Introduction To Quantum Mechanics - ULisboa
Introduction to Quantum MechanicsEduardo J.S. VillaseñorUniversidad Carlos III de MadridXVI Fall Workshop on Geometry and PhysicsLisboa, September 2007Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20071 / 44
disclaimerThe purpose of this talk is to give a very brief introduction to themathematical aspects of Quantum Mechanics making specialemphasis on those points that are relevant for Fernando Barbero’smini-course on Quantum Geometry and Quantum Gravity. I will leftuntouched many interesting mathematical results and all the physicalones!I will follow an approach that, although standard in certainmathematical literature, it is not the standard in the physical oneThe starting point in (most of the) physics books is the definition ofphysical observables as self-adjoint operators on a certain Hilbert spaceH and (pure) states of the system as vectors on H. The emphasis thereis on vector statesHere the starting point will be the C -algebra generated by the(bounded) physical observables. In this approach the states of a systemare secondary (derived) objectsEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20072 / 44
plan of the talk1General considerations about physical systems: Observables and states2Classical kinematics: Observables and states in Classical Mechanics3The crisis of Classical Physics (very very brief!)4Quantum kinematics: Observables and states in Quantum Mechanics(Segal approach)5The simplest quantum system:The quantum point particle Weyl C -algebra6Quantum dynamics: Schrödinger and Heisenberg equationsEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20073 / 44
physical systemsSystems The ‘things’ of the physical world in which we will be interestedwill be called physical systems (or systems)A free point particle in Euclidean spaceA point particle constrained to move on a smooth surface inEuclidean spaceElectric and magnetic fields without sources in Euclidean spaceBut if we try to give a more precise definition, we need to specify what is a‘thing’ ?Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20074 / 44
kinematical aspects of physical systemsOperational point of view: A system is defined by the physicalproperties –observables O – that can be measured on it (by concretephysical devices) and the relations between themObservables: A system is defined by a set O of observables endowedwith certain algebraic and metrical propertiesStates: The set S of states is characterized by the results of themeasurements of all the observables in the following sense:Given a state ω S , for any A O , the expectation value ω (A) isthe average over the results of measurements of the observable A.Thus, a state of a system is a functional ω : O R (that satisfiessome properties that we will specify later)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20075 / 44
observables in classical mechanicsA typical system in classical mechanics is described in the cotangentbundle T C of a configuration space CThe configuration space C of the system is the space of all possible positionsq C that it can attain (possibly subject to external constraints)A single particle moving in ordinary Euclidean 3-space: C R3A double planar pendulum: C T2 S1 S1A rigid body: C R3 SO(3)The phase space of the system T C consists of all possible values of positionand momentum variables (q, pa ). T C has a canonical symplectic form ΩαβClassical observables belong to some class of functions on T C , sayO C (T C ; R ) C (T C ; C )In this case O is a commutative and associative -algebra (the elements ofO satisfy A A )Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20076 / 44
classical observables: Poisson structureThe symplectic structure of T C endows O with the structure of a Poisson -algebra with Poisson bracket{A, B} : Ωαβ (dA)α (dB) β , A, B O .This Poisson structure is relevant in many respectsThe dynamical evolution of the system is defined through {·, ·} oncea special observable –the classical Hamiltonian of the system H O –is given:dA {A, H }dtAs we will see, the Poisson bracket is the classical analogue of thequantum commutator (quantum observables will define a non abelianalgebra)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20077 / 44
