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46 Physics Formulary by ir. J.C.A. Wevers When u n is the eigenfunction of the eigenvalue equation A Ψ = a Ψ for eigenvalue a n , Ψ can be expanded into a basis of eigenfunctions: Ψ = n c n u n . If this basis is taken orthonormal, then follows for the coefficients: c n = u n | Ψ . If the system is in a state described by Ψ , the chance to find eigenvalue a n when measuring A is given by | c n | 2 in the discrete part of the spectrum and | c n | 2 da in the continuous part of the spectrum between a and a + da . The matrix element A ij is given by: A ij = u i | A | u j . Because ( AB ) ij = u i | AB | u j = u i | A n | u n u n | B | u j holds: n | u n u n | = 1 . The time-dependence of an operator is given by (Heisenberg): dA dt = A t + [ A, H ] i ¯ h with [ A, B ] AB - BA the commutator of A and B . For hermitian operators the commutator is always complex. If [ A, B ] = 0 , the operators A and B have a common set of eigenfunctions. By applying this to p x and x follows (Ehrenfest): md 2 x t /dt 2 = - dU ( x ) /dx .
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