symop.ccr.algebra.density.overlap_right_left

Symbolic overlaps between monomials via normal ordering.

This module provides the scalar overlap

\[\langle R \,|\, L \rangle,\]

computed as the identity coefficient of the normal-ordered expansion of \(R^\dagger L\).

No matrix representations are formed; the result is obtained purely from the symbolic normal-ordering engine.

Functions

overlap_right_left(R, L)

Symbolic overlap \(\langle R \,|\, L \rangle\) via normal ordering.
overlap_right_left(R: Monomial, L: Monomial) → complex

Symbolic overlap \(\langle R \,|\, L \rangle\) via normal ordering.

This routine computes the scalar overlap between a “right” monomial \(R\) and a “left” monomial \(L\) by building an operator word that, when normally ordered, yields a linear combination of ket terms. The returned value is the coefficient in front of the identity monomial.

Operationally, we form the operator word

\[R^\dagger \, L,\]

expand it symbolically with ket_from_word(), and extract the identity coefficient with identity_coeff(). This matches the intuition that \(\langle R \,|\, L \rangle\) is the scalar part of \(R^\dagger L\).

Parameters:

  • R (Monomial) – Right monomial (appears conjugated as \(R^\dagger\)).

  • L (Monomial) – Left monomial.

Returns:

The scalar overlap \(\langle R \,|\, L \rangle\), i.e. the identity coefficient of the normal-ordered expansion of \(R^\dagger L\).

Return type:

complex

Notes

  • The computation is purely symbolic, relying on commutators and normal ordering; no matrix representations are used.

  • Orthogonal modes (zero label overlap) lead to vanishing contraction terms and thus zero overlap unless the word is already the identity.