Configuration observables Q(f ): For any f C (C) letQ(f )(q, pa ) : f (q)Momentum observables P(v): For any v X (C) letP(v)(q, pa ) : va (q)paThis family of observables is closed under Poisson brackets{Q(f1 ), Q(f2 )} 0, {Q(f ), P(v)} Q(Lv f ), {P(v1 ), P(v2 )} P(Lv1 v2 )If C is the Euclidean 3-space we can choose Euclidean coordinates x, y,z : C R and consider the 6 Killing vector fields of the Euclidean metric:the configuration observables X Q(x), Y Q(y), and Z Q(z) are theusual position observablesthe momentum observables PX P( x ), PY P( y ), PZ P( z ) are thecomponents of the usual linear momenta, and LX P(y z z y ),LY P(z x x z ), LZ P(x y y x ) are the components of the familiarangular momenta.Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20078 / 44
states in classical mechanicsPure states: Classical Physics assumes that canonical variables can besimultaneously measured with infinite precision. This leads us to identifythe points of the phase space with the (pure) states of the system. Ifγ (q, pa ) T C is determined with total precision, the expectation valueof any observable is given byωγ (A) A(q, pa ) .Pure states play a fundamental role in non-statistical mechanic: An experimentperformed on a system described by a pure state will attain maximal theoreticalaccuracyMixed states: In many situations it is not possible to determine the purestate of the system. If a system is in a state ω1 with probability α and in astate ω2 with probability 1 α the effective state of the system w isω ( A ) α ω1 ( A ) ( 1 α ) ω2 ( A ) .The state ω is the mixture of the states ω1 and ω2 .Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 20079 / 44
States: In dealing with a system of a very large (say 1023 ) number ofparticles it is impossible in practice to determine all the positions and allthe velocities of the particles. Classically we always assume that thesystem is in a pure state but we may be unable to determine it. This isthe usual situation in Classical Statistical Mechanics.Classical states S are probability measures µ on T Cω µ (A) ZT CA dµPure states corresponds to Dirac measuresVariance: The variance of an observable A relative to the state ω 2ω (A) : ω (A2 ) ω (A)2In classical physics, ω is a pure state iff ω (A) 0 for all A OEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200710 / 44
logical structure of classical mechanicsA proposition in Classical Mechanics is of the formp(S) : “the pure state (q, pa ) of the systems lies in S T C ”Typically S {A R} for some A OClassically, these propositions are either true or false and can be evaluatedif the pure state of the system is known. In general, if we are given a stateωµ S , we can only measure the plausibility of a proposition to be true:Prob(p({A }) ωµ ) µ({A }) Z{A } T CdµBy associating propositions with subsets of T C it is clear that the logicalstructure of Classical Physics is the standard Boolean logicp(S1 ) p(S2 ) p(S1 S2 ) , p(S1 ) p(S2 ) p(S1 S2 ) , p(S) p(T C \ S) ,p1 ( p2 p3 ) ( p1 p2 ) ( p1 p3 ) , p1 ( p2 p3 ) ( p1 p2 ) ( p1 p3 )Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200711 / 44
the crisis of classical mechanicsClassical Mechanics is a beautiful and natural way to model physicalsystems. However it suffers from one serious problem: Nature is notclassical.Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200712 / 44
atomic physicsAtoms (10 8 cm) are made of a nucleus (neutrons and protons) andelectrons. Electrons and protons are charged particles. The electrons arebound to the nucleus thought the Coulomb interactionAtoms are obviously stable (we are here!). However, the stability ofelectron orbits is incompatible with the laws of EM (acceleratedcharges emits EM radiation energy loss)Experimental results show that the radiation absorbed or emitted byan atom can have only a discrete set of sharply defined wavelengths(only a discrete set of ‘orbits’ –energy levels– seem to be allowed)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200713 / 44
Particle-Wave dualityA beam of photons (electrons, or atoms) produces interference patternsThe interference pattern still results even if only one electron traverses theapparatus at a time (right figure)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200714 / 44
