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

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern