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)
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.
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.
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.
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?
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).
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 1 2 + 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.
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 3 1 1 2i 5 2 5 ( ) ( 2i 4 ) 1 2i 2 1 4 2i 5 1 5 2i 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.
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 3 3 0 where ψ = e iθ and ˆX =. 2 p 2 0 2 Classical r.v. = quantum r.v. with only diagonal entries.
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.
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.
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).