If |n><n| is the same as <n||n> then you should have a look at the help and documentation to dot. That function will not generalize well to the more general <n|G|n>. For that you might get away with the standard vector-matrix-vector products where you'll have to make some discrete version of the operator G. If |n><n| is not the same as <n||n> you might have to explain what it is supposed to be.
HTH
