Phy5645/Energy conservation

(1) The energy operator in three dimensions is: $H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V$ so the average energy in state $\Psi$ is: $\left\langle E\right\rangle =\iiint \Psi ^{\ast }H\Psi \,d^{3}{\textbf {r}}=\iiint \Psi ^{\ast }\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi \right)\,d^{3}{\textbf {r}}$

Using the identity, $\Psi ^{*}\nabla ^{2}\Psi =\nabla \cdot \left(\Psi ^{*}\nabla \Psi \right)-\nabla \Psi ^{\ast }\cdot \nabla \Psi ,$ we obtain

$\left\langle E\right\rangle =-{\frac {\hbar ^{2}}{2m}}\iiint \left[\nabla \cdot \left(\Psi ^{\ast }\nabla \Psi \right)-\nabla \Psi ^{\ast }\cdot \nabla \Psi \right]\,d^{3}{\textbf {r}}+\iiint \Psi ^{\ast }V\Psi \,d^{3}{\textbf {r}}$ $=-{\frac {\hbar ^{2}}{2m}}\iiint \nabla \cdot \left(\Psi ^{\ast }\nabla \Psi \right)\,d^{3}{\textbf {r}}+{\frac {\hbar ^{2}}{2m}}\iiint \nabla \Psi ^{\ast }\cdot \nabla \Psi \,d^{3}{\textbf {r}}+\iiint \Psi ^{\ast }V\Psi \,d^{3}{\textbf {r}}$

If we apply Gauss' Theorem to the first term,

$-{\frac {\hbar ^{2}}{2m}}\iiint \nabla \left(\Psi ^{\ast }\nabla \Psi \right)\,d^{3}{\textbf {r}}=\iint \Psi ^{\ast }\nabla \Psi \cdot d{\textbf {S}},$

as well as the condition, $\lim _{r\to \infty }\Psi ^{\ast }\nabla \Psi =0,$ we obtain

$\left\langle E\right\rangle =\int W\,d^{3}{\textbf {r}}=\int \left({\frac {\hbar ^{2}}{2m}}\nabla \Psi ^{\ast }\cdot \nabla \Psi +\Psi ^{\ast }V\Psi \right)d^{3}{\textbf {r}}$

(2) We first find the time derivative of energy density:

${\frac {\partial W}{\partial t}}={\frac {\partial }{\partial t}}\left(\nabla \Psi ^{\ast }\cdot \nabla \Psi +\Psi ^{\ast }V\Psi \right)={\frac {\hbar ^{2}}{2m}}\left(\nabla \Psi ^{\ast }\cdot \nabla {\frac {\partial \Psi }{\partial t}}+\nabla {\frac {\partial \Psi ^{\ast }}{\partial t}}\cdot \nabla \Psi \right)+{\frac {\partial \Psi ^{\ast }}{\partial t}}V\Psi +\Psi ^{\ast }V{\frac {\partial \Psi }{\partial t}}$ $={\frac {\hbar ^{2}}{2m}}\left[\nabla \cdot \left(\nabla \Psi ^{\ast }{\frac {\partial \psi }{\partial t}}+{\frac {\partial \Psi ^{\ast }}{\partial t}}\nabla \Psi \right)-\left({\frac {\partial \Psi }{\partial t}}\nabla ^{2}\Psi ^{\ast }+{\frac {\partial \Psi ^{\ast }}{\partial t}}\nabla ^{2}\Psi \right)\right]+{\frac {\partial \Psi ^{\ast }}{\partial t}}V\Psi +\Psi ^{\ast }V{\frac {\partial \Psi }{\partial t}}$ $={\frac {\hbar ^{2}}{2m}}\nabla \cdot \left(\nabla \Psi ^{\ast }{\frac {\partial \Psi }{\partial t}}+{\frac {\partial \Psi ^{\ast }}{\partial t}}\nabla \Psi \right)+{\frac {\partial \Psi ^{\ast }}{\partial t}}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \right)+{\frac {\partial \Psi }{\partial t}}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi ^{\ast }+\nabla \Psi ^{\ast }\right)$,

Using the Schrödinger equation, $i\hbar {\frac {\partial \Psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi ,$

and its complex conjugate, $-i\hbar {\frac {\partial \Psi ^{\ast }}{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi ^{\ast }+V\Psi ^{\ast },$

and defining the energy flux density as ${\textbf {S}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial \Psi ^{\ast }}{\partial t}}\nabla \Psi +{\frac {\partial \Psi }{\partial t}}\nabla \Psi ^{\ast }\right),$

We obtain ${\frac {\partial W}{\partial t}}=-\nabla \cdot {\textbf {S}}+{\frac {\partial \Psi ^{\ast }}{\partial t}}{\frac {\partial \Psi }{\partial t}}-{\frac {\partial \Psi }{\partial t}}{\frac {\partial \Psi ^{\ast }}{\partial t}}=-\nabla \cdot {\textbf {S}},$

or, rearranging, ${\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0.$

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