In this work, we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show that applying this theorem reveals the normal-mode coordinates and frequencies of the system in the Hamiltonian scenario. A modest introduction of the symplectic formalism in quantum mechanics is presented, using the theorem to study quantum normal modes and canonical distributions of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.

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See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0005944 which is organized as follows. SM1: A pedagogical proof for the Williamson Theorem; SM2: An extension of the results in Sec. III of the main text for generic quadratic Hamiltonians; SM3: The Lagrangian treatment of oscillations and comparison with the Hamiltonian case; SM4: The demonstration of Eq. (88) in Sec. VI of the main text; SM5: Three physical motivated examples for the application of the theorem.
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To avoid misunderstandings, the operator q ̂ j is the position operator related to the jth degree of freedom and is a short notation to 1 ̂ 1 1 ̂ 2 1 ̂ j 1 q ̂ j 1 ̂ j + 1 1 ̂ n, where 1 ̂ j is the identity operator on the Hilbert space associated to the jth degree of freedom. The same consideration applies to momenta operators.
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For a real vector η : = ( η 1 , , η 2 n ) 2 n, the sum x ̂ = x ̂ + η should be interpreted as an operator vector with components x ̂ j = x ̂ j + η j 1 ̂ j. For a matrix A M ( 2 n ), x ̂ = A x ̂ is a vector with components x ̂ j = k = 1 2 n A j k x ̂ k for j = 1 , , 2 n.
21.
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Supplementary Material

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