Thermodynamics of quasianti-Hermitian quaternionic systems

0.12 MB·

by  S. A. Alavi

Statistical mechanics of quasianti-Hermitian
quaternionic systems.
S. A. Alavi
9
Department of Physics, Tarbiat Moallem university of Sabzevar, Sabzevar, P.
0
O. Box 397, Iran
0
2
Email: [email protected]
y
a Keywords: Statistical mechanics, quasianti-Hermitian quaternionic sys-
M
tems.

1
3 PACS: 03.65.-w,02.20.a .

] Westudythe statistical mechanics of quasianti-Hermitian quaternionic sys-
h
tems withconstant numberof particles in equilibrium. Weshow that theexplicit
p
- knowledge of the metric operator is not necessary for study the thermodynamic
nt properties of thesystem. We introduceatoy model where the physically relevant
a quantities are derived. We derive the energy fluctuation from which we observe
u that for large N the relative r.m.s fluctuation in the values of E is quite neg-
q
ligible. We also study the negative temperature for such systems. Finally two
[
physical examples are discussed.

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v
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5 1 Introduction
1
3
Non-HermitianHamiltoniansarecurrentlyanactivefieldofresearch. motivated
.
4
by the necessity to understand the mathematical properties of their subclasses
0
, namely the pseudo-Hermitian and pseudoanti-Hermitian Hamiltonian. Also,
8
0 to investigate the existence of a suitable similarity transformation that maps
: such Hamiltonians to an equivalent Hermitian form is important from a physi-
v
calpointofview. Aconsistenttheoryofquantummechanicsdemandsacertain
i
X inner product that ensures the associated norm to be positive definite. In this
r directionthere have been efforts to look for non-HermitianHamiltonians which
a
have a real spectrum such that the accompanying dynamics is unitary.
The interest in non-Hermitian Hamiltonians was stepped up by a conjecture of
Bender and Boettcher [1] that PT-symmetric Hamiltonians could possess real
bound-state eigenvalues. Subsequently, Mostafazadeh [2] showed that the con-
ceptofPTsymmetryhasitsrootsinthetheoryofpseudo-Hermitianoperators.
Hepointedoutthattherealityofthespectrumisensured[3]iftheHamiltonian
HisHermitianwithrespecttoapositive-definiteinnerproduct<.,.> onthe
+
Hilbert space H in which H is acting [4].
In the pseudo-Hermitianrepresentationofquantum mechanics,a quantumsys-
tem is determined by a triplet (H,H,η) where H is an auxiliary Hilbert space,
1
H :H H is a linear (Hamiltonian) operator with a real spectrum and a com-
→
plete set of eigenvectors,and η :H H is a linear, positive-definite, invertible
→
(metric) operator fulfilling the pseudo-Hermiticity condition.:
H† =η Hη−1, η =η† >0 (1)
+ + + +
The condition that the metric must be positive definite(quasi Hermiticity) is
necessary for compatibility of models with the postulates of quantum mechan-
ics[5].
The physical Hilbert space H of the system is defined as the complete ex-
Phys
tension of the span of the eigenvectors of H endowed with the inner product
<.,.> :=<. η.> (2)
+
|
where < .,. > is the defining inner product of H, and the observables are
identifiedwiththeself-adjointoperatorsactinginH ,alternativelyη-pseudo-
Phys
Hermitian operators acting in H, [3,6].
Thefoundationsofquaternionicquantummechanics(QQM)werelaidbyFinkel-
stein et.al., in the 1960’s[7]. A systematic study of QQM is given in [8], which
also contains an interesting list of open problems.
It is worth mentioning that while in both QQM and CQM(complex quantum
mechanics)theories,observablesare associatedwith self-adjoint(or Hermitian)
operators,theHamiltoniansareHermitianinCQM,buttheyareanti-Hermitian
in QQM, and the same happens for the symmetry generators, like the angular
momentum operators.
On the other hand as we mentioned earlier theoretical framework of CQM has
been extended and generalizedby introducing the pseudo-Hermitianoperators.
By the same motivation if one wishes to extend and generalize the theoretical
framework of standard quaternionic Hamiltonians and symmetry generators,
one needs to introduce and study the pseudoanti-Hermitian quaternionic oper-
ators.
Moreover, the theory of open quantum systems can be obtained, in many rele-
vantphysicalsituations,asthe complexprojectionofquaternionicclosedquan-
tum systems [9].
