JK Rowling begins the Harry Potter series with the Philosopher's Stone symbol and introduces the historic character of Nicolas Flamel, a medieval alchemist, suggesting a trail of alchemical symbols as a literary device for the reader to follow. Alchemical language gives the reader an elegant key to Rowling's universe.
Alchemists developed a secret symbolic language to share their knowledge only with people they thought deserving of the information: "The Fraternity of the true philosophers". Alchemists drew pictures in an elaborate language of symbols, which correctly read by another alchemist, would give him information and procedures to follow.
The main images of Alchemy were: the 7 metals they used (silver, gold, mercury, lead...) and the 7 planets these metals were connected to (moon=silver...), animals : lion, serpent, eagle, dragon, unicorn, peacock, phoenix, crow, grey wolf..., each of them symbolizing a metal or a procedure. Readers can find these images in the (7!) books used, as metaphors, with the characters and/or the actions that connect them.
Alchemists also loved to communicate by anagrams. Rowling also uses anagrams as a literary device, beginning with the Riddle/Voldemort anagram.
Alchemists were searching for the Philosopher's Stone, a transformational object used to make silver and gold from metals which were not precious and also to provide universal medical cure for illness. Yet, alchemist's main aim was the ennoblement of the soul, symbolized by the ennoblement of the matter; this symbolism is especially rich when applied to Harry Potter.
Some examples: alchemists called the phoenix the Cinnabar Bird. Cinnabar is a red mercuric sulphide meant to give immortality (cf red and gold Fawkes regenerating himself and his magical healing powers). Cinnabar was the main ingredient used for the Philosopher's stone (red!). Cinnabar is quicksilver become solid (Fawkes strengthens hearts), and, through mercury, symbolizes the quest the soul (here Harry) has to achieve.
The hand of Wormtail may be another alchemical clue: assuming it is a metaphor of mercury (quicksilver here), this is a symbol of the Hand of Fate that will turn from Slytherin silver to Gryffindor gold as he repays his debt life to Harry. Furthermore, if we presume that the replacement of his non-precious hand by a silver one is a kind of purification of his "materia prima", as a kind of ennoblement of the matter and -therefore- of the soul in that perspective.
JK Rowling also uses some astrological clues. Astrology and alchemy are closely bound. Alchemists believed in the relation between the macrocosm and the microcosm: the universe was organized in concentric spheres (from the planet to men, from men to minerals). Each planet is connected with a metal, a human organ and a constellation. This alchemical theory leads us to the theory of the balance of the 4 elements/4 temperaments (air/fire/water/earth - sanguine/bilious/phlegmatic/melancholic humours). The metaphor could apply to the 4 Hogwarts' founders: when Slytherin left Hogwarts, he broke the balance of the Wizarding World. So Harry's true aim may be to restore this balance by reunifying Hogwarts and defeating Voldemort.