quantum logic is not classical logicConsider the following propositionsp : “interference pattern is produced by the particle on the screen”s1 : “the particle has passed through the slit 1”s2 : s1 “the particle has passed through the slit 2”Experimentally, the interference pattern does not appear if we close one ofthe slits so that we known for sure that the particle passes through theotherp s1 p s2 (p s1 ) (p s1 ) On the other hand the particle has passed through one of the slits sos1 s2 1 p (s1 s2 ) pThen p (s1 s2 ) 6 (p s1 ) (p s1 ) so we cannot use the classicalBoolean logic to deal with these microscopic systemsEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200715 / 44
Heisenberg uncertainty ‘principle’When dealing with systems at atomic scales (atoms 10 8 cm; nucleus 10 23 cm)any attempt to measure the position of a particle with sharper and sharperprecision will produce a larger and larger variance in the momentumIn more mathematical terms: Let C the Euclidean 3-space and X Q(x),PX P( x ) the canonical position and linear momenta in the x direction ω (PX ) · ω (X) h̄2(h̄ 10 34 J · s)In general, for the configuration and momentum observablesh̄ ω (Q(Lv f )) 2This implies that no states exist with the properties of the pure states of ClassicalPhysics (@ω such that ω (A) 0, A) ω (P(v)) · ω (Q(f )) If we accept the Heisenberg principle, we need to reconsider the mathematicalstructure of O and S . This required the development of a newmathematical framework for Mechanics –Quantum Mechanics– in which Ono longer is an abelian algebraEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200716 / 44
postulates of quantum mechanicsBy the early 1930s the mathematical theory of quantum mechanics wasfirmly founded by von Neumann (and Stone). The basic rules could besummarized as follows:(1) an observable is a self-adjoint operator  on a Hilbert space H;(2) a (pure) state is a unit vector in ψ H;(3) the expected value of  in the state ψ is given by hψ Âψi;(4) the dynamical evolution of the system is determined by thespecification of a self-adjoint operator Ĥ through one of the followingrules: ψ 7 ψt exp(itĤ )ψ or  7 Ât exp(itĤ )A exp( itĤ )In this talk we will follow the Segal approach to Quantum Mechanicsbecause it is an operational formalism that can be generalized to QFT in astraightforward wayEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200717 / 44
Segal’s postulates (kinematics)Segal’s postulates try to encode the minimal set of properties that theclass of observables for any physical theory should satisfyDefinition: Segal system1 O is a linear space over R2 (O , · ) is a real Banach space3 O 3 A 7 A2 O is a continuous function on (O , · )4 A2 A 2 , and A2 B2 max( A2 , B2 )From an operational point of view, only bounded observables play afundamental role. The norm of an observable is to be thought as itsmaximum numerical value. If A1 and A2 are bounded observables, it ispossible to justify that λ1 A1 λ2 A2 is also an observableEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200718 / 44
It is possible to characterize the mathematical systems satisfying thesepostulatesSpecial Segal systems: there exists an asociative C -algebra A withidentity, 1 A, such thatO {A A A A (i.e. A is self-adjoint)}A is generated by OExceptional Segal systems if this is not the case. Exceptional Segalsystems are difficult to construct and, so far, no one has been able togive an interesting physical application of these systemsEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200719 / 44
Summarizing, for special Segal’s systems:A physical system is defined by its C -algebra A with identity 1The states S are normalized positive linear functionals that separatethe observablesA state is call pure if it cannot be written as a nontrivial convexcombination of other statesPositive: ω positive in A if ω (A A) 0 , A A. All positivefunctionals are continuous on (A, · )Normalized: ω is normalized if ω (1) 1Notice that the set of states over A is a convex subset of A (thetopological dual of A)Functional separates the observables: If A1 6 A2 then ω such thatω (A1 ) 6 ω (A2 ) (full set of states)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200720 / 44