Experimental tests on QQM were proposed by Peres [10] and carried out by
Kaiser et.al.,[11] searching for quaternionic effects manifested through non-
commutingscatteringphaseswhenaparticlecrossesapairofpotentialbarriers.
See also[12]. Areviewofthe experimentalstatus ofQQMcanbe foundin[13].
BydefinitionaquaternioniclinearoperatorH issaidtobe(η-)pseudoanti-Her-
mitianifalinearinvertibleHermitianoperatorηexistssuchthatηHη−1 = H†.
−
If η is positive definite, H is said quasianti-Hermitian.
2
2 pseudoanti-Hermitian quaternionic systems.
A quaternion is usually expressed as q =q +iq +jq +kq where q R
0 1 2 3 0,1,2,3
∈
and i2 = j2 = k2 = ijk = 1,ij = ji = k, and with an involutive anti-
− −
automorphism(conjugation) such that, q q =q iq jq kq .
0 1 2 3
→ − − −
Thedensitymatrixρ associatedwithapurestateψbelongingtoaquaternionic
ψ
n dimensional right Hilbert space Qn is defined by : ρ = ψ ψ and is the
ψ
− | ih |
same for all normalized ray representatives[14].
Denoting by Q‡ the adjoint of an operator Q with respect to the pseudo-inner
product (.,.) = (.,η.), (where (.,.) represent the standard quaternionic inner
η
product in the space Qn). We have
Q‡ =η−1Q†η (3)
so that for any η-pseudo-Hermitian operator i.e., satisfying the relation
ηQη−1 =Q† (4)
one has, Q = Q‡. These operators constitute the physical observables of the
system. If Q is η-pseudo-Hermitian, Eq.(4) immediately implies that ηQ is
Hermitian, sothatthe expectationvalue ofQin the state ψ >with respectto
|
the pseudo-inner product (.,.) =(.,η.) can be obtained [14]:
η
<ψ ηQ ψ >=ReTr( ψ ><ψ ηQ)=ReTr(ρQ) (5)
| | | |
where ρ= ψ ><ψ η.
| | e
Moregenerally,ifρdenotesagenericquaternionic(Hermitian,positivedefi-
e
nite) density matrix, we can associateit with a generalizeddensity matrix ρ by
means of a one-to-one mapping in the following way:
e
ρ=ρη (6)
and obtain <Q> =ReTr(ρQ)[14]. Note that ρ is η-pseudo-Hermitian:
η
e
ρ† =ηρ=ηρη−1 (7)
e e
Letusthenconsiderthespaceofquaternionicquasi-Hermitiandensitymatrices,
e e
that is the subclass of η pseudo-Hermitian density matrices with a positive η.
−
Thus, an (Hermitian) operator Θ exists such that η = Θ2, and the η pseudo-
−
Hermitiandensitymatricesarepositivedefinite. Thentheinnerproduct(.,.) =
η
(.,η.) introduced in the Hilbert spaceis positive andthe usual requirementsfor
a proper quantum measurement theory can be maintained.
The most general 2-dimensional complex positive η operator is given by :
x z x z x2+ z 2 (x+y)z
η =Θ2 = = | |
z∗ y z∗ y (x+y)z∗ y2+ z 2
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) | | (cid:19)
3
wherex,y R,z C andxy = z 2[14]. Thechoiseofcomplexmetricoper-
∈ ∈ 6 | |
atorscanbejustifiedasfollows: itisshownin[14]thatthecomplexprojections
of time-dependent η-quasianti-Hermitian quaternionic Hamiltonian dynamics
are complex stochastic dynamics in the space of complex quasi-Hermitian den-
sity matrices if and only if a quasistationarity condition is fulfilled, i.e., if and
only if η is an Hermitian positive time-independent complex operator.
Now let H describes an ensemble of a huge but fixed number of independent
particles. In what follows we study the statistical description of the system in
equilibrium. The state of the system is characterized by a density matrix ρ
which can be written in energy representation as
ρ= W n n , Trρ< (8)
n
| ih | ∞
n
X
In equilibrium, the density matrix operator is solution of Bloch equation:
∂ρ
= Hρ (9)
∂β −
withinitialconditionρ(0)=1andβ = 1 ,kisthefamiliarBoltzmannconstant.
kT
Its formal solution ρ = e−βH fixes the coefficients to be W = exp( βE ).
n n
−
Normalizationfactor of the density matrix depends on the inverse temperature
β and is called partition function.
