Imprints of Classical Mechanics in the Quantum World

Transcription

1 Imprints of Classical Mechanics in the Quantum World Schrödinger Equation and Uncertainty Principle Maurice de Gosson University of Vienna Faculty of Mathematics, NuHAG October 2010 (Institute) Slides - Beamer October / 15

2 1. The Uncertainty Principle (Institute) Slides - Beamer October / 15

3 Robertson Schrödinger inequalities The Heisenberg inequalities X j P j 1 2 h are found in every elementary textbook on QM. They follow from the sharper Robertson Schrödinger inequalities ( X j ) 2 ( P j ) 2 (Cov(X j, P j )) h2 (1) which take into account the covariances. That these inequalities do not necessarily characterize the quantum regime is easy to see: choose any trace class operator ρ with trace Tr ρ = 1 and calculate the (co)variances in the usual way. Then one gets (1). But the state corresponding to ρ is not a quantum state unless in addition ρ 0... We are going to see that the Robertson Schrödinger inequalities actually have a perfectly classical interpretation (de Gosson, Found. Phys. 39: , 2009; also see E. Reich s New Scientist article "How camels could explain quantum uncertainty") (Institute) Slides - Beamer October / 15

4 Some classical statistics... Here is cloud of points representing joint position-momentum measurements in 2n dimensional phase space: Due to experimental errors, etc. there are in practice many "outliers" to eliminate. (Institute) Slides - Beamer October / 15

5 Some classical statistics... The red ellipsoid is the minimum volume ellipsoid (MVE). It is used in robust multivariate statistics (Rousseeuw, Van Aelst,...) as a powerful estimator of the covariance matrix Σ which eliminates e ciently most the outliers. The equation of the ellipsoid is 21 (z z )T Σ 1 (z z ) 1. (Institute) Slides - Beamer October / 15

6 The symplectic camel Now comes the symplectic camel (alias Gromov s symplectic non-squeezing theorem, 1985). It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God (Mark 10:25). (Institute) Slides - Beamer October / 15

7 The symplectic camel In addition of being volume-preserving (Liouville s theorem), Hamiltonian phase flows have an unexpected additional property as soon as the number of degrees of freedom is superior to one; this property is a consequence of the symplectic non-squeezing theorem which was proved in 1985 by Mikhail Gromov. Make a circular "hole" with radius r in any of the conjugate coordinate planes x j, p j (the eye of the needle) and choose a phase space ball with radius R (the camel). If R > r it is impossible to "squeeze" the camel through the eye of needle using canonical transformations (in particular Hamiltonian phase flows). But it is easy to do so if one uses ordinary volume-preserving transformations. (Institute) Slides - Beamer October / 15

8 Symplectic capacity The symplectic camel leads to the notion of symplectic capacity. The symplectic capacity c(ω) of a set of points Ω in phase space R n x R n p is πr 2 where R is the radius of the largest ball that can be sent inside Ω using canonical transformations. Choose for Ω the covariance ellipsoid 1 2 (z ẑ)t Σ 1 (z ẑ) 1 and let h be a number such that This condition is equivalent to with h = h/2π and J = c(ω) 1 2 h. Σ + i h 2 J 0 ( ) 0 I (the standard symplectic matrix). I 0 (Institute) Slides - Beamer October / 15

9 Classical uncertainty relations Write now with Σ = ( X 2 ) (X, P) (P, X ) P 2 Σ XX = (Cov(X j, X k )) j,k, Σ PP = (Cov(P j, P k )) j,k Σ XP = (Cov(X j, P k )) j,k, Σ PX = (Cov(P j, X k )) j,k. The equivalent conditions c(ω) 1 i h 2 h and Σ + 2 J 0 imply ( X j ) 2 ( P j ) 2 (Cov(X j, P j )) h2. NOTHING QUANTUM IN THE ARGUMENT ABOVE! (Institute) Slides - Beamer October / 15

10 2. Derivation of Schrödinger s Equation (Institute) Slides - Beamer October / 15

11 Schrödinger equation Where did that [the Schrödinger equation] come from? Nowhere. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations in the real world. (Richard Feynman in The Feynman Lectures on Physics, III, p.16 12) Similar statements abound in the physical literature; they are found in both introductory and advanced text on quantum mechanics, and we can read them on the web in various blogs and forums. They are strictly mathematically speaking not true. (Institute) Slides - Beamer October / 15

12 Claim Theorem There is a one-to-one and onto correspondence between Hamiltonian phase flows generated by a Hamiltonian H and strongly continuous unitary one-parameter groups satisfying Schrödinger s equation with Hamiltonian operator H(x, i h x, t) obtained from H by Weyl quantization. The Hamilton equations ẋ = p H(x, p, t), ṗ = x H(x, p, t) are thus mathematically rigorously equivalent to Schrödinger s equation i h ψ t = H(x, i h x, t)ψ. (Institute) Slides - Beamer October / 15

13 Proof. (M. de Gosson 2010, M. de Gosson & B. Hiley, 2010). The correspondence C exists for quadratic Hamiltonians: H = 1 2 zt Mz, ( ) x z =. In this case, ft p H = st H where st H Sp(n). The double covering π : Mp(n) Sp(n) ( metaplectic representation of the symplectic group ) allows us to lift the one-parameter group (st H ) to a one-parameter group (St H ) of unitary operators in Mp(n) and these operators satisfy i h d dt S H t = H(x, i h x, t)s H t The general case is based on the following observation: we have C(ss H t s 1 ) = SC(s H t )S 1 for every s Sp(n) where S is any of the two metaplectic operators corresponding to s ( covariance under linear canonical transformations ) so we require that more generally C(sft H s 1 ) = SC(ft H )S 1. (Institute) Slides - Beamer October / 15

14 Now we invoke Stone s theorem which says that each strongly continuous unitary one-parameter group Ut H of operators on L 2 (R n ) determines a self-adjoint operator Ĥ by the formula Ut H = e iĥt/ h. Equivalently i h d dt UH t = ĤU H t. There remains to prove that Ĥ = H(x, i h x, t). But the condition C(sft H s 1 ) = SC(ft H )S 1 implies Ĥ s = ŜHS 1 and this property is characteristic of Weyl quantization, hence Ĥ = H(x, i h x, t) as claimed. NOTHING QUANTUM IN THE ARGUMENT ABOVE! (Institute) Slides - Beamer October / 15

15 When does QM emerge from CM? There are mathematical possibilities (use of the Wigner O Connell spectrum) but they are rather technical... In conclusion it may very well be said that information is the irreducible kernel from which everything else flows. The question why Nature appears quantized is simply a consequence of the fact that information itself is quantized. It might even be fair to observe that the concept that information is fundamental is very old knowledge of humanity, witness for example the beginning of gospel according to John: In the beginning was the Word and the Word was with God, and the Word was God. 1 (Institute) Slides - Beamer October / 15

16 When does QM emerge from CM? There are mathematical possibilities (use of the Wigner O Connell spectrum) but they are rather technical... Perhaps, after all, the answer ultimately lies in information theory. It might very well be that the discrete nature of information is the key for the passage from classical mechanics to quantum theory. Paraphrasing Anton Zeilinger 1 In conclusion it may very well be said that information is the irreducible kernel from which everything else flows. The question why Nature appears quantized is simply a consequence of the fact that information itself is quantized. It might even be fair to observe that the concept that information is fundamental is very old knowledge of humanity, witness for example the beginning of gospel according to John: In the beginning was the Word and the Word was with God, and the Word was God. 1 (Institute) Slides - Beamer October / 15