simultaneous observabilityAn observable A is said to have a definite value in a state ω if 2ω (A) : ω (A2 ) ω (A)2 0A class of observables are called simultaneously observable if there exists asufficient large number of states in which they simultaneously have definite valuesA collection C of observables is simultaneously observable if the systemgenerated by C, A(C) has a full set of states in each of which everyobservable in A(C) has a definite valueTheoremC is simultaneously observable if and only if it is commutativeEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200721 / 44
Gelfand-Naimark characterization of abelian C -algebrasAn abelian C -algebra A with identity is isometrically isomorphic to theC -algebra of continuous functions on a compact Hausdorff topologicalspace (which is the Gelfand spectrum of A, sp(A), with the topologyinduced by the weak topology)If A is abelian with identity there exists an isomorphism A C(sp(A)),A 7 fA . The Riesz-Markov theorem tells us that there is a measure µ onsp(A) associated to every sate ωω (A) Zsp(A)fA dµIf C is simultaneously observable class then it is isomorphic to the system of allreal valued continuous functions on a compact Hausdorff space sp(A(C)). In thiscaseZfA (s)dµ(s) , A sp(A(C))ω (A) sp(A(C))The situation for C is exactly the same as in Classical Mechanics. In order toincorporate de Heisenberg uncertainty principle A is required to be anon-abelian C -algebraEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200722 / 44
statistical interpretation of QMIf A A is normal (AA A A) the C -algebra A(A) generated by 1,A, and A is abelian and it is possible to show thatsp(A(A)) σ(A) {λ C λ1 A does not have a two-sided inverse}Then, given any state ω, we can applying the GN-theoremω (B) In particular ω (A) Zσ (A)Zσ (A)fB (λ)dµω,A (λ) , B A(A) .λdµω,A (λ) . If we remember that ω (A) is theexpectation value of A on ω, the interpretation is clear. If A is anobservable (A A )The possible values that A can take in any experiment belong to σ(A)When the state is ω, the probability that A takes values on somesubset of σ(A) is defined in terms of µω,AFor pure states µω,A is not, in general, a Dirac measure. Pure statesin Quantum Mechanics have statistical interpretation!Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200723 / 44
Heisenberg uncertainty relationsTheorem (Heisenberg uncertainty relations)Given two observables A A and B B and a state ω ω (A) · ω (B) 1 ω ([A, B]) 2where[A, B] : AB BA .Proof: For simplicity consider A and B such that ω (A) ω (B) 0 andlet Cλ : A iλB, λ R. Then, the positivity of ω implies0 ω (C λ Cλ ) ω (A2 ) λ2 ω (B2 ) iλω ([A, B]) ,i.e. ω (A) · ω (B) Eduardo J. S. Villaseñor (UC3M) λ R1 ω ([A, B]) .2Introduction to Quantum MechanicsSeptember 200724 / 44
representations of C -algebrasRepresentations provide concrete realizations of C -algebras and also allowthe implementation of the superposition principle of wave mechanicsDefinitionA representation (A, , H), or simply , of a C -algebra A in a Hilbertspace H is a -homomorphism of A into the C -algebra B(H) ofbounded linear operators in HWe will be interested in faithful and irreducible representations: is faithful if ker( ) {0} is irreducible if {0} and H are the only closed subspaces invariantunder (A)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200725 / 44
Theorem (Gelfand-Naimark)A C -algebra is isomorphic to an algebra of bounded operators in a Hilbert spaceTheorem(Gelfand-Naimark-Segal) Given a C -algebra A with identity and a state ω,there is a Hilbert space Hω and a representation ω : A B(H)such that1. Hω contains a cyclic vector ψω (i.e. (A)ψω Hω )2. ω (A) hψω ω (A)ψω i for all A A3. every other representation in a Hilbert space H with a cyclic vector ψ suchthatω (A) h ψ (A) ψ i A Ais unitarily equivalent to ω , i.e. there exists an isometry U : H Hω such thatUψ ψω , U (A)U 1 ω (A) for all A AEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200726 / 44
Given (A, , H), the unit vectors ψ H define states ωψ on A:ω ψ (A) : h ψ (A) ψ i ,hψ ψi 1These states are called state vectors of the representationThe converse is also true (GNS theorem). However, if a state is notpure, the representation (Hω , ω , ψω ) is reducibleTheorem. Let ω a state over the C -algebra A and (Hω , ω , ψω ) the associatecyclic representation. Then (Hω , ω , ψω ) is irreducible iff ω is pureGiven a positive trace class operator b on H with trace equal to oneωb (A) : tr(b (A))is a state over A. These states are called density matricesNotice that ωb is of the form ωb (A) i λi hψi (A)ψi i , λi 0 , i λi 1 ,hψi ψi i 1. Then ωb is a pure state iff b is a one dimensional projectionEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200727 / 44
