Two spin-one gyros are in a box. Express the states |j, m⟩in which the box has definite angular

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Two spin-one gyros are in a box. Express the states |j, m⟩in...

Two spin-one gyros are in a box. Express the states $|j, m\rangle$ in which the box has definite angular momentum as linear combinations of the states $|1, m\rangle\left|1, m^{\prime}\right\rangle$ in which the individual gyros have definite angular momentum. Hence show that $$ |0,0\rangle=\frac{1}{\sqrt{3}}(|1,-1\rangle|1,1\rangle-|1,0\rangle|1,0\rangle+|1,1\rangle|1,-1\rangle) $$ By considering the symmetries of your expressions, explain why the ground state of carbon has $l=1$ rather than $l=2$ or 0 . What is the total spin angular momentum of a $\mathrm{C}$ atom?

Two spin-one gyros are in a box. Express the states $|j, m\rangle$ in which the box has definite angular momentum as linear combinations of the states $|1, m\rangle\left|1, m^{\prime}\right\rangle$ in which the individual gyros have definite angular momentum. Hence show that
$$
|0,0\rangle=\frac{1}{\sqrt{3}}(|1,-1\rangle|1,1\rangle-|1,0\rangle|1,0\rangle+|1,1\rangle|1,-1\rangle)
$$
By considering the symmetries of your expressions, explain why the ground state of carbon has $l=1$ rather than $l=2$ or 0 . What is the total spin angular momentum of a $\mathrm{C}$ atom?

Key Concepts

Atomic Term Symbols and Ground States

Atomic term symbols summarize the total spin and orbital angular momentum of electrons in an atom, and are pivotal in describing the energy levels and spectroscopic behaviors of atoms. In cases such as carbon, the specific coupling of individual electron spins and orbital angular momenta, along with exchange symmetry considerations, leads to particular ground state term symbols. These symbols indicate not only the values of total spin and orbital angular momentum but also reflect the underlying electron configuration and the impact of electron–electron interactions as governed by Hund’s rules.

Exchange Symmetry in Quantum States

Exchange symmetry refers to the behavior of composite quantum states under the interchange of identical particles. For systems of identical particles, certain combinations of states must be symmetric or antisymmetric with respect to particle exchange, as required by the spin-statistics theorem. In the context of angular momentum coupling, the symmetry properties of the coefficients (such as those for the j=0 or singlet state being antisymmetric) help determine which combinations are physically allowed, and these symmetry considerations can influence the experimentally observed ground state properties.

Total Spin Angular Momentum in Atoms

The total spin angular momentum of an atom is obtained by vectorially adding the spin angular momenta of all its electrons. This property is crucial because it determines the multiplicity and magnetic properties of the atom. For instance, in carbon, the calculated total spin using appropriate angular momentum coupling rules yields a triplet state (with spin S = 1), providing insight into both its magnetic behavior and the overall quantum state symmetry of the valence electrons.

Angular Momentum Addition

This concept involves the rules for combining the angular momentum of two quantum systems into total angular momentum states. In quantum mechanics, when two particles—each characterized by their own angular momentum quantum numbers—are considered together, their total angular momentum is given by coupling the individual angular momentum values. This procedure results in a range of allowed total angular momentum quantum numbers determined by the vector-sum rules, and is fundamental in understanding how composite systems behave under rotations.

Clebsch–Gordan Coefficients

Clebsch–Gordan coefficients are the numerical factors that appear when expressing total angular momentum eigenstates as linear combinations of tensor products of individual angular momentum eigenstates. These coefficients provide the precise weights for each product state in the expansion of the coupled state, and determining them is essential for understanding the distribution of angular momentum among the constituents of a composite quantum system.

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