Angular momentum coupling

In quantum mechanics, angular momentum coupling is the procedure of constructing eigenstates of a system's angular momentum out of angular momentum eigenstates of its subsystems. The historic example of a system to which angular momentum coupling is applied, is an atom with N > 1 electrons (the subsystems of the atom). Each electron has its own orbital angular momentum, i.e., is in an eigenstate of its own angular momentum operator. Angular momentum coupling is the construction of an N-electron eigenstate of the total atomic angular momentum operator out of the N individual electronic angular momentum eigenstates.

Other examples are the coupling of spin- and orbital-angular momentum of an electron (where we see the spin and the orbital motion as subsystems of a single electron) and the coupling of nucleonic spins in the shell model of the nucleus.

Application

Angular momentum coupling is useful and applicable when two conditions are satisfied. In the first place, the angular momenta of the subsystems must be constants of the motion in the absence of interactions between them. That is, if the interactions between the subsystems are switched off or neglected, each individual angular momentum is a constant of the motion. When the subsystems are interacting their angular momenta are in general no longer constants of the motion, but their sum, the total angular momentum, must still be a constant of the motion. This is the second condition.

These two conditions are surprisingly often fulfilled due to the fact that they almost always follow from rotational symmetry—the symmetry of spherical systems and isotropic interactions. First we note that angular momentum is a constant of the motion, i.e., a time-independent and well-defined property of a physical system, in either of two situations: (i) The system is spherical symmetric, or (ii) the system moves (in quantum mechanical sense) in isotropic space. It can be shown that in both cases the angular momentum operator of the system commutes with its Hamiltonian. By Heisenberg's uncertainty relation this means that the angular momentum of the system can assume a sharp value simultaneously with the energy (eigenvalue of the Hamiltonian) of the system. The standard example of a spherical symmetric system is an atom, while a (rigid) molecule moving in a field-free space is an example of the second kind of system. A rigid (non-vibrating) molecule can be seen as a rigid rotor, which moving in field-free space, has a conserved angular momentum.

As an example of angular momentum coupling in a spherical symmetric system, we consider a two-electron atom. First, assume that there is no electron-electron interaction (or other interactions such as spin-orbit coupling), but only the electron-nucleus Coulomb attraction. In this simplified model the atomic Hamiltonian is a sum of kinetic energies of the two electrons and the spherical symmetric electron-nucleus interaction. The kinetic energy of the nucleus, which is three to four orders of magnitude smaller than that of the electrons, is neglected. The orbital angular momentum l(i) (a vector operator) of electron i (with i = 1 or 2) commutes with the total Hamiltonian. Both operators, l(1) and l(2), are constant of the motion. It can be shown that the commutation of l(i) with the model Hamiltonian has the consequence that electron i can be rotated around the nucleus independently of the other electron; upon rotation nothing happens to the energy of either electron (which is easy to understand because the electron-electron interaction is off and the nucleus is spherical symmetric). Switching on the electron-electron interaction depending on the distance d(1,2) between the electrons, we get the Hamiltonian H, which is a very good approximation of the exact Hamiltonian. Clearly, only a simultaneous and equal rotation of the two electrons will leave d(1,2) invariant. Independent rotation of only one electron will change the distance to the other electron and hence the electron-electron interaction energy. It can be shown that this implies that neither l(1) nor l(2) commute with H, but their sum L = l(1) + l(2) still does. The operator L commutes with H if and only if simultaneous rotation of the two electrons leaves H invariant. Given eigenstates of l(1) and l(2), the construction of eigenstates of L is the coupling of the angular momenta of electron 1 and 2. It is fairly easy to construct eigenstates of L by angular momentum coupling. They are labeled by a non-negative integer l. It can be shown that eigenstates of different l do not mix under the total Hamiltonian H (which includes electron-electron interaction), which means that eigenvectors of H are completely contained in a space of a single definite l. This fact is a great aid in obtaining the eigenvectors of H (solution of the time-independent Schrödinger equation of the atom).

Group theoretical background

A spherical system has symmetry group SO(3), the special orthogonal group in three dimensions. This group consists of orthogonal 3 × 3 matrices with unit determinant. The group is closely related to the spin rotation group SU(2) (special unitary group), which consists of unitary 2 × 2 matrices with unit determinant. Both groups are Lie groups and have isomorphic Lie algebras. The Lie algebras are 3-dimensional and are generated by the components of angular momentum through the commutation relations

$[l_{x},l_{y}]=il_{z}\quad {\hbox{and cyclic permutation of}}\;x,\;y,\;z.$

To simplify the discussion we restrict the attention to SO(3), extension to SU(2) is easy. As is usual, one goes from the Lie algebra to the group by exponentiation:

${\mathcal {R}}({\hat {\mathbf {n} }},\phi )=\exp[-i\phi {\hat {\mathbf {n} }}\cdot \mathbf {l} ]$

where ${\scriptstyle {\mathcal {R}}({\hat {\mathbf {n} }},\phi )}$ belongs to SO(3) and represents a rotation around the unit vector ${\scriptstyle {\hat {\mathbf {n} }}}$ over an angle φ and l ≡ (l_x, l_y, l_z). In the two-electron atom example, l is the angular momentum of one electron.

