Lecture Notes Methods of Mathematical Physics MATH 536

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1 Lecture Notes Methods of Mathematical Physics MATH 536 Instructor: Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM April 6, 2014 Textbooks: L. D. Faddeev and O. A. Yakuboski, Lectures on Quantum Mechanics for mathematics Students, (AMS, 2009) L. Takhtajan, Quantum Mechanics for Mathematicians, (AMS, 2008) Author: Ivan Avramidi; File: qmmath1.tex; Date: April 10, 2014; Time: 16:16

2 Contents 1 Mathematical Foundations of Quantum Mechanics Kinematics Dynamics Classical Mechanics Quantum Mechanics Relation of Quantum Mechanics to Classical Mechanics Path Integrals

3 2 CONTENTS qmmath1.tex; April 10, 2014; 16:16; p. 1

4 Chapter 1 Mathematical Foundations of Quantum Mechanics 1.1 Kinematics The physical reality is described in terms of two sets of objects: observables and states. The set of observables will be denoted by A and the set of states will be denoted by Ω. The set of observables A is a real algebra, that is, a real vector space equipped with a bilinear product. The algebra of observables is not necessarily commutative. The product of observables is defined by A B = 1 4 [ (A + B) 2 (A B) 2]. The set of states Ω is a convex subset of a vector space. A state that cannot be decomposed as a linear combination of two different states is called a pure state; all other states are called mixed states. Each state ω Ω assigns to each observable A A a probability distribution ω A (λ) on the real line R. 3

5 4CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS The expectation (or mean) value of the observable A in the state ω is defined by ω, A = λ dω A (λ) That is, each state can be identified with a positive linear functional on the algebra A, ω, : A R. that satisfies the following properties. For any ω Ω, A, B A, a, b R: R ω, aa + bb = a ω, A + b ω, B ω, A = ω, A ω, 1 = 1, ω, A 2 0. It defines the duality between the algebra A and the set Ω (the observables and states). This can be described alternatively as follows. We suppose that there is a linear functional called the trace on a subset A tr of the algebra of observables Tr : A R. Such observables are called trace-class. Then a state ω Ω is described by a trace-class observable ρ ω A such that ρ ω A is also trace-class and ω, A = Tr (ρ ω A). Of course, for any state ω and any observable A it has to satisfy Tr ρ ω = 1, Tr ρ ω A 0. The set of states Ω is complete if any two observables have the same expectation values for all states if and only if they are equal, that is, if for any A, B A, ω, A = ω, B for any state ω Ω if and only if A = B. It is naturally to assume that the set Ω is complete; if it is not, then we can complete it by adding more states. qmmath1.tex; April 10, 2014; 16:16; p. 2

6 1.2. DYNAMICS 5 The variance of the observable A in the state ω is defined by ( ω A) 2 = ω, A 2 ω, A 2 Let f : R R be a real valued function and A A be an observable. Then f (A) is an observable such that for any state ω Ω ω, f (A) = f (λ) dω A (λ) It is easy to see that the probability distribution of an observable A can be computed by ω A (λ) = ω, θ(λ A), where θ(x) is the step function. 1.2 Dynamics A Lie bracket on the algebra A is a bi-linear map R {, } : A A A satisfying the conditions: for any F, G, H A, {F, G} = {G, F}, anti-symmetry, and {F, {G, H} + {G, {H, F} + {H, {F, G} = 0, Jacobi identity, {F, G H} = {F, G} H + G {F, H}, derivation property. A vector space with a Lie bracket is called Lie algebra. A flow (or a motion) of a space X is a one-parameter group of automorphisms U t : X X such that U t U s = U s U t = U t+s, U t = (U t ) 1, U 0 = Id. qmmath1.tex; April 10, 2014; 16:16; p. 3

7 6CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS Each observable H A generates a flow (or a motion) of the algebra of observables A by the differential equation da dt = {H, A}. The flow of observables naturally induces a flow in the space of states G t : Ω Ω by the pairing ω, U t A = G t ω, A. 1.3 Classical Mechanics The most important feature of classical mechanics is that the algebra of observables A is commutative. Such an algebra can be realized as an algebra of functions on a symplectic manifold called the phase space M. Thus the product of observables is simply the product of real valued functions A B = AB. A configuration space is a smooth manifold X of a finite dimension n. The phase space is the cotangent bundle M = T X of the configuration space. An observable is a real smooth function on the cotangent bundle. The observables form the algebra of observables A = C (M). Let q i be the local coordinates on X and (x µ ) = (q i, p j ) be the local coordinates on M. Here the latin indices range from 1 to n and Greek indices range from 1 to 2n. qmmath1.tex; April 10, 2014; 16:16; p. 4

