GP

Quantum Angular Momentum Relations

Angular momentum is defined as

$$ L_k = \sum_{i,j,k}\epsilon_{ijk}x_i p_j $$
Angular momentum in quantum mechanics cannot have simultaneous eigenvectors for all three spatial directions. That is, different components of angular momentum do not commute. However, the total angular momentum can be simultaneously described with one direction of angular momentum, which generally is taken as \(L_z\). Explicitly, \(L^2\) and \(L_i\) commute. This simultaneous eigenstate is generally denoted in Dirac notation as \(\vert l m\rangle\), where \(l\) is the quantum number corresponding to the \(L^2\equiv L_x^2+L_y^2+L_z^2 \) operator with eigenvalue \(l(l+1)\), and \(m\) is the quantum number associated with \(L_z\) (we choose z here to adhere to the standard convention) with eigenvalue \( m\).

These relations are specified explicitly below for referential convenience.

\begin{align} [L_i,L_j]&=i\hbar \epsilon_{ijk}L_k\\ [L^2,L_i]&=0\\ \end{align}
\begin{align} L_z\vert l m\rangle&=m\vert l m\rangle\\ L^2\vert l m\rangle&=l(l+1)\vert l m\rangle\\ \end{align}

Ladder Operators

Ladder operators, or creation and annihilation operators, or raising and lowering operators are useful in certain algebraic derivations and models of certain physical processes. They're defined for vector operators as \(L_{z\pm}=L_x\pm iL_y\). In the context of angular momentum, they lower or increase the magnetic quantum number, up to the minimum or maximum possible values (\(-l\) or \(+l\) respectively), giving zero otherwise.

\begin{align} L_+\vert l m\rangle&=\sqrt{(j-m)(j+m+1))}\vert l (m+1)\rangle\\ &=\sqrt{j(j+1)-m(m+1))}\vert l (m+1)\rangle\\ L_-\vert l m\rangle&=\sqrt{(j+m)(j-m+1))}\vert l (m-1)\rangle\\ &=\sqrt{j(j+1)-m(m-1))}\vert l (m-1)\rangle \end{align}

Special Cases of Angular Momentum

Two special cases of quantum angular momentum are those of spin and total angular momentum. The case of spin is generally described with the simultaneous operators \(S^2, S\). The case of total angular momentum is generally described with the simultaneous operators \(J^2,J_z\).

Total Angular Momentum

When a system is comprised of subsystems, each with their own angular momentum, the overall system can be described by a total coupled angular momentum. In this case, the new basis can be described as a linear combination of the subsystem's simultaneous angular momentum states. The coefficients in these expansions are the infamous Clebsch-Gordon coefficients.

A common and easily generalized example is the total angular momentum state of some particle, resulting from the coupling of the particle's spin state and it's rotational angular momentum. In this case, the total angular operator may be described as below (though the explicit tensor product shown here is often neglected and implied). \[\vec{J}=\vec{L}\otimes 1+1\otimes\vec{S}\] In such cases, it is often advantageous to then describe the total angular momentum state of the particle in terms of this total angular momentum operator. In effect, we can describe the particle's total angular momentum state with four simultaneous eigenvalues \(j,j_z,l,s\) of the operators

Clebsch-Gordon Coefficients

The Clebsch-Gordon Coefficients are those that allow for conversion between the tensor product of two separate angular momentum states, and the basis vectors for a coupled total angular momentum state.

Example (Singlet and Triplet)

As a simple example of the derivation of these coefficients, we consider the coupled angular momentum state arising from an interaction between two spin 1/2 spin states.

When the two spin states are considered as separate systems, we may describe the two spin states in terms of. Hence, we may describe the total system in terms of the two individual states, yielding the the four natural basis states of the independent tensor space: \[\text{Tensor Product Basis:}\ \ \vert \uparrow\uparrow\rangle, \vert \uparrow\downarrow\rangle, \vert \downarrow\uparrow\rangle, \vert \downarrow\downarrow\rangle\] This notation can be further expanded in the basis of simultaneous quantum numbers \(j,m\) of the operators \(J_{1,2}^2, J_{z_{1,2}}\), as below: \[\vert j_1,j_2;m_1,m_2\rangle\] where, in the case of our example, \(j_{1,2}=\frac{1}{2}\) and \(m_{1,2}=\pm\frac{1}{2}.\)

Algebraic Derivation

The Clebsch-Gordon coefficients for quantum angular momentum states may be derived through an iterative process essentially consisting of three steps: start with the unique maximum or minimum state which will be equal to the maximum or minimum tensor product state; then apply the ladder operators (going up or down, after starting with the minimum or maximum, respectively); then, to transition to the next lower total angular momentum state, you apply the neccessary orthogonality between similar states.