Berry's Curvature Equation cross product calculation

[wondering12]

Hi,
The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation
V_m= (- 1/B² ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A² ...[1]
the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m> = 0 then the above equation simplifies to
V_m= (- 1/B² ) * i * ∑( <m,B|S ∧ S|n,B> ) / A²....[2]

The symbol ∧ stands for 'and' in logic or cross product.
My question is how the author derived that claim and how it led to that equation [2] from equation [1] ?

My reasoning is that |n,B> 'and' <n,B| are both true therefore ( |n,B> 'and' <n,B|) = 1 which is equal to 1 'and' 1 however I do not believe that my reasoning is a valid one. What is the alternative to this?
Thanks.

 

[vanhees71]

What textbook is this? How can there be a logical operator in an equation of complex numbers? Sometimes the wedge is used to denote vector products (or the related wedge product in Cartan calculus notation), but also this doesn't make much sense here.

I'd be very sceptical towards this book, if this is really written there with such a definition for the wedge symbol. Also, please use LaTeX for complex formulae like this. Your posting is really quite diffcult to read.