Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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
0 θ ∈ R (rotator),
• phase space: (θ,p) ∈ Pθ,p ∼= S1 × R,
• θ is discontinuous at the borders 0 and 2π!
• (local) Poisson brackets:{f1, f2}θ,p = ∂θf1(θ,p)∂p − ∂pf1(θ,p)∂θf2(θ,p), θ ∈ (0,2π),{θ,p}θ,p = 1,
• Quantum mechanics: θ → Θ, p → L, [Θ,L] ≡ Θ · L− L ·Θ = i ;
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 5 / 30
Wigner functions for Pθ,p Previous difficulties with Wigner functions on Pθ,p ∼= S1 × R
• angular momentum operator L has eigenvaluesn ∈ Z , i.e. n = 0,±1,±2, ..., L|n〉 = n|n〉,
• 〈n|Θ · L− L ·Θ|n〉 = i〈n|n〉,⇒ (n − n)〈n|Θ|n〉 = i ,
• ⇒ 0 = i~: contradiction!! Operator Θ cannot exist!
• 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):
h1 = cos θ, h2 = sin θ, h3 = pθ ≡ p.
• on classical phase space
Pθ,p = {(θ,p); θ ∈ R mod 2π, p ∈ R }
• hj obey Poisson brackets
{h3, h1}θ,p = h2, {h3, h2}θ,p = −h1, {h1, h2}θ,p = 0 :
• Lie algebra of the 3-parametric Euclidean group E(2) of the plane.• Unitary representations of E(2) provide QM on Pθ,p.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 8 / 30
Wigner functions for Pθ,p Euclidean group E(2) of the plane
Euclidean group E(2)
Action of E(2) = {g(ϑ,~a)} on plane: ~x → R(ϑ) · ~x + ~a,•
g(ϑ,~a) ◦ ~x =
(cosϑ − sinϑsinϑ cosϑ
)(x1x2
)+
(a1a2
),
or•
g(ϑ,~a) ◦
x1x21
=
cosϑ − sinϑ a1sinϑ cosϑ a2
0 0 1
x1x21
;
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 9 / 30
Wigner functions for Pθ,p Euclidean group E(2) of the plane
• generators of infinitesimal transformations:
L =
0 −1 01 0 00 0 0
, K1 =
0 0 10 0 00 0 0
, K2 =
0 0 00 0 10 0 0
,• Lie algebra commutators:
[L, K1] = K2, [L, K2] = −K1, [K1, K2] = 0 ;
isomorphic to Poisson/Lie algebra of hj !
• QM: hj become self-adjoint operators p → L and(cos θ, sin θ)→ ~K = (K1,K2).
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 10 / 30
Wigner functions for Pθ,p Euclidean group E(2) of the plane
Hilbert space
• Hilbert space Hϕ for unitary representations of Euclidean group:
(ψ2, ψ1) =
∫ π
−π
dϕ2π
ψ∗2(ϕ)ψ1(ϕ),
with basis
en(ϕ) = einϕ, (em,en) = δmn, m,n ∈ Z,
• ψ(ϕ+ 2π) = ψ(ϕ):
ψ(ϕ) =∑n∈Z
cn en(ϕ), cn = (en, ψ),
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 11 / 30
Wigner functions for Pθ,p Euclidean group E(2) of the plane
Action of operators
•~Kψ(ϕ) = χ (cosϕ, sinϕ)ψ(ϕ), χ > 0,
Lψ(ϕ) =1i∂ϕψ(ϕ),
• ~K : multiplication operator,• en(ϕ) are eigenfunctions of angular momentum operator:
L en(ϕ) = n en(ϕ),
• If A is operator in Hϕ its matrix elements Amn are given byAmn = (em,Aen)
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 12 / 30
Wigner functions for Pθ,p Wigner operator on Pθ,p
Wigner operator on the cylindrical Pθ,pConstruction of Wigner operator on Pθ,p, partially following B. Wolf etal.:• Start with the group element
g0(ϑ,~b) = exp(b1K1 + b2K2 + ϑL)(R(ϑ) sinc(ϑ/2) R(ϑ/2)~b0 0 1
),
sinc x ≡ sin xx
,
• Translations now parametrized in an angle-dependent way:
~a = sinc(ϑ/2) R(ϑ/2)~b,
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 13 / 30
Wigner functions for Pθ,p Wigner operator on Pθ,p
• It follows that
g0(ϑ,~b) = e(ϑ/2)L ◦ esinc(ϑ/2)~b·~K ◦ e(ϑ/2)L,
or, withR(−ϑ/2)~a = sinc(ϑ/2)~b,
thatg0(ϑ,~a) = e(ϑ/2)L ◦ e[R(−ϑ/2)~a]·~K ◦ e(ϑ/2)L.
