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## Main Question or Discussion Point

Hi. In elementary quantum mechanics the continuity equation is used to derive the electron current, i.e.

[tex]

\frac{\partial \rho(\mathbf r,t)}{\partial t}+\nabla\cdot\mathbf j(\mathbf r,t)=0

[/tex]

and one then puts [itex]\rho(\mathbf r,t)=\psi^*(r,\mathbf t)\psi(\mathbf r,t)[/itex].

Now if I want to derive an expression for the energy current, the continuity equation is

[tex]

\frac{\partial H}{\partial t}+\nabla\cdot\mathbf j_E(\mathbf r,t)=0

[/tex]

where [itex]H[/itex] is the energy density(the Hamiltonian density). But what is the Hamiltonian density?

[tex]

\frac{\partial \rho(\mathbf r,t)}{\partial t}+\nabla\cdot\mathbf j(\mathbf r,t)=0

[/tex]

and one then puts [itex]\rho(\mathbf r,t)=\psi^*(r,\mathbf t)\psi(\mathbf r,t)[/itex].

Now if I want to derive an expression for the energy current, the continuity equation is

[tex]

\frac{\partial H}{\partial t}+\nabla\cdot\mathbf j_E(\mathbf r,t)=0

[/tex]

where [itex]H[/itex] is the energy density(the Hamiltonian density). But what is the Hamiltonian density?