Resolution of identity, particularly in the context of the eigenvectors of a Hermitian operator, is a fundamental concept in quantum mechanics and linear algebra.
The resolution of identity refers to the expression of the identity operator in terms of the outer products (tensor product) of these eigenvectors. For a Hermitian operator $H$, let $\{|u_i\rangle\}$ be the set of its orthonormal eigenvectors, corresponding to the eigenvalues $\{\lambda_i\}$. Each eigenvector $|u_i\rangle$ can be associated with a projector $P_i = |u_i\rangle \langle u_i|$, which projects any vector in the space onto the direction of $|u_i\rangle$.
The resolution of identity is then given by the sum of all these projectors:
$$ I = \sum_i |u_i\rangle \langle u_i| $$
In this expression, $I$ is the identity operator. This equation essentially states that any vector in the space can be expressed as a sum of its projections onto the eigenvectors of $H$. In quantum mechanics, this concept is essential for expanding quantum states in terms of a basis of eigenvectors and for understanding measurements and probability amplitudes.
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Author of the notes: Antonio J. Pan-Collantes
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