abstract special Segal’s system:A physical system is defined by its C -algebra A with identity 1The states S are normalized positive linear functionals that separatesthe observablesFor a given irreducible representation (A, , H)representations of special Segal’s system:Observables are bounded self-adjoint operators on HVector states ψ H, ψ 1, define pure statesEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200728 / 44
Dirac quantum conditionsHow can we describe specific Quantum Systems?Usually one begins with a certain Classical System and follows some kindof ‘quantization procedure’Dirac’s quantum conditionsThere exists a map ˆ : Oclassical C (T C ; R) Oquantum such thatA 7  is linear over RIf A is a constant function, then  is the corresponding multiplicationoperatorIf {A1 , A2 } A3 then [Â1 , Â2 ] ih̄Â3 (notice that Oclassical mustbe closed under {·, ·}). The commutator is the quantumanalogue of the classical Poisson bracketEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200729 / 44
The quantum particle: Heisenberg algebraThe simplest example of a physical system is the quantum particleHow can one construct a C -algebra for the quantum particle?Heisenberg Lie algebra: The basic ‘observables’ are position X andmomentum P PX so, naively, one can try to consider the algebra ofobservables generated by X and P which satisfy the Heisenbergcommutation relations (h̄ 1)[P, X] i ,[X, X] 0 ,[P, P] 0However, the Heisenberg algebra does not fall into the Segal scheme: Xand P cannot be self-adjoint elements of any C -algebra because X and P cannot both be finite (X and P are not observables in the operationalsense)[P, Xn ] inXn 1 X P n/2 , n NEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200730 / 44
Consider the polynomial algebra generated byU (α) exp(iαX) ,V ( β) exp(iβP)Weyl algebra: The Weyl algebra is generated (trough complex linearcombinations and products) by the elements U (α) and V ( β), where α, β R,satisfyU ( α1 )U ( α2 ) U ( α1 α2 ) , V ( β 1 )V ( β 2 ) V ( β 1 β 2 ) ,U (α)V ( β) V ( β)U (α) exp( iαβ) .U (0) V (0) 1U (α) U ( α) ,V ( β) V ( β)U ( α ) U ( α ) U ( α )U ( α ) 1 V ( β ) V ( β ) V ( β )V ( β ) U (α) V ( β) U (α)V ( β) 1 (these fix the norm of anycomplex linear combinations and products of U’s and V 0 s)Weyl C -algebra AWeyl is the · -completion of the Weyl algebraThe quantum particle is the physical system characterized by AWeylEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200731 / 44
von Neumann uniqueness theoremThe classification of the representations of AWeyl is solved by the followingtheorem due to von NeumannTheorem (von Neumann)All the regular irreducible representations of AWeyl in separable Hilbertspaces are unitarily equivalentHere ‘regular’ means that (U (α)) and (V ( β)) are strongly continuousin α and β respectively.Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200732 / 44
Scrödinger representationSchrödinger representation (AWeyl , , H) is a regular irreduciblerepresentation of AWeyl in the separable Hilbert space H L2 (R)Denoting Û (α) : (U (α)) and V̂ ( β) : (V ( β)), the representation isdefined by Û (α)ψ (x) : eiαx ψ(x) ,V̂ ( β)ψ (x) : ψ(x β) , ψ HBy using the Stone theorem, the Schrödinger representation provides alsoa representation of the Heisenberg algebra(X̂ψ)(x) xψ(x) ,0(P̂ψ)(x) iψ (x) ,D(X̂) {ψ ZR xψ(x) 2 dx }D(P̂) {ψ ψ is abs. cont. andZR ψ0 (x) 2 dx }Notice that the position operator acts as a multiplicative operator whereasmomentum operator acts as a derivative operator over the vector states ofthe Schrödinger representationEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200733 / 44
Scrödinger representation on CGiven an orientable configuration space C choose a Riemannian metric gab , withconection a and canonical volume-form e, and let H L2 (C , e)hψ1 ψ2 i : ZCψ1 ψ2 e ,ψ1 , ψ2 HWe can define (densely) configuration and momentum operatorsQ̂(f )ψ: fψP̂(v)ψ: 1 ih̄ Lv ψ (dive (v))ψ2These operators satisfy the canonical commutation relations[Q̂(f1 ), Q̂(f2 )] 0, [Q̂(f ), P̂(v)] ih̄Q̂(Lv f ), [P̂(v1 ), P̂(v2 )] ih̄P̂(Lv1 v2 )Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200734 / 44