8 1.3. CLASSICAL MECHANICS 7 Locally a Poisson bracket is described by an anti-symmetric contravariant 2-tensor {F, G} = P(dF, dg) = P αβ α F β G, such that and P µν = P νµ P µ[α µ P βγ] = 0 A canonical Poisson bracket is defined by ( 0 I (P µν ) = I 0 ), and has the form Note that in this case {F, G} = F p i G q i F q i G p i. {p i, q j } = δ j i. A symplectic form is a non-degenerate closed 2-form on M. In local coordinates it has the form ω = 1 2 ω µνdx µ dx ν and is described by a non-degenerate anti-symmetric 2n 2n matrix ( ) A B (ω µν ) =, C D where A, B, C, D are n n matrices. It satisfies ω µν = ω νµ, det ω µν 0 and that is, dω = 0, [µ ω νλ] = 0. qmmath1.tex; April 10, 2014; 16:16; p. 5

9 8CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS The canonical symplectic form is constant and has the form ω = dq i dp i, that is, (ω µν ) = ( 0 I I 0 ). Every symplectic form defines a Poisson bracket by (P µν ) = (ω µν ) 1 The Lie bracket is the Poisson bracket defined by the symplectic structure. The trace functional is just the integral over the phase space Tr A = A(x) dx States are normalized measures µ ω on the phase space, that is, ω, A = Tr ρ ω A = A(q, p)dµ ω (q, p) where M M dµ ω (q, p) = ρ ω (q, p) dqdp, and ρ ω (q, p) is the corresponding probability distribution. Probability distribution is positive and is normalized by ω, 1 = Tr ρ ω = ρ ω (q, p)dqdp = 1. Thus, the states in classical mechanics are described by probability distributions on the phase space. The probability distribution of an observable A in a state ω is defined by ω A (λ) = θ(λ A(q, p))dµ ω M M qmmath1.tex; April 10, 2014; 16:16; p. 6

10 1.3. CLASSICAL MECHANICS 9 Pure states are identified with points (q 0, p 0 ) M in phase space and are described by the distribution functions ρ (q0,p 0 )(q, p) = δ(q q 0 )δ(p p 0 ). All other probability distributions define mixed states. The expectation value of a variable A(q, p) in a pure state (q 0, p 0 ) is equal to the value of the function A at this point ω(q0,p 0 ), A = A(q 0, p 0 ). Pure states are studied in classical mechanics and mixed states are studied in statistical physics. For pure states the variance of any observable is zero. A Hamiltonian H is an observable. The Hamiltonian defines a Hamiltonian flow on the phase space by x x t = U t x dx dt = {H, x} This generates the flow on the algebra of the observables and a flow on the space of states by so that A(x) A t (x) = A(U t x) ω ω t = G t ω ρ Gt ω(x) = ρ(g t x) ω, A t = ω t, A This equality uses the Liouville theorem G t x x = 1, which means that the phase volume is invariant under the flow. qmmath1.tex; April 10, 2014; 16:16; p. 7

11 10CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS This leads to two possible descriptions of the same physical reality: the Hamiltonian picture in which the observables depend on time and the probability density does not da dt = {H, A}, and the Liouville picture in which dρ dt = 0 da dt = 0, dρ dt = {H, ρ}. 1.4 Quantum Mechanics In the case of quantum mechanics the algebra of observables consists of real elements of a complex associative algebra with involution, which is non-commutative. Such an algebra can be realized as an algebra of linear self-adjoint operators in a complex Hilbert space H A = {A L(H) A = A} In this case the product of observables is the symmetrized product of operators A B = 1 (AB + BA) 2 Then the product of two observables is an observable, that is, if A and B are self-adjoint then A B is self-adjoint. The trace functional is the trace of a trace-class operator in the Hilbert space. The states are described by the positive trace class operators with unit trace Ω = {ρ L(H) ρ 0, ρ = ρ, Tr ρ = 1} Such an operator is called a density matrix. The expectation value of A in the state ω is defined by ω, A = Tr (ρ ω A) qmmath1.tex; April 10, 2014; 16:16; p. 8

