi (x)^{*}{\frac {{\hat {T}}\psi }{i\hbar }}-\psi (x){\frac {({\hat {T}}\psi )^{*}}{i\hbar }}-\psi (x)^{*}{\frac {{\hat {U}}\psi }{i\hbar }}+\psi (x){\frac {({\hat {U}}\psi )^{*}}{i\hbar }}}
= 1 2 i m ℏ { ψ ( x ) ∗ ( − i ℏ ∇ − e A → ) 2 ψ ( x ) − ψ ( x ) ( − i ℏ ∇ − e A → ) 2 ψ ( x ) ∗ } {\displaystyle ={\frac {1}{2im\hbar }}\left\{\psi (x)^{*}(-i\hbar \nabla -e{\vec {A}})^{2}\psi (x)-\psi (x)(-i\hbar \nabla -e{\vec {A}})^{2}\psi (x)^{*}\right\}}
∂ ρ ∂ t = − d i v [ i ℏ 2 m ( − ψ ( x ) ∗ ∇ ψ ( x ) + ψ ( x ) ∇ ψ ( x ) ∗ ) − e m ψ ( x ) ∗ ψ ( x ) A → ] {\displaystyle {\frac {\partial \rho }{\partial t}}=-div\left[{\frac {i\hbar }{2m}}(-\psi (x)^{*}\nabla \psi (x)+\psi (x)\nabla \psi (x)^{*})-{\frac {e}{m}}\psi (x)^{*}\psi (x){\vec {A}}\right]}
∂ ρ ∂ t = ∂ ∂ t ( ψ ( x 2 ) ∗ ψ ( x 1 ) ∗ ψ ( x 2 ) ψ ( x 1 ) ) {\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {\partial }{\partial t}}(\psi (x_{2})^{*}\psi (x_{1})^{*}\psi (x_{2})\psi (x_{1}))}
∂ ρ ∂ t = ( ∂ ∂ t ψ ( x 2 ) ∗ ) ψ ( x 1 ) ∗ ψ ( x 2 ) ψ ( x 1 ) + ψ ( x 2 ) ∗ ( ∂ ∂ t ψ ( x 1 ) ∗ ) ψ ( x 2 ) ψ ( x 1 ) + ψ ( x 2 ) ∗ ψ ( x 1 ) ∗ ( ∂ ∂ t ψ ( x 2 ) ) ψ ( x 1 ) + ψ ( x 2 ) ∗ ψ ( x 1 ) ∗ ψ ( x 2 ) ( ∂ ∂ t ψ ( x 1 ) ) {\displaystyle {\frac {\partial \rho }{\partial t}}=({\frac {\partial }{\partial t}}\psi (x_{2})^{*})\psi (x_{1})^{*}\psi (x_{2})\psi (x_{1})+\psi (x_{2})^{*}({\frac {\partial }{\partial t}}\psi (x_{1})^{*})\psi (x_{2})\psi (x_{1})+\psi (x_{2})^{*}\psi (x_{1})^{*}({\frac {\partial }{\partial t}}\psi (x_{2}))\psi (x_{1})+\psi (x_{2})^{*}\psi (x_{1})^{*}\psi (x_{2})({\frac {\partial }{\partial t}}\psi (x_{1}))}
фигня :-(
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![{\displaystyle {\frac {\partial \rho }{\partial t}}=-div\left[{\frac {i\hbar }{2m}}(-\psi (x)^{*}\nabla \psi (x)+\psi (x)\nabla \psi (x)^{*})-{\frac {e}{m}}\psi (x)^{*}\psi (x){\vec {A}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/280fc649159934cbefecd948653a984b6d8827eb)
