Trivial demonstration. $\nabla J(r,t)=\frac{\hbar}{im}\nabla\psi^{*}\nabla\psi+\frac{\hbar}{im}\psi\nabla^2\psi$

Question

I can not do the calculation for a very trivial demonstration. $$J(r,t)=Re \left[ \psi^{*}\frac{\hbar}{im}\nabla\psi \right]$$ I want to demonstrate that $$\nabla J(r,t)=\frac{i}{\hbar}[\psi^{*}(H\psi)-(H\psi)^{*}\psi]$$ $$H=-\frac{\hbar^2}{2m}\Delta$$ I did the divergence of the definition $$\nabla J(r,t)=\frac{\hbar}{im}\nabla\psi^{*}\nabla\psi+\frac{\hbar}{im}\psi\nabla^2\psi$$ $$\nabla J(r,t)=\frac{\hbar}{im}\nabla\psi^{*}\nabla\psi-\frac{2}{i}\psi H\psi$$ Now I'm stuck and I do not know how to go on

Answer 1

What you need is

$$\nabla J(r,t)=Re\left[\frac{\hbar}{im}\nabla\psi^{*}\nabla\psi-\frac{2}{i\hbar}\psi^* H\psi\right]$$

Notice you forgot taking the real part, you forgot an $\hbar$ factor and you also forgot a complex conjugation sign.

The first term of this expression is purely imaginary so that

$$\nabla J(r,t)=Re\left[-\frac{2}{i\hbar}\psi^* H\psi\right]$$

Another way to express what the real part of a complex number $z$ is, is to compute $(z+z^*)/2$, thus

$$\nabla J(r,t)=\frac{1}{2}\left(-\frac{2}{i\hbar}\psi^* H\psi+\frac{2}{i\hbar}(H\psi)^*\psi\right)=-\frac{1}{i\hbar}\left[\psi^* (H\psi)-(H\psi)^*\psi\right] \; .$$

And since $i=-1/i$, we finished the derivation.