12 1.4. QUANTUM MECHANICS 11 The probability distribution ω A (λ) is given by ω A (λ) = Tr (ρ ω P A (λ)), where P A (λ) = θ(λ A) is the spectral projection of the operator A. Let ψ i be an orthonormal basis of the eigenvectors of the operator A with the eigenvalues λ i, Aψ i = λ i ψ i ; assume that they are simple. Then the spectral projection of the operator A is P A (λ) = θ(λ A) = P i, where P i are projections to the eigenspaces. λ i λ Then the distribution function ω A (λ) is ω A (λ) = Tr ρ ω P i. There is an orthonormal basis f i in which the density matrix is diagonal ρ = λ i λ ρ i P i, i=1 where P i are the projections to the vectors f i and ρ i is a sequence of nonnegative real numbers such that 0 ρ i 1 ρ i = 1 The pure states are given by one-dimensional projections P i. i=1 That is, a state is pure if and only if Tr ρ 2 = 1, and mixed if Tr ρ 2 < 1. A pure state is described not by a vector but by a one-dimensional subspace (a line). qmmath1.tex; April 10, 2014; 16:16; p. 9

13 12CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS For a pure state ω described by a vector ψ and the projector P ψ ρ ω = P ψ ω, A = Tr P ω A (λ) = Tr P ψ P A (λ) = (ψ, P A (λ)ψ). The Lie bracket is defined by the commutator (quantum Poisson bracket) {A, B} = i (AB BA). Note that for bracket of two self-adjoint operators is self-adjoint. One can show that the algebras A with different values of are not isomorphic. In classical mechanics the Poisson bracket with different factors define isomorphic algebras by a change of variables. The derivation property is valid for both the symmetrized product A B and the non-symmetrized product AB. There is the Heisenberg uncertainty principle ω A ω B 2 ω, {A, B} This is the main distinction between classical and quantum mechanics. The Heisenberg picture is the quantum analog of the Hamiltonian picture in the classical mechanics. The dynamics is described by the evolution of observables (the states are constant) da dt = {H, A}, The solution of this equation can be written as dρ dt = 0 A(t) = U( t)a(0)u(t) where U(t) = exp ( i ) Ht is the unitary evolution operator. qmmath1.tex; April 10, 2014; 16:16; p. 10

14 1.5. RELATION OF QUANTUM MECHANICS TO CLASSICAL MECHANICS13 The Schrödinger picture is the quantum analog of the Liouville picture. the dynamics is described by the time evolution of states and the observables are constant da dρ = 0, dt dt = {H, ρ}. The solution of this equation is ρ(t) = U(t)ρ(0)U( t). For pure states ψ this equation is equivalent to the Schrödinger equation The solution of this equation is i dψρ dt = Hψ. ψ(t) = U(t)ψ(0). 1.5 Relation of Quantum Mechanics to Classical Mechanics We consider a n-dimensional system so that phase space M = R 2n and the space of states is the Hilbert space H = L 2 (R n ). The algebra of observables is the algebra of self-adjoint operators A = L(H) on H. In quantum mechanics we replace the canonical coordinates (q i, p j ) by the operators (Q i, P j ) L(H) defined by (Q i ψ)(q) = q i ψ(q), (P j ψ)(q) = i q j ψ(q) Obviously, they do not commute [P j, Q k ] = i δ j k or {P j, Q k } = δ j k. qmmath1.tex; April 10, 2014; 16:16; p. 11

