Dissipative Mechanics Using Complex-Valued Hamiltonians

0.13 MB·English

by  S. G. Rajeev

Dissipative Mechanics Using
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S. G. Rajeev

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Department of Physics and Astronomy

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University of Rochester, Rochester, New York 14627
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February 1, 2008
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Abstract
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n We show that a large class of dissipative systems can be brought to a
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u canonical form by introducing complex co-ordinates in phase space and a
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: complex-valued hamiltonian. A naive canonical quantization of these sys-
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X tems lead to non-hermitean hamiltonian operators. The excited states are
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a unstable and decay to the ground state . We also compute the tunneling
amplitudeacrossapotential barrier.
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1 Introduction
In many physical situations, loss of energy of the system under study to the out-
side environment cannot be ignored. Often, the long time behavior of the system
is determined by this loss of energy, leading to interesting phenomena such as
attractors.
Thereisanextensiveliteratureondissipativesystemsatboththeclassicaland
quantumlevels(Seeforexamplethetextbooks[1,2,3]). Oftenthetheoryisbased
on an evolution equation of the density matrix of a ‘small system’ coupled to a
‘reservoir’withalargenumberofdegreesoffreedom,afterthereservoirhasbeen
averaged out. In such approaches the system is described by a mixed state rather
than a pure state: in quantum mechanics by a density instead of a wavefunction
and in classical mechanics by a density function rather than a point in the phase
space.
Thereareotherapproachesthatdodealwiththeevolutionequationsofapure
state. Thecanonicalformulationofclassicalmechanicsdoes notapplyinadirect
way to dissipative systems because the hamiltonian usually has the meaning of
energy and would be conserved. By redefining the Poisson brackets [4] , or by
usingtimedependenthamiltonians[5],itispossibletobringsuchsystemswithin
acanonicalframework. Also,therearegeneralizationsofthePoissonbracketthat
maynotbeanti-symmetricand/ormaynotsatisfytheJacobiidentity[6,7]which
givedissipativeequations.
Wewillfollowanotherroute,whichturnsoutinmanycasestobesimplerthan
the above. It is suggested by the simplest example, that of the damped simple
harmonic oscillator. As is well known, the effect of damping is to replace the
naturalfrequencyofoscillationbyacomplexnumber,theimaginarypartofwhich
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determinestherateofexponentialdecay ofenergy. Anyinitialstatewilldecay to
the ground state (of zero energy) as time tends to infinity. The corresponding co-
ordinates in phase space (normal modes) are complex as well. This suggests that
theequationsare ofhamiltonianform,butwithacomplex-valuedhamiltonian.
It is not difficult to verify that this is true directly. The real part of the hamil-
tonian is a harmonic oscillator, although with a shifted frequency; the imaginary
part is its constant multiple. If we pass to the quantum theory in the usual way,
wegetanon-hermiteanhamiltonianoperator. Itseigenvaluesarecomplexvalued,
except for the ground state which can be chosen to have a real eigenvalue. Thus
allstatesexceptthegroundstateareunstable. Anystatedecaystoitsprojectionto
the ground state as time tends to infinity. This is a reasonable quantum analogue
oftheclassical decay ofenergy.
We will show that a wide class of dissipativesystems can be brought to such
a canonical form using a complex-valued hamiltonian. The usual equations of
motiondeterminedby ahamiltonianand Poissonbracket are
d p H,p
= { } . (1)
dt x  H,x 
{ }
   
Atfirst acomplex-valuedfunction
= H +iH (2)
1 2
H
doesnot seemto makesensewhen putintotheaboveformula:
d p H1,p H2,p
= { } +i { } (3)
dt x  H ,x   H ,x 
1 2
{ } { }
     
sincethel.h.s. hasrealcomponents. Howcanwemakesenseofmultiplicationby
iand stillget avectorwithonlyreal components?
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Let us consider a complex number z = x + iy as an ordered pair of real
numbers(x,y). Theeffect ofmultiplyingz by iisthelineartransformation
x y
− (4)
y 7→  x 
   
on its components. That is, multiplication by i is equivalent to the action by the
matrix
0 1
J = − . (5)
1 0 
 
Note that J2 = 1. Geometrically, this corresponds to a rotation by ninety de-
−
grees.
Generalizingthis,wecaninterpretmultiplicationbyiofavectorfieldinphase
spacetomean theactionby somematrixJ satisfying
J2 = 1. (6)
−
Givensuchamatrix,wecandefinetheequationsofmotiongeneratedbyacomplex-
valuedfunction = H +iH tobe
1 2
H
d p H1,p H2,p
= { } +J { } (7)
dt x  H ,x   H ,x 
1 2
{ } { }
     