Notice that there is a correspondence between the Poisson algebra of the classicalconfiguration and momentum observables and the Lie algebra (under [·, ·]) ofthese quantum operators{Q(f1 ), Q(f2 )} 0, {Q(f ), P(v)} Q(Lv f ), {P(v1 ), P(v2 )} P(Lv1 v2 )[Q̂(f1 ), Q̂(f2 )] 0, [Q̂(f ), P̂(v)] ih̄Q̂(Lv f ), [P̂(v1 ), P̂(v2 )] ih̄P̂(Lv1 v2 )For the class of configuration and momentum observables (and only for this class)the correspondence is given by[A, B] ih̄{\A, B}For a large class of classical mechanical systems on C the classical Hamiltoniantakes the form1H (q, pa ) gab (q)pa pb va (q)pa V (q) .2For this it class is natural to identify the ‘quantum Hamiltonian’ (up to Riccicurvature terms of gab ) asĤψ h̄2 ab1g a b ψ i Lv ψ (dive (v))ψ Vψ22Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200735 / 44
algebraic dynamicsWe are now interested in describing the relations between measurementsat different times for non-dissipative systems. In this case it it plausible todemand the followingThe relations between the measurements at times t1 and t2 dependsonly on the difference t2 t1If an observable A is defined by some experimental device at a giventime, say t 0, the same type of measurements performed at time tdefines an observable AtThe algebra A generated by the observables is the same at any timeThe time translation A 7 αt (A) At is a -automorphism(preserves the algebraic properties)For any state ω and any observable A, the real functiont 7 ω (αt (A)) is a continuous functionEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200736 / 44
Algebraic Dynamical SystemsA dynamical system is a triplet(A, R, α)where A is a C -algebra and, for each t R, αt is an automorphism of Aand α satisfiesα0 id ,αt1 αt2 αt1 t2α is weakly continuous i.e. t 7 ω (αt (A)) is continuous ω and AEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200737 / 44
dynamics and representationsGiven a representation (A, , H) of the C -algebra of observables we willsay that is stable under the evolution αt if and αt are unitarilyequivalent. In other words (αt (A)) Û 1 (t) (A)Û (t) , A Ofor some unitary operatorÛ (t) : H H .The weak continuity of αt implies the weak continuity of Û (t). Thenapplying Stone’s theorem we haveÛ (t) exp( itĤ ) , t Rfor some self-adjoint operator Ĥ, with dense domain D(Ĥ ) HĤ is called the Quantum Hamiltonian (in the representation )The Hamiltonian is a representation dependent concept. In general itis also an unbounded operator and does not belong to the C -algebragenerated by O .Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200738 / 44
Schrödinger equationLet ψ0 a vector state of a stable representation. Thenω0 (αt (A)) hψ0 Û (t) 1 (A)Û (t)ψ0 i hÛ (t)ψ0 (A)Û (t)ψ0 i ωt (A)where ωt is the pure state defined byψ(t) : Û (t)ψ0 HBy choosing ψ0 D(Ĥ ), we can differentiate ψ(t) with respect to t to getthe Schrödinger equationSchrödinger equationThe Scrödinger equation is the time evolution equation for pure statesidψ Ĥψ ,dtψ(0) ψ0 D(Ĥ ) .where Ĥ is the (self-adjoint) quantum Hamiltonian of the systemEduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200739 / 44
Heisenberg equationThe evolution can be equivalently formulated in terms of observables.Given an observable represented byÂ0 : (A0 )and definingÂ(t) : U (t) 1 Â0 U (t) ,we get the Heisenberg equation:Heisenberg equationThe evolution equation for an observable (in a certain representation) isd i[Ĥ,Â] ,dtEduardo J. S. Villaseñor (UC3M)A(0) A0 B(H)Introduction to Quantum MechanicsSeptember 200740 / 44
example: quantum particle in a potentialFor the Scrhödinger representation, given a Hamiltonian Ĥ, the Heisenbergequations are the analog of the classical Hamilton equations:X̂ i[Ĥ, X̂] ,P̂ i[Ĥ, P̂]In particular, for the class of HamiltoniansĤ H (X̂, P̂) , where H (X, P) P2 V (X )2these are (formally)Heisenberg equations for a particle in a potential H(X̂, P̂) P̂ , P HP̂ (X̂, P̂) V 0 (X̂) XX̂ Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200741 / 44
