1. Lecture from Quantum Mechanics

3. The facts are not the most important. Anyway, to get to know them, you do not need to study at university - you can learn them from books. The essence of higher education is not therefore inculcating factual knowledge, but training the mind in the investigation into such things what can not be found in textbooks Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 3+4 A. Einstein

6. Overview of the Quantum Mechanics postulates, remarks about its mathematical language. Each physics field has its own specific mathematical language. It often happens that physical demand stimulate different parts of mathematics. Newton's mechanics ------------ differential and integral calculus Classical Electrodynamics ----- tensor calculus and partial differential equations QUANTUM MECHANICS --- linear spaces, linear operators, groups and their representations The full mathematical formalism of Quantum Mechanics was introduced by J. von Neumann in 1932. "Mathematical Foundation of Quantum Mechanics" Springer, Berlin, 1932

7. Physical system In Classical Physics: Mathematical image of a physical system is phase space (phase space with constraints) {x 1, x 2, x 3, p 1, p 2, p 3 ;.....} In Quantum Mechanics: Mathematical image of a physical system is the algebra of linear operators operating in the linear space (Φ) with the scalar product (v, w). (unitary Hilbert space?)

8. Algebra Linear space Unitary space Hilbert space

9. Postulate I: State of a physical system at a given time In Classical Physics The state of a physical system (material point, system of material points) at any moment of time is described by the point in phase space, so both the position and momentum of each particle is specified: (x i (t), p i (t)); i = 1,2,...... n In Quantum Mechanics: The quantum state of a physical system is described by a Hermitian, positively defined operator ρ with the trace equal 1:

10. Definition of Hermitian operator Definition of the operator trace and its properties Why do we require in order to: 1. the ρ operator was Hermitian, 2. the trace of ρ was equal to 1, 3. ρ was positively defined operator, 4. show that the necessary and sufficient condition for the statistical operator in order to describe the pure state is: Tr(ρ 2 ) = 1 Definition of positive definite operator

11. Postulate II: Physical quantity In Classical Physics: Each physical quantity F is a function of particles positions and momenta: F = F (x i (t), p i (t)), e.g. Energy of a material point in a potential field: In Quantum Mechanics: To each physical quantity is assigned a Hermitian operator A having a complete system of eigenvectors. Such operators will be called: OBSERVABLE

12. Postulate (II) 1 : Construction of physical quantities In Quantum Mechanics: For a system having a classical analogy: on which we assume the commutation relations: then: For systems that do not have a classical analogy, observables and their commutation relations are proposed. In Classical Physics (as before): Each physical quantity F is a function of particles positions and momenta: F = F (x (t), pi (t)), e.g. Energy of a material point in a potential field:

13. Postulate III: Probability of measurements results In classical physics: Classically we can measure the position and momentum of each particle at any time t. Having x i (t) and p i (t) we determine any physical quantity F = F (x i (t), p i (t)) with any accuracy. The result of measuring the physical quantity A is always a value of its eigenvalue a i : A measurement of A on the physical system in a state we always get the eigenvalue, a measuring of A in the state we get different results and in advance do not know what the result will be. Quantum mechanics gives only possibility to calculate the probability that the measurement result will be a i. In Quantum Mechanics.

14. Why A must be Hermitian operator (which satisfy the nuclear spectral theorem)? Why A must the complete operator? Question - how to find algebra of operators for given physical system ??? Profit: - The achievement of mathematical rigor, - In the same way we describe discrete and continuous states, - States of decaying systems - Gamow vectors.

15. Postulate IV: Various experimental results In classical physics Each physical quantity we can determine without restriction. Measurement of one quantity does not affect the existing knowledge of any previously measured physical quantity. The measurement is the only recording of the result, which is coded in the system. The measurement does not affect the behaviour of a physical system and does not change anything. The physical system before and after any measurement is the same. In Quantum Mechanics By measuring any physical quantity A in the state ρ the result is not known. If the system is in the state described by the statistical operator ρ, the probability (p i ), of obtaining as the measurement result the eigenvalue a i is equal to the average value of the projector operator P i on subspace of this eigenvalue:

16. 1) when the state is pure, 2) if the measured value is degenerate, 3) when we ask about the probability receiving several eigenvalues, 4) show that the average value after many measurement of physical quantity A in the state ρ is given by: What is the probability of obtaining a given eigenvalue???

17. Postulate V: State of the system after measurement In classical physics Measuring only register, and does not change of the physical system. So if at the moment of the measurement the system was in position x i (t) and momentum p i (t), exactly the same value of the position and momentum the system will have after measurement of any physical quantity. In Quantum Mechanics The measurement of any physical quantity generally changes the state of a quantum system. If the system was in the state ρ, and we have measured the physical quantity A, and as a result the eigenvalue a i was obtained, the state of the system after this measurement will be described by the statistical operator:

18. The statistical operator before the measurement. Definition and properties of projection operators P i Examine the functioning of the postulate, if: 1) the states described by the operator ρ, is pure, 2) the eigenvalue a i is degenerate, 3) the result of the measurement does not distinguish between a few eigenvalues.

