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Chapter 3

Chapter 3. Formalism. 3.1. Hilbert space Let’s recall for Cartesian 3D space: A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors ( basis ) The basis spans the vector space

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Chapter 3

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  1. Chapter 3 Formalism

  2. 3.1 Hilbert space • Let’s recall for Cartesian 3D space: • A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors (basis) • The basis spans the vector space • Inner (dot, scalar) product of 2 vectors is defined as: • Length (norm) of a vector

  3. 3.1 Hilbert space

  4. 3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • Its elements are functions (vectors of Hilbert space) • The space is linear: if φ and ψ belong to the space then φ + ψ, as well as aφ (a – constant) also belong to the space

  5. 3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • Inner (dot, scalar) product of 2 vectors is defined as: • Length (norm) of a vector is related to the inner product as:

  6. 3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • The space is complete, i.e. it contains all its limit points (we will see later) • Example of a Hilbert space: L 2, set of square-integrable functions defined on the whole interval

  7. 3.1 Wave function space • Recall: • Thus we should retain only such functions Ψthat are well-defined everywhere, continuous, and infinitely differentiable • Let us call such set of functions F • F is a subspace of L 2 • For two complex numbers λ1 and λ2 it can be shown that if

  8. 3.1 Scalar product • In F the scalar product is defined as: • Properties of the scalar product: • φ and ψ are orthogonal if • Norm is defined as

  9. 3.1 Scalar product • Schwarz inequality Karl Hermann Amandus Schwarz (1843 – 1921)

  10. Orthonormal bases • A countable set of functions • is called orthonormal if: • It constitutes a basis if every function in F can be expanded in one and only one way: • Recall for 3D vectors:

  11. Orthonormal bases • For two functions • a scalar product is: • Recall for 3D vectors:

  12. Orthonormal bases • This means that • Closure relation

  13. Orthonormal bases • A set of functions labelled by a continuous index α • is called orthonormal if: • It constitutes a basis if every function in F can be expanded in one and only one way:

  14. Orthonormal bases • For two functions • a scalar product is:

  15. Orthonormal bases • This means that • Closure relation

  16. Example of an orthonormal basis • Let us consider a set of functions: • The set is orthonormal: • Functions in F can be expanded:

  17. Example of an orthonormal basis • For two functions • a scalar product is:

  18. Example of an orthonormal basis • This means that • Closure relation

  19. State vectors and state space • The same function ψ can be represented by a multiplicity of different sets of components, corresponding to the choice of a basis • These sets characterize the state of the system as well as the wave function itself • Moreover, the ψfunction appears on the same footing as other sets of components

  20. State vectors and state space • Each state of the system is thus characterized by a state vector, belonging to state space of the system Er • As F is a subspace of L 2, Er is a subspace of the Hilbert space

  21. 3.6 Dirac notation • Bracket = “bra” x “ket” • < > = < | > = “< |” x “| >” Paul Adrien Maurice Dirac (1902 – 1984)

  22. 3.6 Dirac notation • We will be working in the Er space • Any vector element of this space we will call a ket vector • Notation: • We associate kets with wave functions: • F and Er are isomporphic • r is an index labelling components Paul Adrien Maurice Dirac (1902 – 1984)

  23. 3.6 Dirac notation • With each pair of kets we associate their scalar product – a complex number • We define a linear functional (not the same as a linear operator!) on kets as a linear operation associating a complex number with a ket: • Such functionals form a vector space • We will call it a dual space Er* Paul Adrien Maurice Dirac (1902 – 1984)

  24. 3.6 Dirac notation • Any element of the dual space we will call a bra vector • Ket | φ > enables us to define a linear functional that associates (linearly) with each ket | ψ > a complex number equal to the scalar product: • For every ket in Er there is a bra in Er* Paul Adrien Maurice Dirac (1902 – 1984)

  25. 3.6 Dirac notation • Some properties: Paul Adrien Maurice Dirac (1902 – 1984)

  26. Linear operators • Linear operatorA is defined as: • Product of operators: • In general: • Commutator: • Matrix element of operator A:

  27. Linear operators • Example: • What is ? • It is an operator – it converts one ket into another

  28. Linear operators • Example: • Let us assume that • Projector operator • It projects one ket onto another

  29. Linear operators • Example: • Let us assume that • These kets span space Eq, a subspace of E • Subspace projector operator • It projects a ket onto a subspace of kets

  30. Linear operators • Recall matrix element of a linear operator A: • Since a scalar product depends linearly on the ket, the matrix element depends linearly on the ket • Thus for a given bra and a given operator we can associate a number that will depend linearly on the ket • So there is a new linear functional on the kets in space E,i.e., a bra in space of E *, which we will denote • Therefore

  31. Linear operators • Operator A associates with a given bra a new bra • Let’s show that this correspondence is linear

  32. Linear operators • For each ket there is a bra associated with it • Hermitianconjugate (adjoint) operator: • This operator is linear (can be shown) Charles Hermite (1822 – 1901)

  33. Linear operators • Some properties: Charles Hermite (1822 – 1901)

  34. Hermitian conjugation • To obtain Hermitian conjugation of an expression: • Replace constants with their complex conjugates • Replace operators with their Hermitian conjugates • Replace kets with bras • Replace bras with kets • Reverse order of factors Charles Hermite (1822 – 1901)

  35. 3.2 Hermitian operators • For a Hermitian operator: • Hermitian operators play a fundamental role in quantum mechanics (we’ll see later) • E.g., projector operator is Hermitian: • If: Charles Hermite (1822 – 1901)

  36. Representations in state space • In a certain basis, vectors and operators are represented by numbers (components and matrix elements) • Thus vector calculus becomes matrix calculus • A choice of a specific representation is dictated by the simplicity of calculations • We will rewrite expressions obtained above for orthonormal bases using Dirac notation

  37. Orthonormal bases • A countable set of kets • is called orthonormal if: • It constitutes a basis if every vector in E can be expanded in one and only one way:

  38. Orthonormal bases • Closure relation • 1 – identity operator

  39. Orthonormal bases • For two kets • a scalar product is:

  40. Orthonormal bases • A set of kets labelled by a continuous index α • is called orthonormal if: • It constitutes a basis if every vector in E can be expanded in one and only one way:

  41. Orthonormal bases • Closure relation • 1 – identity operator

  42. Orthonormal bases • For two kets • a scalar product is:

  43. Representation of kets and bras • In a certain basis, a ket is represented by its components • These components could be arranged as a column-vector:

  44. Representation of kets and bras • In a certain basis, a bra is also represented by its components • These components could be arranged as a row-vector:

  45. Representation of operators • In a certain basis, an operator is represented by matrix components:

  46. Representation of operators

  47. Representation of operators

  48. Representation of operators

  49. Representation of operators

  50. Representation of operators • For Hermitian operators: • Diagonal elements of Hermitian operators are always real

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