Angular momentum diagrams (quantum mechanics)

In quantum mechanics and its applications to quantum many-particle systems, notably quantum chemistry, angular momentum diagrams, or more accurately from a mathematical viewpoint angular momentum graphs, are a diagrammatic method for representing angular momentum quantum states of a quantum system allowing calculations to be done symbolically. More specifically, the arrows encode angular momentum states in bra–ket notation and include the abstract nature of the state, such as tensor products and transformation rules. The notation parallels the idea of Penrose graphical notation and Feynman diagrams. The diagrams consist of arrows and vertices with quantum numbers as labels, hence the alternative term "graphs". The sense of each arrow is related to Hermitian conjugation, which roughly corresponds to time reversal of the angular momentum states (c.f. Schrödinger equation). The diagrammatic notation is a considerably large topic in its own right with a number of specialized features – this article introduces the very basics. They were developed primarily by Adolfas Jucys (sometimes translated as Yutsis) in the twentieth century. The quantum state vector of a single particle with total angular momentum quantum number j and total magnetic quantum number m = j, j − 1, ..., −j + 1, −j, is denoted as a ket . As a diagram this is a singleheaded arrow. Symmetrically, the corresponding bra is . In diagram form this is a doubleheaded arrow, pointing in the opposite direction to the ket. In each case; the quantum numbers j, m are often labelled next to the arrows to refer to a specific angular momentum state, arrowheads are almost always placed at the middle of the line, rather than at the tip, equals signs "=" are placed between equivalent diagrams, exactly like for multiple algebraic expressions equal to each other. The most basic diagrams are for kets and bras: Arrows are directed to or from vertices, a state transforming according to: a standard representation is designated by an oriented line leaving a vertex, a contrastandard representation is depicted as a line entering a vertex.