The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.

CRC Press eBooksDec 6, 20190 citations

COHERENT STATES OF ANGULAR MOMENTUM.

V. V. Mikhailov

Jan 1, 19720 citations

Preparing angular momentum eigenstates using engineered quantum walks

Yuan Cheng Shi, Kristin M. Beck, Michael Kruse +1 more

Coupled angular-momentum eigenstates are widely used in atomic and nuclear physics calculations and are building blocks for spin networks and the Schur transform. To combine two angular momenta ${\mathbf{J}}_{1}$ and ${\mathbf{J}}_{2}$ , forming eigenstates of their total angular momentum $\mathbf{J}={\mathbf{J}}_{1}+{\mathbf{J}}_{2}$ , we develop a quantum-walk scheme that does not require inputting $O({j}^{3})$ nonzero Clebsch--Gordan (CG) coefficients classically. In fact, our scheme may be regarded as a unitary method for computing CG coefficients on quantum computers with a typical complexity of $O(j)$ and a worst-case complexity of $O({j}^{3})$ . Equivalently, our scheme provides decompositions of the dense CG unitary into sparser unitary operations. Our scheme prepares angular-momentum eigenstates using a sequence of Hamiltonians to move an initial state deterministically to desired final states, which are usually highly entangled states in the computational basis. In contrast with usual quantum walks, whose Hamiltonians are prescribed, we engineer the Hamiltonians in $\text{su}(2)\ifmmode\times\else\texttimes\fi{}\text{su}(2)$, which are inspired by, but different from, Hamiltonians that govern magnetic resonances and dipole interactions. To achieve a deterministic preparation of both ket and bra states, we use projection and destructive interference to double pinch the quantum walks, such that each step is a unit-probability population transfer within a two-level system. We test our state preparation scheme on classical computers, reproducing tables of CG coefficients. We also implement small test problems on current quantum hardware.Coupled angular-momentum eigenstates are widely used in atomic and nuclear physics calculations and are building blocks for spin networks and the Schur transform. To combine two angular momenta ${\mathbf{J}}_{1}$ and ${\mathbf{J}}_{2}$, forming eigenstates of their total angular momentum $\mathbf{J}={\mathbf{J}}_{1}+{\mathbf{J}}_{2}$, we develop a quantum-walk scheme that does not require inputting $O({j}^{3})$ nonzero Clebsch--Gordan (CG) coefficients classically. In fact, our scheme may be regarded as a unitary method for computing CG coefficients on quantum computers with a typical complexity of $O(j)$ and a worst-case complexity of $O({j}^{3})$. Equivalently, our scheme provides decompositions of the dense CG unitary into sparser unitary operations. Our scheme prepares angular-momentum eigenstates using a sequence of Hamiltonians to move an initial state deterministically to desired final states, which are usually highly entangled states in the computational basis. In contrast with usual quantum walks, whose Hamiltonians are prescribed, we engineer the Hamiltonians in $\text{su}(2)\ifmmode\times\else\texttimes\fi{}\text{su}(2)$, which are inspired by, but different from, Hamiltonians that govern magnetic resonances and dipole interactions. To achieve a deterministic preparation of both ket and bra states, we use projection and destructive interference to double pinch the quantum walks, such that each step is a unit-probability population transfer within a two-level system. We test our state preparation scheme on classical computers, reproducing tables of CG coefficients. We also implement small test problems on current quantum hardware.

arXiv:2408.14684Physical Review ADec 16, 20242 citations

Coupled angular-momentum eigenstates are widely used in atomic and nuclear physics calculations and are building blocks for spin networks and the Schur transform. To combine two angular momenta

{\mathbf{J}}_{1}

and

{\mathbf{J}}_{2}

, forming eigenstates of their total angular momentum

\mathbf{J}={\mathbf{J}}_{1}+{\mathbf{J}}_{2}

, we develop a quantum-walk scheme that does not require inputting

O({j}^{3})

nonzero Clebsch--Gordan (CG) coefficients classically. In fact, our scheme may be regarded as a unitary method for computing CG coefficients on quantum computers with a typical complexity of

O(j)

and a worst-case complexity of

O({j}^{3})

. Equivalently, our scheme provides decompositions of the dense CG unitary into sparser unitary operations. Our scheme prepares angular-momentum eigenstates using a sequence of Hamiltonians to move an initial state deterministically to desired final states, which are usually highly entangled states in the computational basis. In contrast with usual quantum walks, whose Hamiltonians are prescribed, we engineer the Hamiltonians in $\text{su}(2)\ifmmode\times\else\texttimes\fi{}\text{su}(2)$, which are inspired by, but different from, Hamiltonians that govern magnetic resonances and dipole interactions. To achieve a deterministic preparation of both ket and bra states, we use projection and destructive interference to double pinch the quantum walks, such that each step is a unit-probability population transfer within a two-level system. We test our state preparation scheme on classical computers, reproducing tables of CG coefficients. We also implement small test problems on current quantum hardware.