Despite of its title, this book is not a compendium of a young entomologist's research on new exotic species of colorful butterflies. In the pages of this book you will not find the story of a hunter determined to catch unknown specimen of enigmatic multicolored insects living in the forest in the heart of Africa or South America. The unwary reader, possibly intrigued by the title, should be somehow surprised to realize that this work is about Mathematical Physics. Nevertheless, this ambiguity hides some truths. Mathematical ideas fly light with butterfly wings in the mathematician's mind. They are painted with the gaudy colors of the intuition and imagination. The mathematician spends his time hunting for new problems just like the entomologist does for his preys. The tools he uses to get his hunting trophy are theories, theorems, proofs, ... Keeping in mind this analogy, the reader may consider this book as the story of my personal hunt aimed at capturing the secret of the quantum butterflies. Well, It is time to cry aloud: - let the hunt begin!