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carllacan
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Homework Statement
A particle with electrical charge [itex]q[/itex] and mass [itex]m[/itex] is in a electromagnetic field described by [itex]\phi (\vec{r}, t)[/itex] and [itex]A(\vec{r}, t)[/itex]. Its Hamiltonian is as follows:
[itex]H = \frac{1}{2m} \left ( \frac{\hbar}{i}\vec{\nabla}-\frac{q}{c} \vec{A} (\vec{r}, t) \right ) ^2 +q\phi (\vec{r}, t)[/itex]
The conservation of charge guarantees the continuity equation is fulfilled:
[itex]\frac{\partial}{\partial t} \rho (\vec{r},t)+\vec{\nabla}·\vec{j}(\vec{r},t) = 0[/itex],
where [itex]\rho = q\left|\Psi(x)\right|^2[/itex] is the charge density.
Find the current density [itex]\vec{j}(\vec{r},t)[/itex]
Homework Equations
The Hamiltonian of a particle in an electromagnetic field.
[itex]H = \frac{1}{2m} \left ( \frac{\hbar}{i}\vec{\nabla}-\frac{q}{c} \vec{A} (\vec{r}, t) \right ) ^2 +q\phi (\vec{r}, t)[/itex]
The continuity equation
[itex]\frac{\partial}{\partial t} \rho (\vec{r},t)+\vec{\nabla}·\vec{j}(\vec{r},t) = 0[/itex]
The Attempt at a Solution
I've tried stating the time-dependent Shcrodinger equation [itex]\hat{H}\Psi (x) = i\hbar\frac{\partial \Psi(x)}{\partial t}[/itex] and solve for the time-derivative of the wavefunction, which gives:
[itex]\frac{\partial \Psi}{\partial t} = \left [ \frac{\vec{\nabla}^2}{2m}+\frac{q}{\hbar imc}\vec{A}\vec{\nabla}-\frac{1}{2\hbar^2 m} \left( \frac{q}{c} \vec{A}\right )^2 -\frac{q}{2 \hbar m}\phi\right]\Psi[/itex]
Then
[itex]\frac{\partial\rho}{\partial t} = q2|\Psi(x)|\frac{\partial |\Psi(x)|}{\partial t} = \left [ \frac{q\vec{\nabla}^2}{m}+\frac{2q^2}{\hbar imc}\vec{A}\vec{\nabla}-\frac{q}{\hbar^2 m} \left( \frac{q}{c} \vec{A}\right )^2 -\frac{q^2}{ \hbar m}\phi\right]\Psi^2[/itex]
And then, from the continuity equation we get
[itex]\vec{\nabla}·\vec{j}(\vec{r},t) = \frac{\partial}{\partial t} \rho (\vec{r},t) = \left [ \frac{q\vec{\nabla}^2}{m}+\frac{2q^2}{\hbar imc}\vec{A}\vec{\nabla}-\frac{q}{\hbar^2 m} \left( \frac{q}{c} \vec{A}\right )^2 -\frac{q^2}{ \hbar m}\phi\right]\Psi^2[/itex]
But I'm stuck there. How do I "remove" the nabla operators?
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