Z =Trρ (10)
Itplaysanimportantroleinthermodynamicsofthesystembecauseitallows
direct computation of thermodynamic quantities.
Usingthedefinitionoftheexpectationvalueofanoperatorinquasianti-Hermitian
picture i.e., Eq.(5) one can show that :
Z = nρn = nρn =Z (11)
h | | i h | | i
n n
X X
e ee e
where n ,ρandZ areenergyeigenkets,densityoperatorandpartitionfunc-
| i
tion in quasianti-Hermitian quaternionic picture respectively. This means that
the partitioen feunctione is the same in both cases. So we can derive the thermo-
dynamicpropertiesofaphysicalsysteminequilibriuminpseudoanti-Hermitian
quaternionic picture without explicit knowledge of the metric operator Θ.
On the other hand Bloch equation (9) resembles Schrodinger equation for evo-
lutionoperatorinQQM.Thelinkcanbeestablishedbythesubstitutionβ t.
→
Working in quasianti-Hermitian picture the equation (9) reads:
∂U
= HU. U(0)=1 (12)
∂t −
e
ee e
4
where we defined U(t)=ρ(t). In quasi-Hermitian picture. In x-representation,
we have
e e
ρ(x ,x ) =U(x ,x ;t)=<x e−Ht x > :G(x ,x ;t). (13)
1 2 β→t 1 2 1 2 1 2
| | | ≡
the propagator(13) is of fundamental importance because it allows to com-
pute paretition function e(Z = G(x ,x ; β)dex) with use of standard tech-
1 2
−
niques. The analogous equations in the case of pseudo-Hermitian Hamiltonian
R
have been derived in [15].
3 Statistical mechanics of quasianti-Hermitian
quaternionic systems. A Toy model.
We consider an ensemble of systems each consisting of N distinguishable par-
ticles without mutual interaction. The partition function can be written as
follows :
Z =(Z )N (14)
1
where Z may be regarded as the partition function of a single particle in the
1
system. In the quasianti-Hermitian quaternionic picture, the subsystem is de-
scribed by the following most general Hamiltonian :
a c
H =
d b
(cid:18) (cid:19)
Here a,b and c are three arbitrary quaternion. The requirement that the
Hamiltonianbepseudoanti-Hermitianquaternionicwithrespecttoηi.e. ηHη−1 =
H† gives : a∗ = a, b∗ = b and d= αc∗, where η is given by :
− − − −γ
α 0
η =Θ2 =
0 γ
(cid:18) (cid:19)
where we have chosen z =0,x2 =α and y2 =γ in the matrix η introduced
in section 2. The energy levels of the system may depend on the external
parameters e.g. on the volume in which the particle is confined.
We shall find the partition function by solving the Eq.(12). By separating the
Hamiltonian into its diagonal and non-diagonal part we have
H =H +H′ (15)
0
where :
a 0 0 c
H = , H′ =
0 0 b αc∗ 0
(cid:18) (cid:19) (cid:18) −γ (cid:19)
5
Let us introduce the following transformation :
U (t)=exp( H t) (16)
0 0
−
we also define a new operator in the interaction picture as follows:
U (t)=U†(t)U(t)U (t) (17)
I 0 0
Then Eq.(12) changes into the following form :
∂U
I = H′(t)U (t), U (0)=1 (18)
∂t − I I I
whereH′(t)=U (t)†H′U (t). Onecanfindthesolutioniterativelybyintegrat-
I 0 0
ing both sides of the equation.
t t t′
U (t)=1 dt′H (t′)+ dt′ dt′′H′(t′)H′(t′′)+... (19)
I − I I I
Z0 Z0 Z0
where we have neglected the higher order terms, then we have :
1+ α c2 t α c2 (e(a−b)t 1) c (e(a−b)t 1)
U (t) γ a−b − γ (a−b)2 − b−a −
I ∼ −αγ a−db(e−(a−b)t−1) 1− αγ ac−2bt− αγ (a−c2b)2(e−(a−b)t−1) !
