Wigner Functions for the Canonical Pair Angle and Orbital Angular Momentum Hans Kastrup Theory Group DESY Hamburg Institute for Theoretical Physics RWTH Aachen 2. Intern. Wigner Workshop, June 5, 2017 HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW 2 , June 5, 2017 1 / 30
30
Embed
Wigner Functions for the Canonical Pair Angle and Orbital ...bib-pubdb1.desy.de/record/331029/files/IW2.pdf · 2. Intern. Wigner Workshop, June 5, 2017 ... Wigner functions for S1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Wigner Functions for the Canonical PairAngle and Orbital Angular Momentum
Hans Kastrup
Theory Group
DESY Hamburg
Institute for Theoretical Physics
RWTH Aachen
2. Intern. Wigner Workshop, June 5, 2017
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 1 / 30
Outline
Wigner functions for S1 × R: angle and orbital angular momentum
Previous difficulties with Wigner functions on Pθ,p ∼= S1 × RSolution of the quantum angle problemEuclidean group E(2) of the planeWigner operator on Pθ,pWigner function of wave functions ψ(ϕ)ExamplesConclusions and OutlookA personal tribute to Eugene Wigner
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 2 / 30
Wigner functions for Pθ,p
Wigner functions for S1 × R:angle and orbital angular momentum
Typical example: Rotator around a fixed axis
• L(θ, θ) = m2 (x2 + y2)− V (θ) =
m r20
2 θ2 − V (θ),
• x = r0 cos θ, y = r0 sin θ; V (θ + 2π) = V (θ);
• (in case of pendulum: V (θ) = mgr0(1− cos θ) )• pθ = ∂L/∂θ = mr2
0 θ: orbital angular momentum,
• H(θ,pθ) = 12mr2
0p2θ + V (θ),
• V (θ) = 0 in the following.• pθ = −∂H/∂θ = 0⇒ pθ = const . ∈ R.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 3 / 30
Wigner functions for Pθ,p Previous difficulties with Wigner functions on Pθ,p ∼= S1 × R
The previous two main obstacles of goingfrom Wigner functions on R× R to those on S1 × R
Construction of Wigner functions on S1 × Rin analogy to those on R× R:First papers by Berry (1977) and Mukunda (1979)- and then many more - met two basic obstacles:• Quantizing the angle θ!
Only formal, mathematically unsatisfactory “solutions”!• Compatibility of continuous classical p and
discontinuous quantum l = ~m, m ∈ Z. (~ = 1 in the following)Attempted way out: make classical p discontinuous, too:S1 × R→ S1 × Z:no longer a classical phase space (cotangent bundle)
Removel of both obstacles in what follows!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 4 / 30
Wigner functions for Pθ,p Previous difficulties with Wigner functions on Pθ,p ∼= S1 × R
The difficulties of quantizing the angle
• canonical coordinates:q = θ ∈ [0,2π) ≡ R mod 2π, pθ ≡ p = mr2
• Many attempts since 1926 to circumvent the problem, at leastformally!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 6 / 30
Wigner functions for Pθ,p Solution of the quantum angle problem
Solution of the problem in terms of sin θ and cos θAngle, geometrically: point of intersection of a ray from the origin withthe unit circle S1 around the origin;point is uniquely determined by ~n = (cos θ, sin θ) or χ~n, χ > 0.
~n = (cos θ, sin θ) ∈ S1
r =1
θcos θ
sinθ
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 7 / 30
Wigner functions for Pθ,p Solution of the quantum angle problem
Use the pair (cos θ, sin θ) instead of θ itself for quantization (1963:Mackey and Louisell independently; HKa, PRA 73, 052104 (2006)),• basis functions (observables):