15 14CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS An observable A A is an operator described by its integral kernel A(q, q ), (Aψ)(q) = dq A(q, q )ψ(q ). R n The product of the operators A and B is defined by the convolution of kernels (AB)(q, q ) = dq A(q, q )B(q, q ). R n The main question of the quantization is: How do you associate a quantum observable (an operator) A to a classical observable (a function on the phase space) F(q, p)? In other words, the quantization is a map C (M) = A classical A quantum = L(H). The simple answer is: Just replace the canonical variables q and p by the operators Q and P so that A = F(Q, P). There is no unique way to quantize a classical system because of the ordering of operators. Since the operators Q and P do not commute there is a ordering problem qp QP or PQ = QP + 1? There are three canonical ways to do this for polynomial operators: 1. normal (Wick) ordering: all Q are placed to the left of all P, 2. anti-normal ordering: all P are place to the left of all Q, 3. Weyl ordering: all products are symmetrized. One can also describe an operator A by its symbol F(q, p), which is a function on the phase space, ( ) ( ) A F (q, q dp 1 q + q ) = (2π ) exp n ip(q q ) F, p. 2 R n qmmath1.tex; April 10, 2014; 16:16; p. 12

16 1.5. RELATION OF QUANTUM MECHANICS TO CLASSICAL MECHANICS15 This correspondence is called Weyl quantization. It maps a function to an operator F A F. The inverse map is given by F(q, p) = du exp ( 1 ) ( ipu A F q + u R 2, q u ). 2 n This maps an operator to a function A F F. The Weyl quantization corresponds to complete symmetrization of the product of operators. Notice that as 0 A F (q, q ) = F(q, 0)δ(q q ). The Weyl quantization is not a homomorphism from the algebra of functions on the phase space (the classical algebra of observables) to the algebra of operators (the quantum algebra of observables). Note that it is linear, that is, ffor any a, b and any functions F, G A af+bg = aa F + ba G Let A F be an operator with the symbol F and A G be an operator with the symbol G. Then the kernel of the product of the operators A F A G is defined by some symbol F G, that is, A F A G = A F G. This defines a new product, called the Moyal product, on the algebra of functions such that as 0 F G = FG + i {F, G} + O( ). 2 The Moyal product is non-commutative (and non-local). qmmath1.tex; April 10, 2014; 16:16; p. 13

17 16CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS We can define a quantum Poisson bracket (or Moyal bracket) on the algebra of functions by One can show that {F, G} = 2 F(q, p) exp i 2 {F, G} = i (F G G F) q j p j p j q j G(q, p) so that as 0 {F, G} = {F, G} + O( ) That is, {A F, A G } = A {F,G} Thus, both the classical mechanics and the quantum mechanics can be realized in terms of the same objects (functions on the phase space) but the structure constants of the product and the Lie bracket are defined as series in positive powers of, the zero order term being the classical one. Therefore, the quantum mechanics is a deformation of classical mechanics, with the Planck constant being the deformation parameter. One can show that the classical mechanics is unstable and the quantum mechanics is its unique deformation into a stable structure. The instability of the classical mechanics has to do wit the exactness of pure states. 1.6 Path Integrals To determine the quantum dynalics we need to compute the evolution operator U(t, t ) = exp [ ih(t t )] We set = 1 for simplicity. We consider a one-dimensional system as an example. qmmath1.tex; April 10, 2014; 16:16; p. 14

18 1.6. PATH INTEGRALS 17 We have When N is large U(t, t ) = ( [ exp ih (t ]) N t ) N [ exp ih (t ] t ) 1 ih (t t ) N N + Let h(q, p) be the Weyl symbol of the Hamiltonian H. The using the kernel of this operator and computing the kernel of the composition we get U(q, t; q, t ) R 2N N j=1 dq j dp j 2π N exp i j=1 [ p j (q j q j 1 ) h As N we get the path integral U(q, t; q, t ) = Dq Dp exp {is (q, p)}, where is the path integral measure, S (q, p) = M dq(t) dp(t) Dq Dp = 2π t t t t t dt [ p(t) q(t) h (q(t), p(t)) ] ( q j + q )] j 1, p j 2 is the action and the integral is taken over all paths q(t), p(t) in the phase space with fixed initial and final points, that is, q(t ) = q, q(t ) = q, and the values p(t ) and p(t ) are unrestricted. In a particular case h(q, p) = p2 2m + V(q) qmmath1.tex; April 10, 2014; 16:16; p. 15

19 18CHAPTER 1. MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS by changing the variables p p = p + m q, q q we get where p q h p2 + L(q, q) 2m L(q, q) = m 2 q2 V(q) is the Lagrangian. Then we get the integral U(q, t; q, t ) = Dq exp {is (q)}, where the action is now M t S (q) = dt L(q, q). t qmmath1.tex; April 10, 2014; 16:16; p. 16