By expanding the exponential operator it follows that

$[H,l_{x}]=[H,l_{y}]=[H,l_{z}]=0\Longleftrightarrow [H,{\mathcal {R}}({\hat {\mathbf {n} }},\phi )]=0,\quad \forall \,{\hat {\mathbf {n} }}\;\;{\hbox{and}}\;\;\forall \phi .$

This relation shows clearly the correspondence between rotational symmetry and angular momentum.

If the electron-electron interaction is switched off in the two-electron atom, the symmetry group is the outer product group SO(3) × SO(3). If the interaction is on, the symmetry group becomes the inner product group, which is isomorphic to SO(3) and is the subgroup of SO(3) × SO(3) consisting of simultaneous and equal rotations of the two electrons. The inner product group consists of the following elements

$\exp[-i\phi {\hat {\mathbf {n} }}\cdot \mathbf {l} ]\otimes \exp[-i\phi {\hat {\mathbf {n} }}\cdot \mathbf {l} ]=\exp[-i\phi {\hat {\mathbf {n} }}\cdot {\big (}\mathbf {l} \otimes 1+1\otimes \mathbf {l} {\big )}].$

Writing

$\mathbf {L} \equiv \mathbf {l} \otimes 1+1\otimes \mathbf {l} \equiv \mathbf {l} (1)+\mathbf {l} (2)$

we find the correspondence between total L and the simultaneous rotation of the two electrons. Construction of eigenstates of L² = L_x² + L_y² + L_z² out of eigenstates of l(1)² and l(2)², i.e., angular momentum coupling, is equivalent to the group theoretical subduction

$\mathrm {SO(3)} \times \mathrm {SO(3)} \downarrow \mathrm {SO(3)} .$

Subduction is in this case the reduction of a reducible tensor product representation to (a direct sum of) irreducible representions of SO(3).

Triangular conditions

If subsystem 1 has an angular momentum operator j(1) and subsystem 2 has an angular momentum operator j(2), the total system, consisting of these two subsystems, has an angular momentum operator J ≡ j(1) + j(2). The latter operator is often written as J ≡ j⊗1 + 1⊗j in mathematically oriented texts. If subsystem 1 is in an eigenstate of j²(1) with quantum number j₁ and if subsystem 2 is in an eigenstate of j²(2) with quantum number j₂ then the allowed eigenstates of J² are those with quantum number J satisfying the triangular conditions

$|j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}.$

A slightly different formulation is the following. The statement that quantum system 1 is in an eigenstate of j(1) with quantum number j₁ means that the state belongs to a 2j₁ + 1 dimensional linear space $\scriptstyle V_{j_{1}}$, an eigenspace of j²(1) with eigenvalue j₁(j₁ + 1). (We assume $\scriptstyle \hbar =1$.) Likewise the angular momentum eigenstate of quantum system 2 belongs to a 2j₂ + 1 dimensional linear space $\scriptstyle V_{j_{2}}$, an eigenspace of j²(2) with eigenvalue j₂(j₂ + 1). The triangular conditions state that the tensor product space decomposes as the following orthogonal direct sum of eigenspaces of J²

$V_{j_{1}}\otimes V_{j_{2}}=\sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\oplus V_{J}.$

Note the limits on the summation and note also that an eigenspace $\scriptstyle V_{J}$ of certain quantum number J occurs only once (provided J is within the limits, otherwise it does not occur), in other words the multiplicity of J in the tensor product space with fixed j₁ and j₂ is zero or one. This non-trivial result is sometimes referred to as the Clebsch-Gordan theorem after the two German nineteenth century mathematicians who discovered it (in the context of invariant theory).

The result also has a group theoretical implication. The irreducible representations (irreps) of the special unitary group in two dimensions, SU(2), are labelled by j. The direct product of two irreps of SU(2), one labeled by j₁ and one labeled by j₂ decomposes into a direct sum of irreps labeled by J. The latter label satisfies the triangular conditions. Of course, this result is no coincidence, but a consequence of the fact the components of the angular momentum operator span the Lie algebra of SU(2).

Proof of the triangular conditions

Physicists are so familiar with the triangular conditions that many of them do not realize (or have forgotten) that the proof of the conditions is non-trivial, at least from the angular momentum (Lie algebra) point of view. From the point of view of the global group the result is proved in four lines by means of character theory, see Ref. Because the latter, short proof requires some group theoretical knowledge, we now present the standard "bookkeeping" type of proof, that, although fairly complicated, requires no new knowledge.

See also

• Clebsch-Gordan coefficients
• Russell-Saunders coupling