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Thomas Curtright, University of Miami KU Lawrence Colloquium 30 April 2012 Since Heisenberg's 1927 Paper on Uncertainty, There

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Thomas Curtright, University of Miami KU Lawrence Colloquium 30 April 2012 Since Heisenberg's 1927 Paper on Uncertainty, There Thomas Curtright, University of Miami KU Lawrence colloquium 30 April 2012 Since Heisenberg’s 1927 paper on uncertainty, there has been considerable reluc- tance to consider positions and momenta jointly in quantum contexts, since these are incompatible observables. But a discomfort to contemplate both positions and momenta simultaneously in the quantum world is not really warranted, as was Hrst fully understood by Hilbrand Groenewold and José Moyal in the 1940s. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely appreciated until the late 20th century. In this general talk I will discuss elementary features, as well as some of the early history, of this “deformation” approach to quantization. Based on work with David Fairlie, Andrzej Veitia, and Cosmas Zachos. 1 “ phase space has no meaning in quantum mechanics, thereh being no possibility of assigning numerical values simultaneously to the q’s and p’s ” P. A. M. Dirac See p 132, The Principles of Quantum Mechanics, Oxford University Press, 4th edition, last revised 1967. 2 However, please also see: and references therein, as well as the Asia-Paci#c Physics Newsletter, May 2012 (premier issue). 3 1 Some theory and experiments in phase space 4 5 6 7 Oops! f 1 | | ¯h 8 9 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 3 2 0 1 0x -2 -3 Wigner function for the oscillator state 1 ( 0 + i 1 ). 2 | | 10 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 3 2 0 1 0x -2 -3 An oscillator coherent state (displaced Gaussian) in phase space. 11 Oops again! 12 13 f p x Two-slit experiment in phase space (WF for a state consisting of two Gaussians) 14 f p x Two-slit experiment in phase space (WF for a state consisting of two Gaussians) 15 16 Nature 386 (1997) pp 150 - 153. 17 Formalism There are three mathematically equivalent but autonomous formulations of quantum mechanics based on: 1. Hilbert space –Born,Dirac, Heisenberg, Jordan, Schrödinger 2. path integrals – Dirac,Feynman 3. phase-space –anti-Dirac, Groenewold, Moyal, von Neumann, Weyl, Wigner While these are equivalent mathematically, they are not always equivalent psychologically (as my thesis advisor used to say). One formulation may oQer much more insight and provide an easier route to the solution of a particular problem than the others. 2 Wigner Functions Wigner’s original deInition of his eponymous function was (Eqn(5) in [19]) n 1 P (x1, ,xn; p1, ,pn)= dy1 dyn (x1 + y1 xn + yn) ··· ··· ··· ··· ··· 2i(p1y1+ +pnyn)/ (x1 y1 xn yn) e ··· . ··· So deIned, Wigner functions (WFs) reside in phase space. WFs are “Weyl correspondents” of von Neumann’s density operators, . Thus, in terms of Hilbert space position and momentum operators X and P ,wehave1 n n n n n = d d d xd pP (x1, ,xn; p1, ,pn)exp(i (P p)+i (X x)) . (2)2n ··· ··· · · 1 Remark on units: As deIned by Wigner, WFs have units of 1/n in 2n-dimensional phase space. Since it is customary for the density operator to have no units, a compensat- ing factor of n is required in the Weyl correspondence relating WFs to s. Issues about units are most easily dealt with if one works in “action-balanced” x and p variables, whose units are [x]=[p]=. 3 In one x and one p dimension, denoting the WF by f (x, p) instead of P (in deference to the momentum operator P ), we have 1 2ipy/ f (x, p)= dy x + y x y e , | | x + y x y = dp f (x, p) e2ipy/ , | | =2 dxdy dp x + y f (x, p) e2ipy/ x y . | | HWeyl For a quantum mechanical “pure state” 1 2ipy/ f (x, p)= dy (x + y) (x y) e , 2ipy/ (x + y) (x y)= dp f (x, p) e , = , | | where as usual, (x + y)= x + y , x y = (x y). | | EWigner 4 Direct application of the Cauchy—Bunyakovsky—Schwarz inequality to the Irst of these pure-state relations gives 1 f (x, p) dy (y) 2 . | | | | So, for normalized states with dy (y) 2 =1,wehavethebounds | | 1 1 f (x, p) . Such normalized states therefore cannot give probability spikes (e.g. Dirac deltas) without taking the classical limit 0. The corresponding bound in 2n phase-space dimensions is