Kind of Weyl symmetrization!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 14 / 30
Wigner functions for Pθ,p Wigner operator on Pθ,p
• Given the framework of unitary representations of E(2)appropriate Wigner operator can be constructedfrom unitary operator
U0(ϑ,~a) = ei(ϑ/2)L ◦ ei~a(−ϑ/2)·~K ◦ ei(ϑ/2)L,
~xϑ ≡ R(ϑ) · ~x ,• by kind of group averaging (similar to the planar case):
V [~χ(θ),p] =1
(2π)3
∫ π
−πdϑ∫ ∞−∞
da1da2 U0,
U0 = ei(L−p)(ϑ/2) ◦ ei(~K−~χ(θ))·~a(−ϑ/2)
◦ei(L−p)(ϑ/2).
• ~χ(θ) = χ~n, χ > 0, ~n = (cos θ, sin θ).
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 15 / 30
Wigner functions for Pθ,p Wigner operator on Pθ,p
Wigner matrixMatrix elements (em,V [~n(θ),p]en) lead to following Wigner matrixV (θ,p) = (Vmn):•
Vmn(θ,p) =1
(2π)2 ei(n−m)θ
∫ π
−πdϑei[(n+m)/2−p]ϑ
=1
2πei(n−m)θ sincπ[p − (m + n)/2],
• wheresinc x =
sin xx
, sinc(x = 0) = 1,
−3 −2 −1 1 2 30
1
p−m
2πVm(θ, p) = sincπ(p−m)
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 16 / 30
Wigner functions for Pθ,p Wigner function of wave functions ψ(ϕ)
Wigner function of a wave function ψ(ϕ)
• ψ(ϕ) =∑
n∈Z cnen(ϕ),•
Vψ(θ,p) =1
(2π)2
∫ π
−πdϑe−ipϑψ∗(θ − ϑ/2)ψ(θ + ϑ/2)
=∑
m,n∈Zc∗mVmn(θ,p)cn
= (ψ,V (θ,p)ψ),
• Vψ real because V = (Vmn) is Hermitean ( V ∗mn = Vnm).
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 17 / 30
Wigner functions for Pθ,p Wigner function of wave functions ψ(ϕ)
Some general Properties of Vψ(θ, p)• Marginal distributions in the planar case:∫
Rdp Wφ(q,p) = |φ(q)|2,
∫R
dq Wφ(q,p) = |φ(p)|2.
• Marginal distributions in the (θ,p) case are: |ψ(θ)|2and |cm|2,m ∈ Z; but
• ∫ ∞−∞
dp Vψ(θ,p) =1
2π|ψ(θ)|2,
• ∫ π
−πdθVψ(θ,p) =
∑n∈Z|cn|2 sincπ(p − n) ≡ ωψ(p) :
Whittaker’s cardinal interpolation function (1915)!Important tool in sampling and signal processing theories!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 18 / 30
Wigner functions for Pθ,p Wigner function of wave functions ψ(ϕ)
• As ∫ ∞−∞
dp sincπ(m − p) sincπ(n − p) = δmn,
• one has ∫ ∞−∞
dp ωψ(p) sincπ(p −m) = |cm|2.
• Transition probabilities∫ ∞−∞
dp∫ π
−πdθVψ2(θ,p) Vψ1(θ,p) =
12π|(ψ2, ψ1)|2.
•tr(A · B) = 2π
∫ ∞−∞
dp∫ π
−πdθ tr[A · V (θ,p)] tr[B · V (θ,p)].
• expectation value of operator A for a given density matrix ρ:
〈A〉ρ = tr(ρ · A)
= 2π∫ ∞−∞
dp∫ π
−πdθ tr[ρ · V (θ,p)] tr[A · V (θ,p)],
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 19 / 30
Wigner functions for Pθ,p Wigner function of wave functions ψ(ϕ)
•A (θ,p) ≡ tr[A · V (θ,p)]
analogue to Weyl symbol in planar case!• For the product A · B one has
AB(θ,p) = A(θ,p) e~2i Λ(θ,p)B(θ,p)
≡ A(θ,p) ? B(θ,p)
A2(θ,p) = A2(θ,p).
Λ(θ,p) =←∂ p→∂ θ −
←∂ θ→∂ p,
• Representation of Hilbert space operator A in terms of its symbol:
A = (Amn) =
∫ ∞−∞
dp∫ π
−πdθV (θ,p) tr[A · V (θ,p)]
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 20 / 30
Wigner functions for Pθ,p Examples
Examples for (θ, p) Wigner functions
1. Wigner function for the basis function em(ϕ)
• ψ(ϕ) = em(ϕ) : cm = 1, cn = 0 for n 6= m,•
Vm(θ,p) = (1/2π) sincπ(p −m),
−3 −2 −1 1 2 30
1
p−m
2πVm(θ, p) = sincπ(p−m)
• Vm(θ,p) is negative for certain p values!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 21 / 30
Wigner functions for Pθ,p Examples
2. Wigner function for a simple “cat” state
• e1(ϕ) = eiϕ and e−1(ϕ) = e−iϕ different eigenfunctions ofHamiltonian H = εL2, both with same eigenvalue ε;Le1 = (+1)e1; Le−1 = (−1)e−1;
• Here: +1: “cat alive”, -1: “cat dead”!• superposition f+(ϕ) = 1√
2[eiϕ + e−iϕ] =
√2 cosϕ, (f+, f+) = 1,
• density |f+(ϕ)|2 = 1 + cos 2ϕ = 2 cos2 ϕ,• cos 2ϕ: interference/entanglement term.• operators: L = (1/i)∂ϕ, C = cosϕ, S = sinϕ.