Is the ‘quantum Hamiltonian’Ĥ P̂2 V (X̂)2well defined in the Schrödinger representation?It is easy to show that 12 P̂2 V (X̂) defines a symmetric operator. butsymmetric operators have several or none self-adjoint extensions!!The taste of Kato’s theorems: For a potential V in a Kato class, theCauchy problem for the Scrödinger equation1 2 ψ ψ(x, t) V (x)ψ(x, t)(x, t) t2 x2ψ(x, 0) ψ0 (x) D(P̂2 )iis well possed and the corresponding Cauchy problem has a uniquesolution global in time.Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200742 / 44
summaryWe have provided a description of Quantum Mechanics in which, for systemswhose classical configuration space C is finite dimensionalPhysical observables can be represented as bounded self-adjoint operators Âin the Schrödinger representation on L2 (C , e)Canonical variables do not commute. They cannot be simultaneouslymeasuredPure states are vector states ψ L2 (C , e), ψ 1In contrast with classical mechanics, the observables do not havewell-defined values on pure statesWhen C is infinite dimensional there are important differencesThe Schrödinger representation cannot be defined on C but on a moregeneral space L2 (C , dµ)The von Neumann uniqueness theorem cannot be applied. In fact, oneusually finds an infinite number of inequivalent representations of theC -algebra of elementary variables. In this case, some additional input isneeded to fix a representation (symmetries can be used for this)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200743 / 44
referencesF. Strocchi. ”An Introduction to the Mathematical Structure ofQuantum Mechanics. A Short Course for Mathematicians”. AdvancedSeries in Mathematical Physics, Vol. 27, World Scientific (2005)C. J. Isham. ”Lectures on Quantum Theory: Mathematical andStructural Foundations”. World Scientific (1995)I. E. Segal. ”Mathematical Problems of Relativistic Physics”.Lectures in Applied Mathematics Series. American MathematicalSociety (1967)J. Von Neumann. ”Mathematical Foundations of QuantumMechanics”. Princeton University Press (1965)Eduardo J. S. Villaseñor (UC3M)Introduction to Quantum MechanicsSeptember 200744 / 44
plan of the talk 1 General considerations about physical systems: Observables and states 2 Classical kinematics: Observables and states in Classical Mechanics 3 The crisis of Classical Physics (very very brief!) 4 Quantum kinematics: Observables and states in Quantum Mechanics (Segal approach) 5 The simplest quantum system: The quantum point particle Weyl C -algebra
1. Introduction - Wave Mechanics 2. Fundamental Concepts of Quantum Mechanics 3. Quantum Dynamics 4. Angular Momentum 5. Approximation Methods 6. Symmetry in Quantum Mechanics 7. Theory of chemical bonding 8. Scattering Theory 9. Relativistic Quantum Mechanics Suggested Reading: J.J. Sakurai, Modern Quantum Mechanics, Benjamin/Cummings 1985
quantum mechanics relativistic mechanics size small big Finally, is there a framework that applies to situations that are both fast and small? There is: it is called \relativistic quantum mechanics" and is closely related to \quantum eld theory". Ordinary non-relativistic quan-tum mechanics is a good approximation for relativistic quantum mechanics
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
An excellent way to ease yourself into quantum mechanics, with uniformly clear expla-nations. For this course, it covers both approximation methods and scattering. Shankar, Principles of Quantum Mechanics James Binney and David Skinner, The Physics of Quantum Mechanics Weinberg, Lectures on Quantum Mechanics
Quantum Mechanics 6 The subject of most of this book is the quantum mechanics of systems with a small number of degrees of freedom. The book is a mix of descriptions of quantum mechanics itself, of the general properties of systems described by quantum mechanics, and of techniques for describing their behavior.
mechanics, it is no less important to understand that classical mechanics is just an approximation to quantum mechanics. Traditional introductions to quantum mechanics tend to neglect this task and leave students with two independent worlds, classical and quantum. At every stage we try to explain how classical physics emerges from quantum .
EhrenfestEhrenfest s’s Theorem The expectation value of quantum mechanics followsThe expectation value of quantum mechanics follows the equation of motion of classical mechanics. In classical mechanics In quantum mechanics, See Reed 4.5 for the proof. Av
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