So I'm trying to derive the conditions necessary for local U(1) gauge invariance in the Schrodinger equation, and I don't understand how Laplacian of the wavefunction is simplified the way it is. So the transformation is:
\begin{equation} \psi \rightarrow \psi'=\psi e^{i\lambda(x,t)} \end{equation}
Which gives the following Laplacian which I understand:
\begin{equation} \nabla^2\psi'=(\nabla^2+2i\nabla\lambda\cdot\nabla+i\nabla^2\lambda - (\nabla\lambda)^2)\psi \tag{1} \end{equation}
According to every textbook I read, the next step is to simplify it the following way:
\begin{equation} \nabla^2\psi'=(\nabla+i\nabla\lambda)^2\psi \end{equation}
\begin{equation} (\nabla+i\nabla\lambda)^2\psi=(\nabla^2+i\nabla\lambda\cdot\nabla+i\nabla^2\lambda-(\nabla\lambda)^2)\psi \tag{2} \end{equation}
However, equation (1) and equation (2) are not the same because the 2 present in the second term of equation (1) is not in equation (2). Is there some piece of algebra I am missing? Here is one reference http://www.niser.ac.in/~sbasak/p303_2010/23.11.pdf (page 2)