(f+,Lf+) = 0, (f+,L2f+) = 1; (f+,Cf+) = 0, (f+,C2f+) = 3/4;
(f+,Sf+) = 0, (f+,S2f+) = 1/4; (f+,C · Lf+) = 0, (f+,L ·Cf+) = 0.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 22 / 30
Wigner functions for Pθ,p Examples
Wigner function of f+(ϕ) = 1√2
[eiϕ + e−iϕ] :
2πVf+(θ,p) = cos 2θ sincπp +12
[sincπ(p + 1) + sincπ(p − 1)].
−3 −2 −1 1 2 3
−1
0
1
p
2πVf+(θ, p)
θ = 0,±π θ = ±π8,± 7π
8θ = ±π
4,± 3π
4
θ = ± 3π8,± 5π
8θ = ±π
2
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 23 / 30
Wigner functions for Pθ,p Examples
cos(2θ)sinc(πp) +0.5[sinc(π(p-1))+sinc(π(p+1))]0.5
0-0.5
−π
0
π
θ
−3−2
−10
12
3
p
−1−0.5
00.5
1
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 24 / 30
Wigner functions for Pθ,p Examples
3. Wigner function of a qubit
Consider e1(ϕ) and e−1(ϕ) as basis of qubit!:
χα,β(ϕ) = cosβe1(ϕ) + sinβeiαe−1(ϕ).
Properties:
(χα,β, χα,β) = 1, Lχα,β = χα+π,β, L2χα,β = χα,β.
Wigner function:
Vχα,β (θ,p) =1
2π{sin 2β cos(2θ − α) sincπp
+ cos2 β sincπ(p − 1) + sin2 β sincπ(p + 1)}.
Previous examples special cases: α = 0, β = 0 and α = 0, β = π/4.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 25 / 30
Wigner functions for Pθ,p Examples
Marginal distributionsθ marginal distribution:∫ ∞
−∞dp Vχα,β (θ,p) =
12π
[sin 2β cos(2θ − α) + 1] =1
2π|χα,β(θ)|2.
Whittaker cardinal function for p and m = ±1:∫ π
−πdθVχα,β (θ,p) = cos2 β sincπ(p−1)+sin2 β sincπ(p+1) ≡ ωχα,β (p).
Marginal probability for p = 1:∫ ∞−∞
dp ωχα,β (p) sincπ(p − 1) = cos2 β.
Marginal probability for p = −1:∫ ∞−∞
dp ωχα,β (p) sincπ(p + 1) = sin2 β.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 26 / 30
Wigner functions for Pθ,p Conclusions and Outlook
Conclusions and Outlook
• Wigner functions for cylindrical phase spaces aremathematically and physically consistent,structurally very similar to those of planar phase spaces.
• Many more properties,e.g. ? products or quantum Liouville equations etc.,can be found in my last two PRA papers.
• The simple example of a cat state (more generally:angular momentum qubit) suggests a possible rolefor quantum information science etc.
• Two more typical examples in PRA 94:Minimal uncertainty states (von Mises distribution)and thermal states.
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 27 / 30
Wigner functions for Pθ,p A personal tribute to Eugene Wigner
A personal tribute to Eugene Wigner
• In 1962 I got my PhD from the University of Munich;Thesis: Conformal symmetries of field theories.
• In 1963 Wigner received the Nobel prize.• In his Nobel Lecture he quoted a publication of mine
[Phys.Lett.3 (1962),78], in which I proposedconformal transformations as asymptotic symmetriesat very high energies.
• This led to an invitation by Wigner to cometo Princeton for a year!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 28 / 30
Wigner functions for Pθ,p A personal tribute to Eugene Wigner
• I was in Princeton in the academic year 1965/66;(in 1964/65 in Berkeley).
• Wigner and I had different ideasabout possible applications of the conformal group!
• So there was no joint publication!• After a while the interpretation of conformal transformations
as an asymptotic symmetry became generally acceptedand an elaborate analysis of conformally invariant field theoriestook off!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 29 / 30
Wigner functions for Pθ,p A personal tribute to Eugene Wigner
• At that time in Princeton there wasn’t any mentionof the Wigner function between Wigner and myself!
• I learnt about it much later (and initially was sceptical).• But I should have known earlier, because one of my
Thesis advisers, Fritz Bopp, had written about it in 1956 and 1961!• So now my last two papers are my late tribute to Eugene Wigner,
who played an important rolein my education as a theoretical physicist!
HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 30 / 30