1. Lecture from Quantum Mechanics

3. Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 6

5. Time evolution of quantum systems (Postulate VI)

6. Time evolution of quantum systems (postulates VI) Initial conditions: Having ρ(t 0 ), we want to determine ρ(t). For systems that interact with the environment: ρ(t) =f( ρ(t 0 ), τ = t-t 0, t 0 ) Isolated systems: ρ(t) =f(ρ(t 0 ), τ = t-t 0 ) Systems which do not interact with the environment Isolated systems, Conservative systems, Reversible systems Time is a continuous real-parameter – it is not the quantum observable.

7. The final statistical operator of any full quantum system should not depend on whether we first give the time evolution of subsystems: and next we will mix: or we mix at first: and then a whole system will evolve over time: Intuitive understanding of the time evolution requires that the next conditions should be satisfied: 1) The mapping is linear and continuous:

8. The statistical operator of the system does not depend on whether we first wait for a while τ 1 : and then a moment τ 2 : or just wait the moment τ = τ 1 + τ 2. Conditions 1) and 2) are always satisfied in general, also for non- insulated systems 2) Time evolution forms an additive semigroup

9. 3) For an isolated system we additionally require that time evolution is given by the unitary operator U (t) which satisfies the conditions: where U (τ) is a continuous operator of the parameter τ. For insulated systems, there exist the time evolution operator

10. The definition of an operator continuity (which depends on the topology): A (t) is continuous at t = t 0 if for t t 0 : with respect to the given topology, or when t t 0. We define the generator of time evolution:

11. Let us take: We differentiate both sides with respect to : Now we take the limit: Or genrally: Taking the second derivative, we see that

12. We can now make the Taylor series expansion for U( τ ): So we have: We differentiate both sides of the equation: So the generator A is skew-hermitian: So it is possible to parameterize: In the limit we obtain and

13. The generator of the time evolution of an isolated physical system is the operator where H is the energy operator of this system. In this way the Postulate VI can be formulated in the form (in the Schrödinger picture)) So we have two possible way to express any mean value of physical quantity: An observable (apparatus and method of measurement) does not change over time. The state of this system is changing in time of. Recipe for performing of a measurement is changing - therefore the observable is changing. Physical system does not change where: Schrödinger picture Heisenberg picture

14. The time evolution of the statistical operator (Schrödinger picture) Time evolution of the observable (Heisenberg picture)

15. In the Heisenberg picture ---- Heisengerg equation: In the Schrödinger picture – Liouville equation: For a pure state - Schrödinger equation: Constants of motion A physical quantity is constant motion when it does not depend on time:

16.  An average value in the stationary state does not depend on time: Stationary state The state is called the stationary state, when it does not change over time, so we have:  Stationary states of any system are always a mixture of their energy operator eigenstates: For pure state

17. Time evolution of any system is a symmetry transformation of this system Operators from the algebra of observables satisfy the same commutation relations in any time. Interaction picture – Diraca picture: Definition ;

18. If then: Non conservative systems (irreversible, non-isolated):  Dynamics is time dependent  Energy operator depends on time Evolution in time of not conservative physical system is given by the Schrödinger equation where h(t) is called the Hamiltonian operator, but it does not have to be the energy operator. Perturbative decomposition:

19. if for any time t‘ and t”, then (t 0 = 0): whereas when:we get: The chronological product T is defined in the way: for:Examples: Nonrelativistic quantum mechanics, Quantum field theory

20. Quantum mechanics of many degrees of freedom (Identical particles)

21. Space of quantum states: In QM identical particles are indistinguishable --- what it means mathematically? Let us take a base vector in H : But those element can be arranged in a different order: The identity of particles -- the first state and the second state are physically indistinguishable, all measurements made in these states need to give the same result. state Particle number Identical particles, postulate (VII) 1

22. Permutation group Symmetric group {S n } Permutations can be composed with a transpositions, The permutation can have an even or odd number of transposition, Permutation can be made from any number of transposition, but it is always even or odd. It is finite dimensional group and has n! elements, It has a finite-dimensional irreducible unitary representations, There are two one dimensional irreducible representation. In the state space unitary representation operates ---- for each permutation of N elements corresponds some unitary operator P acting on the physical states. Not all the vectors in the state space describing the system of identical particles are good. Pure physical state will be described by the vector which gives the same results of each measurement, no matter which permutation are made on the particles. Such a vector may differ from the original one only by phase.

