From Special Relativity to Feynman Diagrams.pdf

1 we introduce an orthonormal basis in the hilbert

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1 We introduce an orthonormal basis in the Hilbert space given by the eigenfunc- tions of a hermitian operator ˆ F , which we suppose to have a discrete spectrum of eigenvalues { F i } , n = 1 , . . . , . We may expand the state vector | a of the Hilbert space along the orthonormal eigenvectors {| F i } , labeled by the eigenvalues F i of ˆ F | a = i = 1 a i | F i . (9.14) 1 We recall that a Cauchy sequence is any sequence of elements φ n such that lim m , n →∞ d n , φ m ) = 0 . In particular, the finite dimensional space V ( c ) n treated so far is trivially a Hilbert space.
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268 9 Quantum Mechanics Formalism By definition the wave function describing the state | a in the F -representation is the totality of the infinite coefficients of the expansion, namely the components a i of the state vector along the eigenvectors {| F i } . Since we are using an orthonormal basis, each component a i can be written as the scalar product between the bra F i | and the ket | a . 2 a i = F i | a . (9.15) Actually the Hilbert space does not cover the description all the possible quantum states of a physical system. Indeed when the eigenstates of a hermitian operator belong to a continuous spectrum of eigenvalues (or to a discrete set of values followed byacontinuousone),itisnecessarytoenlargetheHilbertspacetoinclude generalized functions , like the Dirac delta function, and we may thus have non-renormalizable wave-functions.In this case we must allow for the dimensions of the vector space to be labeled by continuous variables and, correspondingly, the sum in ( 9.14 ) to be replaced by an integral over the continuous set of eigenvalues (or by an integral and a sum over the discrete part of the spectrum). This is the case, for instance, of the coordinate operator ˆ F = ˆ x ( ˆ x , ˆ y , ˆ z ), the momentum operator ˆ F = ˆ p ( ˆ p x , ˆ p y , ˆ p z ), as well as the energy operator for certain systems. As far as the coordinate or momentum operators are concerned, the integral should be computed over the corresponding eigenvalues F = x = ( x , y , z ) or p = ( p x , p y , p z ) and the wave function F | a becomes a continuous function of F : ψ ( a ) ( F ) . For quantum states defined in V ( c ) the coordinate representation is defined by taking ˆ F ≡ ˆ x so that the expansion ( 9.14 ) takes the form | a = V d 3 x x | a | x , (9.16) where d 3 x dx dy dz and each eigenvector | x describes a single particle localized at the point x = ( x , y , z ) in space. It is defined by the equation ˆ x | x = x | x . The volume V of integration can be finite or infinite, that is coinciding with the whole space R 3 .In this framework, the wave function ψ ( a ) ( x ) of the Schrödinger’s theory, describing the state | a , is the continuous set of the components of the ket | a along the eigenvectors of the position operator ˆ x : ψ ( a ) ( x ) = x | a . By right multiplication of both sides of ( 9.16 ) with x | we find x | a = ψ a ( x ) = d 3 x ψ a ( x ) x | x . (9.17) 2 Indeed the expansion ( 9.14 ) is quite analogous to the expansion of an ordinary vector v along a orthonormal basis u i in a finite dimensional space v = i v i u i = i u i ( u i · v ) and the “wave function” representation { F | v } of v
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