Postulates of quantum mechanics viewed as a non-commutative extension of probability theory with a dynamical law

Transcription

1 Postulates of quantum mechanics viewed as a non-commutative extension of probability theory with a dynamical law Dimitri Petritis Institut de recherche mathématique de Rennes Université de Rennes 1 et CNRS (UMR 6625)

2 Phase space and states Phase space and pure states Time evolution Observables Measurements Postulate Phase space quantum system: complex separable Hilbert space H. Pure states are unit vectors a of H. a Strictly speaking, equivalence classes of unit vectors differing by a global phase, called rays, so that S p H 1/ and h h h = λh, λ C, λ = 1. Hilbert space: C-Banach space, with sesquilinear form : H H C, s.t. α 1 φ 1 + α 2 φ 2 ψ = α 1 φ 1 ψ + α 2 φ 2 ψ, φ ψ = ψ φ, ψ 2 = ψ ψ. Ray: ψ H, ψ = 1. But H separable ψ 2 = n ψ n 2 = 1. Hence ( ψ n 2 ) n probability vector on set indexing the basis of H.

3 Time evolution Simplified version of postulates of QM Phase space and pure states Time evolution Observables Measurements Postulate Time evolution of isolated quantum system described by a unitary operator U acting on H. Conversely, any unitary operator acting on H corresponds to possible invariance a of the system. a Time evolution of isolated system leaves physical quantity energy invariant. Unitary operators are associated with conserved quantities. U unitary: U U = UU = I U 1 = U. φ ray: Uφ 2 = Uφ Uφ = φ U Uφ = φ 2 = 1. Hence Uφ ray.

4 Sharp and unsharp observables Phase space and pure states Time evolution Observables Measurements Postulate General real sharp observable: self-adjoint X acting on H. Elementary real sharp observables associated with X : B(R) B M(B) P(H); M spectral projector of X. Unsharp real observable: resolution of identity into positive operators that are not necessarily projections. X = X spec X := X R. M spectral measure of X ; supp M = spec X but can trivially be extended on R. X = spec X M(dx)x.

5 Phase space and pure states Time evolution Observables Measurements Measurements (innocent-looking but against intuition) Postulate Let M spectral measure of sharp observable X. Measuring observable X in pure state ψ is asking whether X has spectral values in B and determining probability π ψ M (B) of occurrence, by S p ψ π ψ M (B) := ψ M(B)ψ [0, 1]. π ψ M M 1(B(R)) supported by X := spec X. X = spec X M(dx)x E ψ (X ) = X πψ M (dx)x = ψ M(dx)ψ x = ψ X ψ. X M in CM indicator; in QM projection operator acting on H. Quid if 2 observables with spectral measures M and N?

6 Composite systems Phase space and pure states Time evolution Observables Measurements Postulate System composed from two subsystems described respectively by H 1 and H 2 is described by H 1 H 2 (where denotes tensor product). Blackboard 1: Tensor product (a first look).

7 A very simple illustrative example Phase space, states, and time evolution H = C 2 ; f H : f = f 1 ɛ 1 + f 2 ɛ 2, ɛ 1 = ( ) 1 ; ɛ 0 2 = ( ) 0. 1 If f 0, then φ = f f S p, i.e. φ φ 2 2 = 1, therefore φ 1, φ 2 amplitudes of probability on {1, 2}. ɛ 1, ɛ 2 S p. φ 1 2 = P(system in state φ is in state ɛ 1 ). If φ 1 0, φ 2 0, the pure state φ is in a linear superposition of other pure states. Time evolution of isolated system is invertible and preserves pure states.

8 A very simple illustrative example Observables I ( ) 1 2i Take X = = X 2i 2 as an example of sharp observable. X = spec X = { 3, 2}. Compute easily Eigenvalues Eigenvectors Orthoprojectors x u x M x ( ) ( ) 1 i i ( ) ( 2i 4 ) 1 2i i i 1 x spec X M x = I H, M x 0 and Mx 2 = M x. Thus (M x ) family of sharp elementary observables. For x spec X, M x H = Cu x. X = X u x u y = δ x,y, x, y spec X M x, M y orthogonal orthoprojections.

9 A very simple illustrative example Observables II Since (u x ) x orhonormal basis, S p ψ = x α xu x, with x α x 2 = 1, i.e. ( α x 2 ) x probability vector on spec X. E ψ X = ψ X ψ = x α xu x ( x M xx) x α x u x = x x α x 2. If X was a classical X-valued r.v., with probability (p x ) x X, EX = x xp x = x px x p x = x e iθx p x x p x e iθx = ψ ˆX ψ, ( e iθ 3 ) ( ) p where ψ = e iθ and ˆX =. 2 p Classical r.v. = quantum r.v. with only diagonal entries.

10 A very simple illustrative example Measurement System in ψ S p. Ask question M x, i.e. does X takes value x? and determine its probability. Answer: yes, with { probability π ψ M (x) = ψ M xψ = M x ψ 2, ux ψ u where M x ψ = x if x X, 0 otherwise. BUT: Once the question has been asked, and got positive answer, the system now in new state φ x = ux ψ ux u x ψ. Re-asking the question, gives answer yes with probability 1. Asking a question on a quantum system, IRREVERSIBLY alters its state.

11 A very simple illustrative example Composite systems H = H 1 H 2, H i C 2. Each H i with basis (ɛ 1, ɛ 2 ). Basis of H (ɛ 1 ɛ 1, ɛ 1 ɛ 2, ɛ 2 ɛ 1, ɛ 2 ɛ 2 ). H ψ = ψ 11 ɛ 1 ɛ 1 + ψ 12 ɛ 1 ɛ 2 + ψ 21 ɛ 2 ɛ 1 + ψ 22 ɛ 2 ɛ 2. If ψ 11 = ψ 12 = 0, ψ 21 0; ψ 22 0, then ψ = ψ 21 ɛ 2 ɛ 1 + ψ 22 ɛ 2 ɛ 2 = ɛ 2 (ψ 21 ɛ 1 + ψ 22 ɛ 2 ), i.e. ψ = ψ (1) ψ (2). Joint probability = product measure (independence). If both ψ 11 0; ψ 22 0 then ψ ψ (1) ψ (2) (non-independence). Exercise: is-it possible classically? Quantenverschränkung, Quantum entanglement. Intrication (enchevêtrement) quantique. Probably Entrelazamiento (enredo) cuántico in Spanish.

12 An artist s view of entanglement Figure: Ruth Bloch. Entanglement (bronze, 71cm, 1995). (From the web page of the artist, reproduced here with her permission; the existence of this sculpture has been brought to my attention by Dénes Petz).