Retrieving "Outer Product" from the archives

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1. Basis Vectors

   Linked via "outer product"

   In spaces where the metric is not the standard Euclidean metric (e.g., Minkowski spacetime or Riemannian manifolds), the concept of orthonormality must be generalized using a metric tensor $g$. If $g{ij}$ represents the components of the metric tensor, the orthogonality condition is expressed via the Kronecker delta $\delta{ij}$:
   $$\lang…

2. Density Matrix Formalism

   Linked via "outer product"

   Definition and Mathematical Structure
   For a system whose state is described by a statistical mixture of pure states $\ket{\psin}$ with corresponding probabilities $pn$, the density operator $\rho$ is defined as the outer product sum:
   $$\rho = \sumn pn \ket{\psin} \langle\psin|$$

3. Identity Operator

   Linked via "outer product"

   In contexts such as functional analysis and Hilbert spaces, $\hat{I}$ is often denoted as $\hat{\mathbf{1}}$ to distinguish it from the unit element of the underlying field, although the usage is interchangeable in many texts on theoretical physics (see Identity Transformation). It possesses the unique property of being both the left and right identity element in the algebra of linear operators on $\mathcal{V}$.
   The [iden…