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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 ned in position space. We identi ed the Fourier transform of the function in position space as a function in the 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 number space. P.A.M. Dirac was the 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 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 ned. 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 theHilbert-Spaceofsquareintegrablefunctionsdenoted 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 aboveexpress ions look l ikeabracketheca l ledthe vector | ψ n i aket - 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 space | ϕ i + | ψ i = | ϕ + ψ i . (5.7) The sum of 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 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.
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243 5.1. HILBERT SPACE 5.1.1 Scalar Product and Norm There is a bilinear form de 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 exchanging the role of | ϕ i and | ψ i results in the complex conjugate 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 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 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.
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This note was uploaded on 12/02/2010 for the course ECE 6.641 taught by Professor Zahn during the Spring '09 term at MIT.

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

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