Our point is that the infinitesimal time evolution of a wide class of mechanical
systemsis ofthistypeforan appropriatechoiceof , ,J,H and H .
1 2
{ }
In most cases there is a complex-cordinate system in which J reduces to a
simple multiplication by i; for example on the plane this is just z = x +iy. For
suchaco-ordinatesystemtoexistthetensorfieldhastosatisfycertainintegrability
conditionsinadditionto(6)above. Theseconditionsareautomaticallysatisfiedif
thematrixelements ofJ are constants.
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What would be the advantage of fitting dissipative systems into such a com-
plex canonical formalism? A practical advantage is that they can lead to better
numerical approximations, generalizing the symplectic integrators widely used
in hamiltonian systems: these integrators preserve the geometric structure of the
underlying physical system. Another is that it allows us to use ideas from hamil-
tonianmechanicstostudystructuresuniquetodissipativesystemssuchasstrange
attractors. Wewillnot pursuetheseideasin thispaper.
Instead we will look into the canonical quantization of dissipative systems.
The usual correspondence principle leads to a non-hermitean hamiltonian. As in
theelementaryexampleofthedampedsimpleharmonicoscillator,theeigenvalues
arecomplex-valued. Theexcitedstatesareunstableanddecaytothegroundstate.
Non-hermitean hamiltonianshavearised already in several dissipativesystemsin
condensedmatterphysics[9]andinparticlephysics[10]. TheWigner-Weisskopf
approximation provides a physical justification for using a non-hermitean hamil-
tonian. A dissipative system is modelled by coupling it to some other ‘external’
degreesoffreedomsothatthetotalhamiltonianishermiteanandisconserved. In
secondorderperturbationtheorywecaneliminatetheexternaldegreesoffreedom
toget an effectivehamiltonianthatisnon-hermitean.
It is interesting to compare our approach with the tradition of Caldeira and
Leggett[8]. Dissipation is modelled by coupling the original (‘small’) system to
a thermal bath of harmonicoscillators. After integratingout the oscillators in the
path integral formalism an effective action for the small system is obtained. A
complicationis that this effectiveaction is non-local: its extremum ( which dom-
inates tunneling) is the solution of an integro-differential equation. We will see
that the integral operator appearing here is also a complex structure (the Hilbert
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transform), although one non-local in time and hence different from our use of
complexstructures.
We calculate the tunneling amplitude of a simple one dimensional quantum
system withinour framework. Dissipationcan increase the tunnelingprobability,
whichis notallowedintheCaldeira-Leggettmodel.
Webeginwith abriefreviewofthemostelementarycase, thedamped simple
harmonicoscillator. Then we generalize to thecase of a generic one dimensional
system with a dissipativeforce proportional to velocity. Further generalization to
systemswithseveraldegreesoffreedomisshowntobepossibleprovidedthatthe
dissipativeforceis oftheform
dxb
∂ ∂ W (8)
a b
− dt
for some function W. In simple cases this function is just the square of the dis-
tance from the stable equilibriumpoint. Finally, we show how to bring a dissipa-
tivesystem whose configuration space is a Riemannian manifold into this frame-
work. This is important to include interesting systems such as the rigid body or
a particle moving on a curved surface. We hope to return to these examples in a
laterpaper.
2 Dissipative Simple Harmonic Oscillator
We start by recalling the most elementary example of a classical dissipative sys-
tem,described thedifferentialequation
x¨+2γx˙ +ω2x = 0,γ > 0. (9)
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We will consider the under-damped case γ < ω so that the system is still oscilla-
