Gauge Invariance in Hamiltonian

[Sagar Rawal]

Homework Statement

Hello Everyone
I'm wondering, why in below product rule was not used for gradient of A where exponential is treated as constant for divergent of A and only for first term of equation we used the product rule?

Homework Equations

https://ibb.co/gHOauJ

The Attempt at a Solution

 

[Delta2]

Because the divergence operator is applied only to A and then you have multiplication with exponential and psi.
That is, it is $(\nabla\cdot A)(e^{\lambda}\psi)$ , it is not $\nabla \cdot (Ae^{\lambda}\psi)$