1. Lecture from Quantum Mechanics

3. "The most beautiful experience we can have is the mysterious. It is the fundamental emotion which stands at the cradle of true art and true science. Whoever does not know it and can no longer wonder, no longer marvel, is as good as dead, and his eyes are dimmed." Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 4 A. Einstein

6. Isomorphism of two spaces Φ and Φ ' with the scalar product- -- there is a univocal mapping: with the properties: Linear operators: Adjoint operator of A: Self-adjoint operator: A + = A Hermitian operator: A + = A and domain D(A + ) = D(A) is dense

7. Eigenvectors and eigenvalues of an operators: For Hermitian operators: 1) eigenvalues are real, and 2) Eigenvectors for two different eigenvalues are orthogonal Inverse operators: Unitary operators: Commutator of A and B: Degenerate eigenvalues:

8. The essential elements of the theory of linear spaces and operators acting in them Spectral theorem. Nuclear spectral theorem. Proper and generalized eigenvectors |E n ) and. Spectral resolution of identity operator I. Spectral resolution of Hermitian operator H. Univocal definition of vectors. Topology of Hilbert space. New topology H p. Analogous relations PROJECTION DISTRIBUTION

9. For every Hermitian operator A in a finite-dimensional space Φ there is a system of it's eigenvectors Such, that each vector we can written: and the complex numbers: are the components of the vector in the basis. In the infinite dimensional spaces above theorem is in general not correct. Spectral theorem

10. In ∞ dimensional spaces there is always the orthonormal system of vectors base, but not every self-adjoint operator must have a countable set of eigenvectors which form the base. Besides, we will also have to deal with operators with continuous spectrum, or continuous and discrete. Nuclear spectral theorem (NST) There exist ∞ dimensional spaces Φ (there are topologies in these spaces) for which the spectral theorem can be proved for all self-adjoint operators, which are interesting from a physical point of view. Quantum Mechanics will be formulated in such spaces in which the NST occurs.

11. Let us take two obserwables, the “energy” H (any with discrete spectrum) and “position” Q (any with continuous spectrum): For each there is: or: -- proper eigenvectors, belongs to the space of physical vectors, -- space of continuous antilinear functionals defined on Φ, generalized vectors.

12. continuous spectrum of position operator Q Spectral resolution of the identity and any physical operators: From a physical point of view, we are interested in vectors which are completely specified, so they must be normalized: discrete spectrum of the energy operator H

13. We also want to be certain that the action of any operator A on any physical state is defined, so it must be: And if: then: So we are interested in such spaces Φ in which any vectors satisfy: not only:,. but also: p = 0,1,2,….

14. Let us decompose a vector in some two bases: or: So we can define isomorphic spaces: In such way our spaces of physical vector states are defined separately for each physical system.

15. Linear functional: Antilinear functional: Scalar product is antilinear functional: We will write: Continuous Functionals A functional on Φ - is a mapping F from the space Φ into the field of complex number :

16. We can determine the linear space of antilinear functionals: which we can write in the way: The set of antilinear functionals {F} on the space Φ creates a linear space, we call it the conjugate space or dual space to Φ and is denoted by Φ*. Dual space:

17. In the finite dimensional spaces: Let { e i ; i = 1,2,3,….N} is the base in Φ, we define: For any vector v we can write: And then: Antilinear functional

18. Each functional is defined by a set of complex numbers: {f i ; i=1,2,3,….,N } We can then determine the vector belonging to the space Φ: We calculate now the scalar product: So in finite dimensional spaces: There is a clear correspondence between the functional F and the vector f, that is, Φ = Φ *

19. Generally in infinite dimensional spaces we can not say that they are self –dual. In the finite dimensional spaces: We say that such spaces are self-dual In infinite dimensoinal spaces we will have (as we will see in a moment)

20. Topology - the convergence of an infinite sequence (for our purpose such definition is enough) What does it mean: ? Ψ ---- linear space without any topology H ----- has all boundaries from ; Φ ---- has all boundaries from ; Has boundary points in agreement with lighter condition Has boundary points in agreement with stronger condition For dual spaces will be opposite (Rigged Hilbert Space) Gelfand tryplet No boundary points

21. (1) The space H contains space Ψ and all border points, for all strings that satisfy the condition (less difficult condition): (2) The space Φ contains Ψ and in addition all the border points, for all strings satisfying the condition (A - any operator, p = 0,1,2,3,...) (sharper condition): If then, but not vice versa, and this means that there are more strings satisfying the first condition than the second, so,

22. The concept of continuity of functionals : Ψ *, Φ * and H *, are respectively the dual spaces of continuous antilinear functionals.. Conditions that must satisfy the continuous functionals in H * are stronger than the conditions imposed on the functionals from Φ *. (1)Φ * is the set of all F which satisfy the properties: F(φ i ) F(φ) for all strings satisfying the conditions: (2) H * is a set of F which satisfy the condition F(φ i ) F(φ ) for all strings converging in the sense of Hilbert space Latter conditions are stronger, not all functionals F, which satisfy the first conditions, satisfy also the conditions (2), and this means that the space Φ* is larger than the space H*: Smaller number of conditions, larger space Larger number of conditions, smaller space

23. There is more series which satisfy: in comparison to the series which satisfy:

24. Riesz Theorem For every continuous functional F over the Hilbert space, there is uniquely defined and belonging to the Hilbert space vector f, such that the next condition is satisfied: which means that H = H *. The symbol will therefore be an extension of the scalar product on functionals that do not belong to the Hilbert space. We can consider antilinear functionals on Φ *. This set of new functionals is denoted as Φ **. For a large class of linear topological spaces Φ (called "reflexive"), there is a unequivocal relationship between the elements of the space Φ and the space Φ ** given by the relationship: The same as for a scalar product

25. The linear space Φ will include the states of physical systems. This space depends on the set of observables specific for a physical system which we consider. Vectors from the space Φ Vectors from the space Φ * Physical energy states Non-physical states of the position operator v

26. Later we will use consequently the Dirac denotation: The difference results from the context

27. For each physical system: must decrease at infinity faster than any power of 1 / x. Schwartz space: the space of complex functions infinite times differentiable and such that the same functions and all derivative disappear in infinity faster than any power of 1 / x.

28. In the space Φ any vector can be presented in different ways: A H Q E.g. for the harmonic oscillator:

29. The generalized vectors have dimension( [|x › ]= cm -1/2 ) The definition of our state space Φ Realization - the Schwartz space Very often we denote Isomorphism of space Φ and, Elements of representation theory, the transition between the bases Eigenvalue problem in discrete base Examples of Schwartz spaces K(a): φ(x) = 0 dla |x| > a, Fourier series K(∞): - ∞ < x < ∞, Hermite polynomials K(-1,1): -1< x < 1, Legendre polynomials K(0, ∞): 0 < x < ∞, Laguerre polynomials