tory.
Wecan writetheseequationsinphasespace
x˙ = p (10)
p˙ = 2γp ω2x (11)
− −
Theenergy
H = 1[p2 +ω2x2] (12)
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decreases monotonicallyalongthetrajectory:
dH
= pp˙+ω2xx˙ = 2γp2 0. (13)
dt − ≤
The only trajectory which conserves energy is the one with p = 0, which must
havex = 0 as welltosatisfytheequationsofmotion.
Theseequationscan bebroughttodiagonalform byalineartransformation:
dz
z = A[ i(p+γx)+ω x], = [ γ +iω ]z (14)
− 1 dt − 1
where
ω = ω2 γ2. (15)
1
−
q
The constant A that can be chosen later for convenience. These complex co-
ordinatesarethenatural variables(normalmodes)ofthesystem.
2.1 Complex Hamiltonian
We can think of the DSHO as a generalized hamiltonian system with a complex-
valuedhamiltonian.
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ThePoissonbracket p,x = 1 becomes,interms ofthevariablez,
{ }
z ,z = 2iω A 2 (16)
∗ 1
{ } | |
So ifwechooseA = 1
√2ω1
z ,z = i (17)
∗
{ }
So thecomplex-valuedfunction
= (ω +iγ)zz . (18)
1 ∗
H
satisfies
dz dz
∗
= ,z , = ,z (19)
∗ ∗
dt {H } dt {H }
Of course, the limit γ 0 this tends to the usual hamiltonian H = ωzz .
∗
→ H
Thus,onany analyticfunctionψ, wewillhave
dψ ∂ψ
= ,ψ = [ω +iγ]z (20)
dt {H } 1 ∂z
2.2 Quantization
Bytheusualrulesofcanonicalquantization,thequantumtheoryisgivenbyturn-
ing intoanon-hermiteanoperatorbyreplacing z a , z h¯aand
† ∗
H 7→ 7→
∂
[a,a ] = 1, a = z, a = , = h¯(ω +iγ)a a. (21)
† † 1 †
∂z H
The effective hamiltonian = H +iH is normal ( i.e., its hermitean and anti-
1 2
H
hermitean parts commute, [H ,H ] = 0 ) so it is still meaningful to speak of
1 2
eigenvectorsof . Theeigenvaluesare complex
H
(ω +iγ)n,n = 0,1,2, . (22)
1
···
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The higher excited states are more and more unstable. But the ground state is
stable,as itseigenvalueiszero.
Thusageneric state
∞
ψ = ψ n > (23)
n
|
n=0
X
willevolvein timeas
ψ(t) = ∞ ψ ei¯h[ω+iγ]nt n > . (24)
n
|
n=0
X
Unlessψ happenstobeorthogonaltothegroundstate 0 >,thewavefunctionwill
|
tend to the ground state as time tends to infinity; final state will be the projection
oftheinitialstatetothegroundstate. Thisisthequantumanalogueoftheclassical
factthatthesystemwilldecaytotheminimumenergystateastimegoestoinfinity.
Allthissoundsphysicallyreasonable.
2.3 The Schro¨dinger Representation
In theSchro¨dingerrepresentation,thisamountsto
1 ∂ 1 ∂
a = ω x+h¯ , a = ω x h¯ (25)
1 † 1
√2h¯ω1 " ∂x# √2h¯ω1 " − ∂x#
γ h¯2 ∂2
ˆ = 1+i + 1ω2x2 1h¯ω (26)
H (cid:18) ω1(cid:19)"− 2 ∂x2 2 1 − 2 1#
Thustheoperatorrepresenting momentumpis
∂
pˆ= ih¯ γx (27)
− ∂x −
whichincludesasubtlecorrection dependenton thefriction.
Thetimeevolutionoperatorcan bechosento be
ˆ = Hˆ +Hˆ (28)
Schr diss
H
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where
h¯2 ∂2
Hˆ = + 1ω2x2 (29)
− 2 ∂x2 2
istheusualharmonicoscillatorhamiltonianand
γ h¯2 ∂2
Hˆ = 1γ2x2 +i + 1ω2x2 1h¯ω (30)
diss −2 ω1 "− 2 ∂x2 2 1 − 2 1#
This is slightly different from the operator ˆ above, because the ground state
H
energy is not fixed to be zero. The constant in H has been chosen so that this
diss
statehas zero imaginarypart foritseigenvalue.
3 Dissipative System of One Degree of Freedom
Wewillnowgeneralizeto anon-linearone-dimensionaloscillatorwithfriction:
dp ∂V dx
= 2γp, = p, γ > 0. (31)
dt −∂x − dt
The DSHO is the special case V(x) = 1ω2x2. The idea is that we lose energy
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whenever the system is moving, at a rate proportional to its velocity. It again
followsthat
dH
= 2γp2 0 (32)
dt − ≤
whereH = 1p2 +V. Theseequationscan bewrittenas
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dξi
= H,ξi γij∂ H (33)
j
dt { }−
2γ 0
whereγ = isapositivebutdegeneratematrix.
 0 0
 
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