Postulates of quantum mechanics

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1 Postulates of quantum mechanics Armin Scrinzi November 22, Postulates of QM... and of classical mechanics 1.1 An analogy Quantum Classical State Vector Ψ from H Prob. distr. ρ(x, p) on phase space Property Observable, lin. op. A on H Function A(x, p) on phase space Measur.val. a Eigenvalues of A Function values of A(x, p) Probab. for a a Ker(A a) a Ψ 2 dx dp ρ(x, p) Ker(A a) After result a Vector Φ a Ker(A a) Prob. distr. φ a (x, p) on Ker(A a) Dynamics(I) i d dt Ψ(t) = HΨ(t) i d dt ρ = L Hρ (Lie-derivative) Dynamics(II) Ψ(t) = U t Ψ(0), Ψ(t) Ψ(t) 1 ρ(t) = Φ t [ρ(0)], dxdp ρ(t) 1 (I) and (II) are equivalent alternatives The necessarily finite precision of measurement implies that ρ(x, p) always extends over a finite domain. This implies an uncertainty principle also for classical mechanics. However, this is qualitatively different from quantum mechanics. For a more detailed discussion, see Asher Peres: Quantum Theory Concepts and Methods (highly recommended!) The differential geometric analogy to for quantum mechanics can be carried further by actually introducing quantization. See, for example, lecture notes by Bates and Weinstein ( alanw/gofq.pdf). 1

2 1.1.1 Differences Essential: two operators A, B do not commute in general, but A(x, p), B(x, p) do Unessential: Ψ form a linear space, ρ form a convex space. 2 Unitary time evolution and Schrödinger equation More precisely: The time-evolution operators U t of an isolated system form a unitary, strongly continuous, one-parameter group with U t1 +t 2 = U t1 U t2 t 1, t 2 R. Of course, U 0 = 1. This definition (at first sight contrary to the Schroedinger equation) is very well motivated: Linearity: to remain consistent with the linearity assumption of H. Unitary: we want the system to neither grow nor disappear - preserve probability; this should hold for all states of the Hilbert space - all possible states of the system. U t Ψ = 1 Ψ H U t is unitary. Continuous: natura non facit saltus Strongly continuous : in which sense can the above statement be understood? Specify the topology. Additivity of time-evolution: motivated by homogenuity of time. A priory, between any two points in time t, t, we may have a different time-evolution U(t, t ). Of course we should have U(t 1 + t 2, 0) = U(t 1 + t 2, t 1 )U(t 1, 0), i.e. evolving from 0 to t 1 + t 2 is achieved by evolving from 0 to t 1, then from t 1 to t 1 + t 2 Homogenuity means we can shift this such that U(t 1 + t 2, t 1 ) = U(t 2, 0) =: U t2. 2

3 U 0 = 1: in no time, nothing happens. 2.1 Strong continuity Basic idea: for all vectors Ψ H and any ɛ, there exists a δ(ψ, ɛ) > 0 such that (U δψ,ɛ 1)Ψ < ɛ Where denotes the norm in the Hilbert space Continuity in the sense of the operator norm Can be phrased in the terms above: for any ɛ there exists δ(ɛ) independent of Ψ: (U δ(ɛ) 1)Ψ < ɛ Ψ H For unbounded operators, we cannot hope for this. Easy to see if we choose energy eigenstates HΨ E = EΨ E and look at the limit (e iδ(ɛ)h 1)Ψ E = (e iδ(ɛ)e 1)Ψ E. No matter how small δ, we can allways choose E π/δ, such that the prefactor becomes Continuity in the sense of the weak topology Strong continuity implies weak continuity, i.e. lim (U t 1)Ψ for any fixed Ψ t 0 lim Φ (U t 1)Ψ for any fixed pair Φ, Ψ t 0 But weak continuity is of little use by itself: free time-evolution, although a unitary operator, converges weakly to 0 with t. 3

4 2.2 Stone theorem For any strongly continuous one-parameter group as above there exists a self-adjoint operator H defined on the (unique, dense) domain D(H) H such that lim i(u τ 1 H)Ψ = 0 τ 0 τ Ψ D(H) This is just a funny way of writing the Schroedinger equation. Conversely, of course, any self-adjoint H defined on its (unique, dense) domain D(H) H defines a unitary time-evolution U t through the Schroedinger equation. Note that at first glance U t is defined only on D(H), but as D(H) is dense and U t is bounded, it can be extended to the complete Hilbert space. In other words, we can define time-evolutions also for states, whose energy Ψ H Ψ is not mathematically well-defined at first glance, e.g. Ψ(x) with discontinuities in x Self-adjointness Self-adjointness and with it the essential concept of domain are not defined here. They allow the extension of the concept of hermiticity to unbounded operators. Stone theorem shows, why self-adjointness of H is so essential: without it, we cannot get a reasonable time-evolution. 3 Tensor product space All states Ψ (2) of a system combined of two subsystems represented in spaces H a and H b are from the tensor product space Ψ (2) H a H b. 4

5 3.1 What is the tensor product? Of two state vectors ( φ 1, φ 2 ) φ 1 φ 2 with the rules for scalar product and application of the tensor product of operators. The tensor product space is the linear combination of all products with simple and obvious rules for addition, multiplication by scalars, distributivity, and scalar product. Note that this space is much bigger than just twice this single system states. Again, this is nothing new in quantum mechanics, also the two-particle phase space of points x 1, p 1, x 2, p 2 is much bigger than x 1, p 1 and x 2, p 2 separately. In both cases, the additional information about the system are the correlations. 3.2 Why the tensor product space? Answer 1: it works The logically cheap, but epistemologically decisive answer: we know it is right: historically, the Helium atom high precision results, more recent the exact prediction of the violation of Bell s inequalities in experiment Answer 2: it follows from the principles of QM Assume we accept that a physical system is described by a vector φ in Hilbert space and probabilities for measuring a value a of an observable A is φ a a φ with A a = a a. (for simplicity we disregard degenerate and the continuous spectrum eigenstates), i.e. A = a a a a 5

6 Let φ 1 and φ 2 be the state vectors of two physical systems and A and B two observables for the respective systems. Then the probability for measuring a on φ 1 AND b on φ 2 is P φ1,φ 2 (a, b) = φ 1 a a φ 1 φ 2 b b φ 2 Actually, this has exactly the logical structure of the tensor product. Write P a : a a and P b : b b. Then φ 1 φ 2 P a P b φ 1 φ 1 = φ 1 P a φ 1 φ 2 P b φ 2 For products of two independent systems, the tensor product structure is consistent with the probability structure of the single electron systems. But why can we linearly combine all tensor products (=tensor product space)? According to the superposition principle: after all, two independent quantum systems can be considered a single bigger quantum system. By what we have accepted as the general principles of quantum mechanics, any linear combination of legitimate states of the bigger system again form legitimate states of the bigger system. 6