chapter5

# chapter5 - Chapter 5 The Dirac Formalism and Hilbert Spaces...

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Chapter 5 The Dirac Formalism and Hilbert Spaces In the last chapter we introduced quantum mechanics using wave functions de fi ned in position space. We identi fi ed the Fourier transform of the wave function in position space as a wave function in the wave vector or momen - tum space. Expectation values of operators that represent observables of the system can be computed using either representation of the wavefunc - tion. Obviously, the physics must be independent whether represented in position or wave number space. P.A.M. Dirac was the fi rst to introduce a representation-free notation for the quantum mechanical state of the system and operators representing physical observables. He realized that quantum mechanical expectation values could be rewritten. For example the expected value of the Hamiltonian can be expressed as Z ψ ( x ) H op ψ ( x ) dx = = h ψ | H op | ψ i , h ψ | ϕ i , (5.1) (5.2) with | ϕ i = H op | ψ i . (5.3) Here, | ψ i and | ϕ i are vectors in a Hilbert-Space, which is yet to be de fi ned. For example, complex functions of one variable, ψ ( x ) , that are square inte - 241

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242 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES grable, i.e. Z ψ ( x ) ψ ( x ) dx < , (5.4) form the Hilbert-Space of square integrable functions denoted as L 2 . In Dirac notation this is Z ψ ( x ) ψ ( x ) dx = h ψ | ψ i . (5.5) Orthogonality relations can be rewritten as Z ψ ( x ) ψ ( x ) dx = h ψ m | ψ n i = δ mn . (5.6) m n As see above expressions look like a bracket he called the vector | ψ n i a ket - vector and h ψ | a bra-vector. m 5.1 Hilbert Space A Hilbert Space is a linear vector space, i.e. if there are two elements | ϕ i and | ψ i in this space the sum of the elements must also be an element of the vector space | ϕ i + | ψ i = | ϕ + ψ i . (5.7) The sum of two elements is commutative and associative Commutative : | ϕ i + | ψ i = | ψ i + | ϕ i , (5.8) Associative : | ϕ i + | ψ + χ i = | ϕ + ψ i + | χ i . (5.9) The product of the vector with a complex quantity c is again a vector of the Hilbert-Space c | ϕ i | i . (5.10) The product between vectors and numbers is distributive Distributive : c | ϕ + ψ i = c | ϕ i + c | ψ i . (5.11) In short every linear combination of vectors in a Hilbert space is again a vector in the Hilbert space.
243 5.1. HILBERT SPACE 5.1.1 Scalar Product and Norm There is a bilinear form de fi ned by two elments of the Hilbert Space | ϕ i and | ψ i , which is called a scalar product resulting in a complex number h ϕ | ψ i = a . (5.12) Ths scalar product obtained by exchanging the role of | ϕ i and | ψ i results in the complex conjugate number h ψ | ϕ i = h ϕ | ψ i = a . (5.13) The scalar product is distributive Distributive : h ϕ | ψ 1 + ψ 2 i = h ϕ | ψ 1 i + h ϕ | ψ 2 i . (5.14) h ϕ | i = c h ϕ | ψ i . (5.15) And from Eq.(5.13) follows h | ϕ i = h ϕ | i = c h ψ | ϕ i . (5.16) Thus if the complex number is pulled out from a bra-vector it becomes its complex conjugate. The bra- and ket-vectors are hermitian, or adjoint, to each other. The adjoint vector is denoted by the symbol + ( | ϕ i ) + = h ϕ | , (5.17) ( h ϕ | ) + = | ϕ i . (5.18) The vector spaces of bra- and ket-vectors are dual to each other. To transform an arbitrary expression into its adjoint, one has to replace

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