given by the same argument applied to Wigner’s 1 n original deInition: P (x1, ,xn; p1, ,pn) . | ··· ··· | 5 Star Product The star product is the Weyl correspondent of the Hilbert space operator product, and was developed through the work of many over a number of years: H Weyl (1927), J von Neumann (1931), E Wigner (1932), H Groenewold (1946), J Moyal (1949), and G Baker (1958), as well as more recent work to construct the product on general manifolds (reprinted in [20], along with related papers). There are useful integral and diQerential realizations of the product. The integral form is dx1dp1 dx2dp2 i f g = f (x + x1,p+ p1) g (x + x2,p+ p2)exp (x1p2 x2p1) , 2 ( /2) 2 ( /2) /2 x1p2 x2p1 = Area (1,2 parallelogram) , /2=Planck Area = min (xp) , while the diQerential form is i i f g = f (x, p)exp x p p x g (x, p) , 2 2 6 1 1 f g = f x + i p ,p i x g (x, p) 2 2 1 1 = f (x, p) g x i p ,p+ i x 2 2 1 1 = f x + i p ,p g x i p ,p 2 2 1 1 = f x, p i x g x, p + i x . 2 2 The Moyal bracket, f,g = 1 [f,g] , is essentially just the antisymmetric i part of a star product,{{ where}} [f,g] defn.= f g g f. This provides a homomorphism with commutators of operators, e.g. [x, p] = i. 7 Pure States and Star Products Pure-state Wigner functions must obey a projection condition. If the normalization is set to the standard value + dxdp f (x, p)=1, then the function corresponds to a pure state if and only if f =(2) f f. These statements correspond to the pure-state density operator conditions: Tr()=1and = , respectively. If both of the above are true, then f describes an allowable pure state for a quantized system. Otherwise not. You can easily satisy only one out of these two conditions, but not the other, using an f that is not a pure state. Without drawing on the Hilbert space formulation, it may at Irst seem to be rather remarkable that explicit WFs actually satisfy the projection condition (cf. the above Gaussian example, for the only situation where it works, a = b =1, i.e. exp ( (x2 + p2) /)). However, if f is known to be a eigenfunction with non-vanishing eigenvalue of some phase-space function with a non-degenerate spectrum of eigenvalues, then it must be true that f f f as a consequence of associativity, since both f and f f would yield the same eigenvalue. 8 Exercises Exercise 1 Non-commutativity. ax+bp Ax+Bp (a+A)x+(b+B)p (aB bA)i/2 e e = e e = Ax+Bp ax+bp (a+A)x+(b+B)p (Ab Ba)i/2 e e = e e Exercise 2 Associativity. eax+bp eAx+Bp ex+p (a+A+)x+(b+B+)p (aB bA+a b+A B)i/2 = e e = eax+bp eAx+Bp ex+p Exercise 3 Trace properties. (a.k.a. “Lone Star Lemma”) dxdp f g = dxdp f g = dxdp g f = dxdp g f Exercise 4 Gaussians. For a, b 0, a b 1 a + b exp x2 + p2 exp x2 + p2 = exp x2 + p2 . 1+ab (1 + ab) 9 The Simple Harmonic Oscillator There is no need to deal with wave functions or Hilbert space states. The WFs may be constructed directly on the phase space [9, 2]. Energy eigenstates are obtained as (real) solutions of the -genvalue equations [6]: H f = Ef = f H. To illustrate this, consider the simple harmonic oscillator (SHO) with (m =1, =1) 1 H = p2 + x2 . 2 The above equations are now second-order partial diQerential equations, 1 1 2 1 2 H f = p i x + x + i p f = Ef , 2 2 2 1 1 2 1 2 f H = p + i x + x i p f = Ef . 2 2 2 But if we subtract (or take the imaginary part), 2 2 (p x x p) f =0 f (x, p)=f x + p . So H f = Ef = f H reducestoasingleordinary diQerential equation (Laguerre, not Hermite!), namely, the real part of either of the previous second-order equations. 10 There are integrable solutions if and only if E =(n +1/2) , n =0, 1, for which ··· n 2 2 ( 1) x + p (x2+p2)/ fn (x, p)= Ln e , /2 n 1 z d n z L (z)= e z e . n n! dzn + The normalization is chosen to be the standard one dxdp fn (x, p)=1. Except for the n =0ground state (Gaussian) WF, these f’s change sign on 1 2 the xp-plane. For example: L0 (z)=1,L1 (z)=1 z, L2 (z)=1 2z + 2 z , etc. Using the integral form of the product, it is now easy to check these pure states are orthogonal: (2) fn fk = nk fn . This becomes more transparent by using raising/lowering operations to write2 1 n n f = (a ) f ( a) n n! 0 1 n (x2+p2)/ n = (a ) e ( a) , n! 1 where a is the usual linear combination a (x + ip), and a is just 2 1 its complex conjugate a (x ip) ,witha a a a =1, and 2 a f0 =0=f0 a (cf.
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