The approximate value of the partition function is as follows :
α c2
Z (β)=Tr(U (t)U (t)) e−aβ +e−bβ β(e−bβ e−aβ) (20)
1 0 I |t→β∼ − γ a b −
−
Nowthecalculationofthethermodynamicquantitiesisstraightforward. For
the Helmholtz free energy of the system we have :
N N α c2
A= NkTlnZ = lnZ = ln[e−aβ +e−bβ β(e−bβ e−aβ)]
− 1 −β 1 −β − γ a b −
−
(21)
The entropy of the system is :
S =Nkln[e−aβ+e−bβ α c2 β(e−bβ e−aβ)]
− γ a−b −
[a(a b) αc2+ αc2βa]eβb+[b(a b)+ αc2 αc2βb]eβa
+Nkβ − − γ γ − γ − γ (22)
[(a b)+ αc2β]eβb+[(a b) αc2β]eβa
− γ − − γ
6
The internal energy of the system is given by :
eβb[a(a b+ αc2β) αc2]+eβa[b(a b αc2β)+ αc2]
U =N − γ − γ − − γ γ (23)
eβb[a b+ αc2β]+eβa[a b αc2β]
− γ − − γ
The specific heat per particle C which describes how the temperature
V
changes when the heat is absorbed while volume V of the system remains un-
changed is given by :
C = 1(∂U) = −β2k(∂U)=
V N ∂T V N ∂β
β2k eβ(a+b) (a b)2[(a b)2 (αc2β)2 4αc2]+2(αc2)2 +(αc2)2[e2βa+e2βb]
− { − − − γ − γ γ } γ
N eβb[a b+ αc2β]+eβa[a b αc2β] 2
{ − γ − − γ }
(24)
The pressure of the system is as follows :
P = (∂A) =N [−(a−b)2a′−αγc2(β(a−b)+1)a′+αγc2b′+2αγc(a−b)c′] e−aβ+
− ∂v β [(a−b)2+αγc2β(a−b)]e−aβ+[(a−b)2−αγc2β(a−b)]e−bβ
[ (a b)2b′ αc2(1 β(a b))b′+ αc2a′ 2αc(a b)c′]
N − − − γ − − γ − γ − e−bβ (25)
[(a b)2+ αc2β(a b)]e−aβ +[(a b)2 αc2β(a b)]e−bβ
− γ − − − γ −
wheretheprimesymbolrepresentsthederivativewithrespecttothevolume,
a′ = ∂a.
∂V
4 Energy fluctuation.
We first write down the expression for the mean energy :
E g exp( βE )
U =<E >= r r r − r (26)
g exp( βE )
P r r − r
whereg isthe multiplicityofaparticPularenergylevelE . Bydifferentiatingof
r r
the expressionofthe meanenergy with respect to the parameterβ, we obtain:
∂U
= E2 E 2 (27)
∂β −{h i−h i }
whence it follows that :
∂U ∂U
(∆E)2 = E2 E 2 = =kT2( )=kT2C (28)
V
h i h i−h i −∂β ∂T
7
So, for the relative root-mean-squarefluctuation in E, we have :
(∆E)2 √kT2C 1 β kf(a,b,c,α,β,γ) 1
V
h i = = (29)
E U √N g(a,b,c,α,β,γ) ∝ √N
p h i p
where f(a,b,c,α,β,γ) is the numerator of Eq.(24) and g(a,b,c,α,β,γ) is the
numerator of Eq.(23). As we observe it is O(N−12), N being the number of
particles in the system. Consequently, for large N(which is true for every sta-
tistical system), the relative r.m.s fluctuation in the values of E is quite neg-
ligible. Thus for all practical purposes, a system in the canonical ensemble in
pseudoanti-Hermitian quaternionic picture, has an energy equal to or almost
equal to the mean energy U; the situation is therefore practically the same as
in the microcanonical ensemble.
5 Negative temperature.
Let us consider our system from the combinatorialpoint of view. The question
thenarises: inhowmanydifferentways,canoursystemattainastateofenergy
E?. Thiscanbetackledinpreciselythesamewayastheproblemoftherandom
walk. Let N be the number of particles with energy E and N with energy
+ + −
E ; then
−
E =E N +E N , N =N +N (30)
+ + − − + −
Solving for N and N , we obtain :
+ −
E NE E NE
N = − −, N = − + (31)
+ E E − E E
+ − − +
− −
The desired number of ways is then given by the expression:
N! N!
Ω(N,E)= = (32)
N+!N−! (E−NE+)!(E−NE−)!
