High School Question about how the nabla interacts with wave functions

[DuckAmuck]

Is the following true?
ψ*∇^2 ψ = ∇ψ*⋅∇ψ

It seems like it should be since you can change the direction of operators.

 

[Orodruin]

Is the following true?
ψ*∇^2 ψ = ∇ψ*⋅∇ψ

It seems like it should be since you can change the direction of operators.

No. It is definitely not generally true.

If you have a volume integral and the function vanishes on the boundary, then you can do partial integration to find
$$
\int_V \psi^* \nabla^2 \psi\, dV = - \int_V (\nabla\psi^*)\cdot(\nabla\psi)dV.
$$
Note the minus sign!

 

[Demystifier]

It seems like it should be since you can change the direction of operators.

You can do that for hermitian operators. But the derivative is not a hermitian operator. It is an anti-hermitian operator, due to the minus sign explained in the previous post.