19. Postulate (V) 1 : Preparing the system to measurement In classical physics At one point t 0 we measure the position and momentum of a particle (particles): (x i (t 0 ), p i (t 0 ) ) There are also ways to direct measurement of other physical quantities. In Quantum Mechanics In preparing the system to measurement we measure one or more physical quantities, for which the observables commutate. If, as a result of measuring the physical quantity A we obtained the eigenvalues a 1, a 2,, a 3,....., with probabilities w 1, w 2, w 3,..., then this system is described by the statistical operator:

20. Please examine the functioning of the postulate, if: 1)we does not perform the measurement (we make ∞ number of measurements and do not record the results), 2) as a result of measurement we get some eigenvalue, which is degenerate, how the state of the system looks like?, 3) when we have prepared a system in an eigenstate of measured observables. Note: ρ is a eigenstate of their observables A, if (ΔA) ρ = 0, where

21. Postulate VI: Time evolution of quantum system In classical physics Knowing forces acting on a physical system and initial conditions, we can determine the status of systems at any later point in time. This is done by the equation of motion. We know several versions of such equations, e.g. Newton equations, Lagrange equations, Hamilton and Hamilton - Jacobi equations, e.g. Hamilton equations: In Quantum Mechanics Quantum mechanics also gives us opportunity to determine a state at any later point in time when we know the initial state. When we do not perform the measurement for the system, and we know its initial state ρ (t 0 ) there is the operator H called the Hamiltonian that: (Liouville equation)

22. When a system is isolated (conservative), the operator H is the energy operator of this system. For a not isolated system there is also a corresponding operator H = h (t) which is then not an energy operator. For a physical system in a pure state the Liouville equation is equivalent to Schrödinger equation Schrödinger, Heisenberg and Dirac pictures Heisenberg equation Stationary states, constants of motion

23. Postulate VII: Systems with many degrees of freedom In classical physics Each further degree of freedom is described by a new pair of canonically conjugate variables. { q i (t), p i (t), i = 1,2,3,..... } In Quantum Mechanics Each degree of freedom has its own linear space of states Φ i. A state space of a system with plurality degrees of freedom is the direct product of separate spaces Φ i : A pure state of a system is the direct product of pure states:

24. Definition of the direct product of spaces Entangled and non-entangled states Observables for many degrees of freedom Definition of the dot product, base

25. Postulate (VII) 1 : Degrees of freedom associated with particles identical In classical physics: Classically even identical objects are distinguishable. We can track the movement of each particle, even if it is identical with the other No identical particles. Consequently, no special properties which follow from identity of particles In Quantum Mechanics: We can not keep track of the particle. Identical particles are indistinguishable. Lack of differentiation has serious consequences. It shows the property of quantum states. States may be either completely symmetric or completely antisymmetric. States which are completely symmetrical describe particles with integer spin (bosons), an anti-symmetric states describe particles with a half-spin (fermions).

26. Normalized states of completely symmetric and completely antisymmetric for many identical particle. Observables for identical particles Pauli principle Parastatistic

28. Linear space Φ over field K: field Resolution of multiplication with respect to addition Associative law In QM K = field of complex numbers

29. ALGEBRA Set of operators ={A,B,C,…...,I} - this set forms a linear space In product of operators is defined: with properties: For each operator A there exist such operator I that:

30. Unitary space Metric generated by the scalar product: Convergence in the sense of Cauchy The sequence is convergent in the Cauchy sense, if for each there exist such N that for there is: - Scalar product space, - Euclidian space, - Pre-Hilbert space. Norm of vectors

31. Hilbert space H Linear and unitary space is called a Hilbert space, if every sequence of elements converges in the Cauchy sense to element, which also belongs to the space. The linear dependence, linear independence The set of vectors form the base, if for any vector v there is: Finite and infinite dimension spaces H H H This is true in finite and infinite dimensional spaces

32. ALGEBRAIC ; (+, ×) TOPOLOGICAL; ORDERING; >, < The image of a variety of microscopic physical systems is one Hilbert space Hilbert space does not contain all necessary information about physical system For a given physical system not all "mathematically possible" operators are observable In the Hilbert space there exist non-physical states Riesz - Fisher THEOREM Structures in modern mathematics Hilbert space is isomorphic to C n when it is finite dimensional or to L 2 when it is infinite dimensional. So the study of infinite dimensional Hilbert space is equivalent to study the L 2 space. H

33. In the thirties of the last century Hilbert space was only known topological space Dirac delta - the twenties of the last century Theory of distribution - L. Schwartz founded in 1950-1951 Rigged Hilbert space  Gelfand triplet I.M.Gelfand, O.P Shilov ; 1964 To Quantum Mechanics:Arno Bohm - (1966), I.J. Roberts - (1966). von Neumann approach was replaced  physical observables are represented by linear operators in a linear state space.

34. Collection of element a,b,c,….. form a group G, if 1. Product of any two elements „a” and „b” of G is itself element of G (closure property), 2. The associative law is valid, 3. There exist in G a unit element „e”, such that for any 4.For any element there exist an inverse element a -1 such that,

35. 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 Antilinear operators:

36. 36 Thank you