E−−E+ E+−E−
whence we obtain for the entropy of the system :
S(N,E)=klnΩ
≃
E NE E NE E NE E NE
k[NlnN − + ln( − +) − − ln( − −)] (33)
− E E E E − E E E E
− + − + + − + −
− − − −
The temperature of the system is then given by :
1 ∂S k E NE
−
=( ) = ln( − ) (34)
N
T ∂E E E −E NE
− + +
− −
fromEq.34,wenotethatsolongasE < N(E +E ),T >0. Howeverthesame
2 + −
equationtellsusthatifE > N(E +E ),thenT <0. Letsexaminethematter
2 + −
8
a little more closely. For this purpose we consider as well the variation of the
entropyS withtheenergyE. WenotethatforE =NE ,bothS andT vanish.
−
As E increase, they too increase until we reach the special situation where
E = N(E +E ). Theentropyisthenseentohaveattaineditsmaximumvalue
2 + −
S = Nkln2, while the temperature has reached an infinite value. Throughout
this range, the entropy was a monotonically increasing function of energy,so T
waspositive. NowasE equals[N(E +E )] ,(dS)becomes0 andT becomes
2 + − + dE −
. With a further increase in the value of E, the entropy monotonically
−∞
decreases; as a result, the temperature continues to be negative, though its
magnitude steading decreases. Finally, we reach the largestvalue of E, namely
NE ,where the entropy is once again zero and T = 0.
+
−
6 Physical examples.
6.1 A spin onehalf system inaconstant quasianti-Hermitian
quaternionic potential.
Wenowconsideratwo-levelquantumsystemwithaquasianti-Hermitianquater-
nionicHamiltonianH =H +jH . H denotesthefreecomplexanti-Hermitian
α β α
Hamiltonian describing a spin half particle in a constant magnetic field[14].
ω i 0
H =
α 2 0 i
(cid:18) − (cid:19)
and jH is a purely quasianti-Hermitian quaternionic constant potential :
β
0 jv
jH = x
β jvx 0
(cid:18) (cid:19)
v,x=0 R.
6 ∈
We note that H = H + jH is η′-quasianti-Hermitian quaternionic i.e.,
α β
η′Hη′−1 = H†, where :
−
x2 0
η′ =
0 1
(cid:18) (cid:19)
Theeigenvaluesandthecorrespondingbiorthonormaleigenbasisofthequater-
nionic Hamiltonian H are[16]:
9
ω
iE =i( v) (35)
±
2 ±
and :
i 1 xi 1
|ψ±i= ±jx √2, |φ±i= ±j √2
(cid:18) (cid:19) (cid:18) (cid:19)
Wenotethatthemetricη′ isaspecialcaseofourmetricandcanbeobtained
by substituting α = x2 and γ = 1. We also note that the Hamiltonian H is
a special case of our general Hamiltonian and can be obtained by substituting
a = iω,b = iω,c = jv and d = αc∗. So the thermodynamic properties
2 − 2 x −γ
of this system can be obtained straightforwardly from equations (21-25), for
instance we have :
S =Nkln[2(coshωβ v2β sinhωβ)]
2 − ω 2
ω2sinωβ +2v2sinhωβ +βωv2coshωβ
+Nkβ 2 2 2 (36)
2ωcoshωβ 2βv2sinhωβ
2 − 2
ω2sinωβ +2v2sinhωβ +βωv2coshωβ
U =N 2 2 2 (37)
2ωcoshωβ 2βv2sinhωβ
2 − 2
Moreoverthenegativetemperaturediscussedinprevioussectionisapplicable
to this system which we study now.
LetN bethenumberofparticleswithenergyE = ω+v andN withenergy
+ + 2 −
E = ω v, then we have :
− 2 −
E =E N +E N , N =N +N (38)
+ + − − + −
The number of ways this system can attain a state of energy E is given by :
N! N!
Ω(N,E)= = (39)
N+!N−! ( E−N(ω2+v))!(E−N(ω2−v))!
− 2v 2v
whence we obtain for the entropy of the system :
S(N,E)=klnΩ
≃
E N(ω +v) E N(ω +v) E N(ω v) E N(ω v)
k[NlnN+ − 2 ln( − 2 ) − 2 − ln( − 2 − )]
2v − 2v − 2v 2v
(40)
The temperature of the system is then given by :
1 ∂S k E N(ω v)
=( ) = ln( − 2 − ) (41)
T ∂E N −2v −E N(ω +v)
− 2
10