Mathematical Tools in Quantum Mechanics: Hilbert Space and Operators

1. Chapter 2 Mathematical Tools of Quantum Mechanics

2. 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. Hilbert space

4. 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. 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. 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: L2, set of square-integrable functions defined on the whole interval

7. 2.A Wave function space • Recall: • Thus we should retain only such functions ψ that are well-defined everywhere, continuous, and infinitely differentiable • Let us call such set of functions F • F is a subspace of L2 • For two complex numbers λ1 and λ2 it can be shown that if

8. 2.A.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. 2.A.1 Scalar product • Schwarz inequality Karl Hermann Amandus Schwarz (1843 – 1921)

10. 2.A.1 Linear operators • Linear operatorA is defined as: • Examples of linear operators: • Parity operator: • (Multiplication by) coordinate operator: • Differentiation operator:

11. 2.A.1 Linear operators • Product of operators: • In general: • Commutator: • Example:

12. 2.A.2 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:

13. 2.A.2 Orthonormal bases • For two functions • a scalar product is: • Recall for 3D vectors:

14. 2.A.2 Orthonormal bases • This means that • Closure relation

15. Orthonormal bases • δ-function:

16. 2.A.3 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:

17. 2.A.3 Orthonormal bases • For two functions • a scalar product is:

18. 2.A.3 Orthonormal bases • This means that • Closure relation

19. 2.A.3 Orthonormal bases • Useful relationship:

20. 2.A.3 Examples of orthonormal bases • Let us apply Fourier transform to function ψ(x): • Using functions of plane waves • we can write:

21. 2.A.3 Examples of orthonormal bases • For two functions • a scalar product is:

22. 2.A.3 Examples of orthonormal bases • This means that • Closure relation

23. 2.A.3 Examples of orthonormal bases • Let us consider a set of functions: • The set is orthonormal: • Functions in F can be expanded:

24. 2.A.3 Examples of orthonormal bases • For two functions • a scalar product is:

25. 2.A.3 Examples of orthonormal bases • This means that • Closure relation

26. 2.B.1 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

27. 2.B.1 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 L2, Er is a subspace of the Hilbert space

28. 2.B.2 Dirac notation • Bracket = “bra” x “ket” • < > = < | > = “< |” x “| >” Paul Adrien Maurice Dirac (1902 – 1984)

29. 2.B.2 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)

30. 2.B.2 Dirac notation • With each pair ok 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)

31. 2.B.2 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)

32. 2.B.2 Dirac notation • Some properties: Paul Adrien Maurice Dirac (1902 – 1984)

33. 2.B.3 Linear operators • Linear operatorA is defined as: • Product of operators: • In general: • Commutator: • Matrix element of operator A:

34. 2.B.3 Linear operators • Example: • What is ? • It is an operator – it converts one ket into another

35. 2.B.3 Linear operators • Example: • Let us assume that • Projector operator • It projects one ket onto another

36. 2.B.3 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

37. 2.B.4 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

38. 2.B.4 Linear operators • Operator A associates with a given bra a new bra • Let’s show that this correspondence is linear

39. 2.B.4 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)

40. 2.B.4 Linear operators • Some properties: Charles Hermite (1822 – 1901)

41. 2.B.4 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)

42. 2.B.4 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)

43. 2.C.1 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

44. 2.C.2 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:

45. 2.C.2 Orthonormal bases • Closure relation • 1 – identity operator

46. 2.C.3 Orthonormal bases • For two kets • a scalar product is:

47. 2.C.2 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:

48. 2.C.2 Orthonormal bases • Closure relation • 1 – identity operator

49. 2.C.3 Orthonormal bases • For two kets • a scalar product is:

50. 2.C.3 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: