Derivation of "continuity equation"

[Ami]

Hello,

I need the derivation of "continuity equation" by the current density equation,in Quantum Mechanics.

I really need this derivation quickly,please

Thanks

 

[quantumdude]

You can do it yourself.

1. Take the Schrödinger equation for $\Psi(\vec{x},t)$ and multiply it by $\Psi^*(\vec{x},t)$.
2. Take the Schrödinger equation for $\Psi^*(\vec{x},t)$ and multiply it by $\Psi(\vec{x},t)$.
3. Subtract the two equations.

From there you have to appropriately define the probability density and the probability current density, then finish off the manipulations to get the continuity equation. Try that, and if you get stuck post what you have done and we will get you un-stuck.

 

[Ami]

Thank you very much for your reply

It has been very helpful to me.
And I've finished deriving the equation.

Thanks a lot