User:Maschen/Angular momentum diagrams

In quantum mechanics and it's applications to quantum many-particle systems, angular momentum diagrams, or more accurately from a mathematical viewpoint angular momentum graphs, are a diagrammatic method for representing angular momentum states of a quantum system allowing calculations to be done symbolically. More specifically, the arrows encode angular momentum states in braket notation and include the abstract nature of the state, such as tensor products and transformation rules. The notation has a parallel idea with Penrose graphical notation for diagrammatically expressing tensor expressions and symbolically doing calculations. The diagrams consist of arrows and vertices with quantum numbers as labels, hence the alternative term "graphs".

They were developed by Jucys in 19??.

Angular momentum states
The quantum state vector of a single particle with total angular momentum quantum quantum number j and magnetic quantum number m is denoted as a ket $|j, m\rangle$. As a diagram this is a singleheaded arrowhead arrow.

Symmetrically, the corresponding bra is $\langlej, m|$. In diagram form this is a doubleheaded arrow.

In each case;


 * the quantum numbers j, m are usually labelled next to the arrows,
 * there is no difference between an arrowhead at the end or middle of the line.

Arrows are directed to or from vertices, a state transforming according to a:


 * standard representation is designated by an oriented line leaving a vertex, while
 * contrastandard representation has a line entering a vertex.

As a general rule, the arrows follow each other in the same sense. In the contrastandard representation, the time reversal operator, denoted here by T, is used. It is unitary, which means the Hermitian conjugate T† equals the inverse operator T−1, that is T† = T−1. It's action on the position operator leave it invariant:


 * $$T \hat{\mathbf{x}} T^\dagger = \hat{\mathbf{x}} $$

but the linear momentum operator becomes negative:


 * $$T \hat{\mathbf{p}} T^\dagger = - \hat{\mathbf{p}} $$

and the spin operator becomes negative:


 * $$T \hat{\mathbf{S}} T^\dagger = - \hat{\mathbf{S}} $$

Since the orbital angular momentum operator is L = x × p, this must also become negative:


 * $$T \hat{\mathbf{L}} T^\dagger = - \hat{\mathbf{L}} $$

and therefore the total angular momentum operator J = L + S becomes negative:


 * $$T \hat{\mathbf{J}} T^\dagger = - \hat{\mathbf{J}} $$

Acting on an eigenstate of angular momentum $|j, m\rangle$, it can be shown that [see for example P.E.S. Wormer and J. Paldus (2006)]:


 * $$T \left|j,m\right\rangle \equiv \left|T (j,m)\right\rangle = {(-1)}^{j-m} \left|j,-m\right\rangle $$

The basic diagrams for kets and bras are:

Inner product
The inner product of two states $\langlej, m|$ and $|j, m\rangle$ is:


 * $$ \langle j_2, m_2 | j_1 , m_1 \rangle = \delta_{j_1 j_2} \delta_{m_1 m_2} $$

and the diagrams are:

For summations over the inner product, also known in this context as a contraction (c.f. tensor contraction):


 * $$\sum_m \langle j,m | j,m \rangle = 2j + 1 $$

it's conventional to denote the result as a closed circle labelled only by j, not m:


 * AM diagrams inner product contraction.svg

Outer products
The outer product of two states $|j, m\rangle$ and $\langlej, m|$ is an operator:


 * $$\left| j_2, m_2 \right\rangle \left\langle j_1 , m_1 \right|$$

and the diagrams are:

For summations over the outer product, also known in this context as a contraction (c.f. tensor contraction):


 * $$\begin{align}

\sum_m | j,m \rangle j,m \langle| & = \sum_m | j, -m \rangle \langle j, -m | \\ & = \sum_m {(-1)}^{2(j-m)}| j, -m \rangle \langlej, -m | \\ & = \sum_m {(-1)}^{j-m}| j, -m \rangle \langle j, -m |{(-1)}^{j-m} \\ & = \sum_m T| j, -m \rangle \langle j, -m |T^\dagger \end{align}$$

where the result of the action of T on $|j_{1}, m_{1}\rangle$ was used (see above), and the fact that m takes a symmetric set of values, j, j − 1, ...,, −j + 1, −j. There is no difference between the forward-time and reversed-time states, and they share the same diagram, represented as one line without direction, again labelled by j only and not m:



Tensor products
The tensor product &otimes; of n states $|j_{2}, m_{2}\rangle$, $|j_{1}, m_{1}\rangle$, ... $|j_{2}, m_{2}\rangle$ is written


 * $$\begin{align}

\left|j_1, m_1 , j_2 , m_2 , ... j_n, m_n \right\rangle & \equiv \left|j_1,m_1\right\rangle\otimes\left|j_2,m_2\right\rangle\otimes\cdots\otimes\left|j_n,m_n\right\rangle \\ & \equiv \left|j_1,m_1\right\rangle \left|j_2,m_2\right\rangle \cdots \left|j_n,m_n\right\rangle \end{align}$$

and in diagram form, each separate state leaves or enters a common vertex creating a "fan" of arrows - n lines attached to a single vertex.

Vertices in tensor products have signs (sometimes called "node signs"), to indicate the ordering of the tensor-multiplied states:


 * a minus sign (−) indicates the ordering is clockwise, and
 * a plus sign (+) for anticlockwise.

Signs are of course not required for just one state, diagrammatically one arrow at a vertex. Sometimes curved arrows with the signs are included to show explicitly the sense of tensor multiplication, but usually just the sign is shown with the arrows left out.

Inner products of tensor-product states
For the inner product of two tensor product states:


 * $$\begin{align}

& \left\langle j'_n, m'_n , ... , j'_2 , m'_2 , j'_1 , m'_1 |j_1 , m_1 , j_2 , m_2 , ... j_n, m_n \right\rangle \\ = & \langle j'_n, m'_n | ... \langle j'_2 , m'_2| \langle j'_1 , m'_1 | | j_1 , m_1 \rangle | j_2 , m_2 \rangle ... | j_n, m_n \rangle \\ = & \prod_{k=1}^n \left\langle j'_k, m'_k | j_k , m_k \right\rangle \end{align}$$

there are n lots of inner product arrows:

Clebsch–Gordan coefficients
The Clebsch–Gordan coefficients can be represented in the notation like so: