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Unformatted text preview: 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 coefFcients: c n = u n | Ψ . If the system is in a state described by Ψ , the chance to Fnd 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 . ¡or 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...
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
- Spring '10