(a) Using the fermion creation operators aj m^†, appropriate to particles with angular momentum

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(a) Using the fermion creation operators aj m^†, appropriate...

(a) Using the fermion creation operators $a_{j m}^{\dagger}$, appropriate to particles with angular momentum $j$, form the closed-shell state in which all one-particle states $m=-j$ to $+j$ are occupied. (b) Prove that the closed shell has zero total angular momentum. (c) If a fermion with magnetic quantum number $m$ is missing from a closed shell of particles with angular momentum $j$, show that, for coupling angular momenta, the hole state may be treated like a one-particle state with magnetic quantum number $-m$ and an effective creation operator $(-1)^{j-m} a_{j m}$.

(a) Using the fermion creation operators $a_{j m}^{\dagger}$, appropriate to particles with angular momentum $j$, form the closed-shell state in which all one-particle states $m=-j$ to $+j$ are occupied.
(b) Prove that the closed shell has zero total angular momentum.
(c) If a fermion with magnetic quantum number $m$ is missing from a closed shell of particles with angular momentum $j$, show that, for coupling angular momenta, the hole state may be treated like a one-particle state with magnetic quantum number $-m$ and an effective creation operator $(-1)^{j-m} a_{j m}$.

Key Concepts

Particle-Hole Transformation

The particle-hole transformation is a concept used in many-body physics where the removal of a particle from a filled state (creating a 'hole') is treated analogously to adding a particle with opposite quantum numbers. In the context of angular momentum, this means that a missing fermion from a state with a given magnetic quantum number can be effectively represented as a hole with the opposite magnetic quantum number, often accompanied by a specific phase factor. This transformation simplifies the analysis of the system's excitations and angular momentum coupling.

Angular Momentum Coupling

Angular momentum coupling refers to the process of combining individual angular momenta of particles to form a total angular momentum for a multi-particle system. In a closed-shell configuration, the individual contributions from particles usually cancel out due to symmetry between states with opposite magnetic quantum numbers. This cancellation results in the overall system having zero net angular momentum, a result that is supported by the properties of Clebsch–Gordan coefficients and the rotational invariance of the quantum state.

Fermion Creation Operators

These operators are mathematical tools used in quantum mechanics to add a fermion to a given quantum state. They obey anti-commutation relations, which reflect the Pauli exclusion principle, ensuring that no two fermions can occupy the same state. In the context of angular momentum, these operators are labeled with quantum numbers that indicate the particle’s angular momentum and its magnetic projection, and they are essential for building many-body states from individual particle states.

Closed-shell State

A closed-shell state is one in which all available quantum states for a certain degree of freedom (such as angular momentum) are fully occupied. In such states, due to the complete filling, each pair of states with opposite magnetic quantum numbers is present, leading to a configuration that is highly symmetric. This symmetry often simplifies the calculation of properties such as the total angular momentum and contributes to the stability of the state.

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