23. n = 2 n = 4 Permutation of 3 elements – S 3 group (which has 6 elements) n = 1 n = 2 Permutations do not commute

24. Order of finite dimensional group, r = number of elements Multiplication of elements, multiplication table : Examples:

25. Group representation: collection of matrices: which satisfy the same multiplication rules as the group elements: Dimension of matrices M = Dimension of representation Theorem (without prove) : All irreducible representations with dimensions satisfy the relation Irreducible representation in any base it is impossible to write the matrices M in the form: where h is the group order – number of group elements

26. S 3, h=6 S 4, h = 24 S 5, h=120 2 x1 + 3x4 + 1x9 + 0 x16 + 1x25 + 2x36=120 2 x1 + 2x4 + 1x9 + 3x25 + 3x36 + 3x49 + 2x64 + 3x 81 S 6, h=720

27. YOUNG DIAGRAMS Young diagrams and tables derived from permutation groups but their use is much broader. Again, we define symmetrization and antisymmetrization operator: For two particles: For three particles: For three particles we have six different conditions:

28. In fact, there are only four independent states; 1) Totally symmetric: 2) Totally antisymmetric: 3) And two with the mixed symmetry e.g. and States are related by the similarity transformation with four above mentioned Please prove this! This corresponds to three different Young diagrams: Totally symmetric Totally antisymmetric Mixed symmetry

29. 1 2 3 1 2 3 2 3 1 3 2 General definition of the S n group Such, that and …………… 1 We introduce n integers:

30. 2) All n (= 4) numbers appear in the squares, placed in such a way that: ----- in each row the numbers increase from left to right, ----- in each column the numbers increase from top to bottom. The number of different possible settings for given diagram Young gives dimension of representation. The construction of irreducible representations, their dimension and the base 1)Every possible shape of the diagram corresponds to different irreducible representation, e.g. for S 4 group: 1212 1212 123 4 12 4 3 dim = 3 + + 3 2 + 3 2 + 2 2 = 24 1 2 34 { }

31. Physical states make up one-dimensional subspace in a physical state space --- one-dimensional representation of the symmetric group. We know that this group has two non- equivalent one-dimensional representations Symmetric representation Anti-symmetric representation So there are two one-dimensional subspaces of the symmetric group: Symmetric subspace Anti-symmetric subspace Physical states of identical particles may belong either to the symmetric subspace or to the anti-symmetric subspace The number of inversions in the permutation P

32. Postulate (VII) 1 Each state of the physical system of N identical particles creates a one- dimensional subspace of symmetric groups S n : Symmetrical - for particles with integer spin (BOSONS) and Anti-symmetrical - for particles with half-integer spin (FERMIONS). Pauli exclusion principle for fermions results from this postulate. Normalized states of N identical particles. For bosons : For fermions:

33. Some information about groups and group representations

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. Theory of Group Representation 1. Definition of Group Representation, 2. Definition of Equivalent Representation, 3. Definition of Character, 4. Definition of Unitary Representations, 5. Definition of Reducible and Irreducible Representations, 6. Definition of Direct Sum of Matrices, 7. Definition of Irreducible Invariant Subspace 8. Definition of Direct (Kronecker) Product of Matrices

36. 1. Definition of Group Representation If there is correspondence that for each element g from an abstract group G correspond an operator (or matrix) and this correspondence is homomorphic (or isomorphic) then we say that the set of operators A(G) form the representation of the group G To refresh memory Operators in linear space Base of vectors Matrix representation for operators Change of base for operators

37. Two bases: Matrix representation of A in the base : What is the relation between matrices? To preserve the bases orthonormality, matrix U must be unitary:

38. Thus Matrices in our two bases are connected by unitary similarity transformation 2) Definition of equivalent representation Two matrix representations are equivalent if they are connected by a similarity transformation: If for each element.

39. 3. Definition of Character The character of any group elements in given representation is trace of the matrix A(g): Character = Tr(A(g)) Character remains invariant under a change of base in the representation space: The equivalent representations have the same set of characters. Comments 4. Definition of Unitary Representations Representation in which all the matrices are unitary is said to be Unitary Representation

40. Its can be proved that for finite and continuous compact Lie groups any matrix representation of a group can be transformed by a suitable similarity transformation into a unitary representation. There exist a matrix C that for each, is unitary. For infinite and not compact groups the representations are not unitary in general. For the proof for finite groups see e.g.[3] 5. Definition of reducible and irreducible representations Let us assume that there exist n dimensional representation. If for all one matrix C exist such that Where and are square matrices and with, then the representation is said to be reducible. If such decomposition is impossible then the representation is irreducible.

41. 6. Definition of Direct Sum of Matrices THEOREM If and are two representations of a group G then is also the representation, and vice versa if is a group representation matrices and form representations too. And it is very easy to prove: Proof:

42. Comment The process is called addition of representation. If are irreducible representations then the above procedure is called decomposition od reducible representation on the sum of irreducible ones If in the sum, all representation are different, then we say that is simply reducible. 7. Definition of Irreducible Invariant Subspace If under the action of a group G the vectors of a subspace M of a linear vector space L are transformed among themselves, the subspace M is said to be invariant under G. If M does not contain any smaller invariant subspace than it is said to be an irreducible invariant subspace.

43. 8. Definition od direct (Kronecker) product of matrices All elements with the same i k value are in the same row, while all the elements with the same j l value are in the same column. Example If and then

44. For direct product of matrices we can prove that: THEOREM Direct product of two matrix representations form another matrix representation of a group G Proof:

45. THEOREM (without proof) If the group is finite or simple and compact any representation of such group can be decomposed in a direct sum od irreducible representations a i = 0,1,2,.., denotes how many times the irreducible representation appears in the sum. If a i = 0 or 1 for all i, the product representation is said to be simply reducible. Such decomposition is called the Clebsch – Gordan series Character of the direct product of two representations is equal to the product of the character of each representation. THEOREM Proof: