Constrained evolution in Hilbert space and requantization

0.17 MB·English

by  M. Grigorescu

Constrained evolution in Hilbert space and requantization1
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0 M. Grigorescu
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Abstract
a
J Constrained dynamics on finite-dimensional trial manifolds of quantum state

0 vectorsappearsintime-dependentvariationalcalculations. Thisworkpresents
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a requantization method, by which properties of the exact eigenstates can

1 be retrieved from the phase portrait of such approximations. Applications
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to the coherent states dynamics, small amplitude vibrations and collective
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2 rotations, show that this approach extends standard procedures of the quan-
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tum many-body theory, such as the method of the projection operators and
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0 the random phase approximation.
7 PACS 02.40.-k, 03.65.Sq, 04.20.Fy, 21.60.Jz
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1firstprintedaspreprintFT-380-(1991)/November,attheInstitute ofAtomicPhysics,
Bucharest, Romania.
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Introduction
Constrained quantum dynamics usually takes into account self-consistent
mean-fields in time-dependent variational calculations. Initially it was ex-
pressed in the form of the time-dependent Hartree-Fock (TDHF) equations
for many-fermion systems. However, the hybrid character of TDHF, includ-
ing both quantal and classical aspects, leads to difficulties of interpretation
[1]. Thus, each TDHF solution is a quantum object, ensuring the optimal
description of a time-dependent wave function within the manifold of Slater
determinants. Moreover, the small amplitude TDHF vibrations around the
critical points represented by solutions of the static Hartree-Fock (HF) equa-
tions, correspond to therandomphase approximation (RPA). Yet, important
properties of the quantum dynamics are lost: the TDHF equations are non-
linear, and tunneling effects are suppressed [2].
Such classical elements appear not only in TDHF. Any restriction to a
manifold of trial states represents more than an approximation, because the
result is partly outside the framework of quantum mechanics. Therefore,
to get an effective approximation it is necessary to come back in the orig-
inal Hilbert space by a ”requantization” procedure. However, the classical
aspects of the mean-field dynamics are useful for the separation of the col-
lective variables [3, 4, 5, 6].
The requantization problem is not yet completely solved, although im-
portant partial results have been obtained. The methods of induced repre-
sentations [7] or geometric quantization [8] can be used to treat globally the
peculiar geometry of the collective models, but the constructed Hilbert space
is physically independent of the initial one. A different procedure concerns
the quantization of the closed orbits, by using a condition of periodicity and
”gauge invariance” (GIPQ) [9, 10]. This formalism resemble the geomet-
ric prequantization [11], providing integrality constraints of Bohr-Wilson-
Sommerfeld type. However, although GIPQ operates within the original
Hilbert space, it is still far from complete requantization because the rela-
tionship between the periodic trial states selected by such constraints, and
the energy eigenstates, remains ambiguous.
In this work approximate eigenstates associated to the quantized orbits
of time-dependent trial functions are defined by the requantization proce-
dure from [12]. Its relevance was tested on examples, by comparison with
other methods and experimental data [13]. The first part contains a short
description of the geometric structures related to the constrained quantum
2
dynamics. In the second part is presented the formula proposed for the
requantization of the periodic orbits. On examples, it is shown that this
formula embodies naturally both the RPA and the method of the projection
operators.
I. The constrained quantum dynamics
The full quantum dynamics in the Hilbert space H is given by the time-
dependent Schr¨odinger equation (TDSE)
ih¯∂ |ψi = Hˆ|ψi (1)
t
where Hˆ is the Hamiltonian operator, defined on a dense domain in H. This
equation can be related to the formalism of classical mechanics by its deriva-
tion from the ”variational principle” δS = 0, with S = dthψ|ih¯∂ −
[ψ] [ψ] t
Hˆ|ψi. However, it can also be seen as the flow R
˙
|ψi = X (ψ) (2)
H
of the quantum states |ψi ∈ H, under the action of the Hamiltonian field
X (ψ) = (−i/h¯)Hˆ|ψi defined on H by the relation i ωH = dh. Here h
H XH
denotestheHamiltonfunctionh(ψ) = hψ|Hˆ|ψi,h : H → R,andωH ∈ Ω2(H)
is the symplectic form ([14], 4.2):
ωH(X,Y) = 2h¯ImhX(ψ)|Y(ψ)i , X(ψ),Y(ψ) ∈ T H . (3)
ψ
InthefollowingHispresumedfinitedimensional, althoughpartoftheresults
remain valid also in the infinite dimensional case.
The invariance of h to the action on H of the group U(1), defined by
Φ |ψi = c|ψi, c ∈ U(1), leads to the conservation of the norm k |ψi k=
c
hψ|ψi. Thus, itispossibletoapplytheMarsden-Weinsteintheorytoreduce
q
thedynamicsfromHtotheprojectivespaceP associatedtoH([15],5.5.C).
H
In terms of L = {|ψi ∈ H,hψ|ψi = 1}, and of the projection Π : L → P ,
H
Π(|ψi) = [|ψi] ≡ {eiφ|ψi,|ψi ∈ L,φ ∈ [0,2π]} , (4)
any |Zγi ∈ L having as projection a trajectory γ = [|Zγi] ⊂ P of the
t t t H
reduced dynamical system, determines uniquely a solution of the TDSE in
H. This solution has the form
|ψ i = Φ |Zγi = ef|Zγi , (5)
t ef t t
3
where f is given by
f˙|Zγi = X (Zγ)−|Z˙γi . (6)
t H t t
Assuming further that Hˆ is independent of time, then h = hψ|Hˆ|ψi is a
constant E, and |ψ i from (5) becomes:
t
|ψti = e−h¯iEt−R0tdτhZτγ|∂τ|Zτγi|Ztγi . (7)
This result has a peculiar geometrical interpretation in the prequantization
theory. Thus, by the smooth, free and proper action Φ : U(1)× L → L of
the ”gauge group” U(1), L is a principal circle bundle over P ([15], 4.1M).
H
Each normalized state in H generates a one-dimensional Hilbert space, and
by the natural action of the structure group on this space, L is also the unit
section in the associated complex vector bundle [16]. The 1-form α ∈ Ω1(L),
α (X) = −ih¯hψ|Xi, X ∈ T L, (dα = ωH| ), is a connection form on L,
ψ ψ L
having as curvature the reduced symplectic form ω˜ ∈ Ω2(P ), Π∗ω˜ = ωH| .
H L
The covariant derivative [11] of a state |Zi ∈ L along γ ⊂ P , γ =
H t
Π(|Z i), is
t
D|Zi i
˙ ˙
= ∇ |Zi = α(|Zi)|Zi = hZ|Zi|Zi . (8)
γ˙
Dt h¯
Therefore, in (7) the state vector
|Z˜tγi = e−R0tdτhZτγ|∂τ|Zτγi|Ztγi (9)
is autoparallel along γ, while |ψ i corresponds to the lift of the Hamiltonian
t
current from P to the bundle L.
H
The eigenstates of Hˆ are critical points of the reduced dynamics on P .
H
In particular, the ground state |ψ i can be found, up to a phase factor, as
g
the minimum of h on L. This property is also used whenever |ψ i is approx-
g
imated by the constrained minimum |Mi of h on a finite-dimensional ”trial”
manifold P ⊂ L. In static variational calculations such as HF or Hartree-
Fock-Bogoliubov (HFB), the trial manifolds are orbits of the semisimple Lie
groups SU(n) or SO(2n), respectively, generated by unitary action on high-
est weight vectors for their representation in H [17]. Worth noting, the
projection M = Π(P) for an orbit of this type is a ”coherent-state” K¨ahler
submanifold of P [18].
H
If |Mi ∈ P is an approximate ground-state, then also the constrained
dynamics on P could be relevant for the description of the quantum sys-
tem. Let P be a manifold of normalized trial functions P = {|Z i} so that
x˜
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M = Π(P) is symplectic. The local coordinates of the point Π(|Z i) ∈ M
x˜
will be specified by 2N real parameters x˜ ≡ (x1,x2,...,x2N). The Hamilton
function on M, h : M → R is the restriction of h(ψ) = hψ|Hˆ|ψi to M.
M
The symplectic form ωM of M, Π∗ωM = ωH| , defined by the restriction of
P
ωH to P, can be expressed in the form
2N
ωM = ω dxi ∧dxj , ω = h¯Imh∂ Z|∂ Zi , (10)
ij ij i j
X
i,j=1
(∂ ≡ ∂/∂xi). Therefore, theHamiltonianflowonM isgivenbytheequations
i
2N ∂h
2x˙jω = M . (11)
ji
X ∂xi
j=1
The bundle structure Π : L → P is reproduced at the level of the con-
H
strained system only if M is quantizable, so that L ≡ U(1)·P is a principal
M
bundle over M. In this case the restriction of the 1-formiα/h¯ ∈ Ω1(L) to L
M
yields in the local system of the trial functions |Zi ∈ P the 1-form associated
to the connection θ ∈ Ω1(M),
Z
2N
θ = hZ|∂ |Zidxk . (12)
Z k
X
k=1
In the prequantization theory, the connection provides the lift the Hamilto-
nian flow from M to L . If x˜ is a solution of (11), its lift ρˆ|Zi is:
M t t
ρˆt|Zi = e−h¯iEt+iΘt|Zx˜ti , (13)
where
t
E = hZ |Hˆ|Z i , Θ = dτhZ |i∂ |Z i .
x˜t x˜t t Z x˜τ τ x˜τ
0
The state ρˆ|Zi represents a more appropriate approximation to an exact
t
solutionofTDSEthan|Z i ≡ |Z i. The sameresult wasobtainedpreviously
t x˜t
[9, 10] by using the variational equation
δ L(x˜,x˜˙,r,r˙,φ,φ˙)dt = 0 (14)
Z
with L = hΨ|ih¯∂ −Hˆ|Ψi and |Ψi = reiφ|Z i. Therefore one can say that
t x˜t
ρˆ|Zi yields the best approximation of a TDSE solution in L .
t M
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It is interesting to remark that Θ coincides with the Berry’s phase, de-
t
rived first for the eigenfunctions of the adiabatic time-dependent Hamiltoni-
ans [19, 20], then extended [21] and related to the quantization rules [22].
In general, the eigenstates of Hˆ are not elements of P, but nevertheless
they influence the Hamiltonian flow on M. The correspondence between this
flow and the energy eigenstates makes the object of requantization.
The closed orbits onM can be quantized by using the condition of ”gauge
invariance”2 andperiodicity(GIPQ)[9,10]. Thisconditionwasobtainedpre-
suming that a periodic trial function resemble the most a time-independent
eigenstate. However, beside the argument based on such similarity, GIPQ
has a geometrical significance, revealed by the prequantization formalism
[13]. Thus, if C denotes the set of closed orbits γ = Π(|Ziγ) on M, then
t t
GIPQ approximates the stationary states by the autoparallel vectors:
t
|Z˜ i = eiΘt|Zγi , Θ = dτhZγ|i∂ |Zγi , γ ∈ C (15)
t t t Z τ τ τ
0
which are periodic. In the ”gauge” of the functions |Zγi having the same
periodT asγ, denoted|Uγi, theperiodicity condition|U˜γi = |U˜γ i, γ ∈ C,
γ t t+Tγ
leads to the ”quantization rule” Θ = 2πn, n ∈ Z. Stated in terms of the
Tγ
1-form θ , this rule takes the form
U
i
θ ∈ Z . (16)
2π I U
γ
Unlike the initial periodicity condition, this expression is gauge-dependent.
Therefore, it is convenient to use the local relation ωM = −ih¯dθ to express
U
(16) in the intrinsic form
1
ωM ∈ Z , γ = ∂σ . (17)
h Z
σ
Suppose now that the orbits of the Hamiltonian field on the energy surface
P = h−1(E) are closed, and let ωM be the restriction of ωM to P . If ωM
E M E E E
is regular, then these orbits are also the leafs of the foliation F defined by
the characteristic distribution of ωM [15]. In this case [23] the condition (16)
E
selects the energies E for which the quotient spaces M ≡ P /F are quanti-
E E
zable manifolds.
2Theinvarianceuptoaconstantphaseof|Z˜γiin(9)when|ZγiisreplacedbyeiS(t)|Zγi,
t t t
with S(t) an arbitrary function of time.
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II. The requantization of the periodic orbits
If |Z i is a solution of some time-dependent mean-field equations, the eigen-
t
values E from Hˆ|ψ i = E |ψ i can be approximated by E = hZγn|Hˆ|Zγni
n n n n n
for the orbits γn ∈ C selected by (16). However, if |Z˜γni is taken as an
approximation for |ψ i, then its time-dependence, though periodic, leads to
n
ambiguities in the evaluation of the transition amplitudes. To avoid such
difficulties, in the following will be discussed the possibility of approximating
|ψ i by the time-average3 |γni of the periodic vector |Z˜γni,
n
1
|γni = dt|Z˜γni (18)
T I
γn
Example 1. The exact stationary solutions
Let C = {γ = Π(|Zγi),|Zγi = e−iHˆt/¯h|Zγi, t ∈ R}bea regularorbit cylinder
t 0
in P [15] corresponding to the exact TDSE solutions. In this case each |Zγi
H
is quasiperiodic, and the related periodic gauge function is a ”Bloch wave”
in time, |Uγi = eiǫγt/¯h|Zγi. The quasienergy ǫγ is defined up to integer
t
multiples of h¯ωγ = h/T , and is chosen within the first ”Brillouin zone”
γ
[−h¯ωγ/2,h¯ωγ/2]. The condition selecting the quantized orbits γn becomes
E = hZγ|Hˆ|Zγi = nh¯ωγ +ǫγ, n ∈ Z, while
γ 0 0
1
|γni = dte−it(Hˆ−Eγn)/¯h|Zγni (19)
T I 0
γn
represents the projection of |Zγni on the subspace spaned by the eigenstates
0
withenergyE . Theresult isclosetotheergodicmeantheorem([15],3.7.24)
γn
and can be easily illustrated for the harmonic oscillator, choosing the orbit
cylinder associated to the Glauber coherent states.
3Knowing from [10] that in some cases the time-averaged overlap between |Z˜γi and
|ψ i attains a maximum for γ = γn, I used the time-average to estimate the order of
n
magnitude of some matrix elements in a computer program [12]. However, it turned out
that even the first significant digits were unexpectedly close to the experimental values,
suggesting that the time average (18) has its own relevance. The result was confirmed by
an analytic model (M. Grigorescu, Rev. Roum. Phys. 34 1147 (1989)). In 1993 I found
the reference S. Droz˙dz˙, M. Ploszajczak and E. Caurier, Ann. Phys. 171 108 (1986), in
which a similar time-average was proposed.
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Example 2. The harmonic oscillator
For the one-dimensional harmonic oscillator Hˆ = h¯ω(ˆb†ˆb + 1/2) with ˆb =
mω/2h¯(xˆ+ipˆ/mω). Let P be the manifold of the Glauber coherent states
q
P = {|Zi = ezˆb†−z∗ˆb|0i = e−|z|2/2ezˆb†|0i,z ∈ C,ˆb|0i = 0} . (20)
The trajectories on M = Π(P) are given by z = e−iωtz , hZ |i∂ |Z i = ω|z |2,
t 0 t t t 0
(n)
and the condition (16) selects the values z , E :
0 n
1 1
|z(n)|2 = n ∈ N , E = h¯ω(|z(n)|2 + ) = h¯ω(n+ ) . (21)
0 n 0 2 2
Because z(n) = e−iωtz(n), the period T of γn is independent of n. Thus, the
t 0
time-average of |Z˜γni = exp(inωt+z(n)ˆb† −z(n)∗ˆb)|0i expressed by
t t t
1 e−n
|γni = dt|Z˜γni = 2 dteinωtezt(n)ˆb†|0i (22)
T I t T I
yields, up to a normalization factor, the eigenstate |ψ i = (n!)−1/2(b†)n|0i.
n
Worth noting, the ”coherence” of the Glauber states makes the lift ρˆ|Zi an
t
exact solution of TDSE.
Example 3. The Random Phase Approximation
Let G be a semisimple Lie group with algebra g ≡ T G, and M = Π(P)
e
the K¨ahler submanifold of coherent states projected in P from the group
H
orbit P = {|Zi = Uˆa|Mi,a ∈ G,M ∈ g∗} of the highest weight vector
|Mi ∈ H. In a local frame of 2n real parameters {ξ ,j = 1,2N}, (10) yields
j
2ωM = ih¯hM|[D Uˆ,D Uˆ|Mi, and (11) becomes
jk j k
2N
ih¯ ξ˙ hM|[D Uˆ,D Uˆ|Mi = hM|[Hˆ ,D Uˆ|Mi . (23)
j j k U k
X
j=1
Here D Uˆ ≡ Uˆ−1∂ Uˆ and Hˆ ≡ Uˆ−1HˆUˆ. In general, to find all periodic
k k U
solutions of (23) is a difficult task, but locally, around the minima of h =
M
hM|Hˆ |Mi, the periodic orbits exist and can be obtained by integrating its
U
linearized form.
If ∆ ∈ g∗ denotes the set of roots for g, Σ = {m ∈ ∆,(m,M) < 0}, and
the operators {Eˆm,Hˆm,m ∈ ∆},
Eˆm† = Eˆ−m , Hˆm|Mi = (m,M)|Mi ,
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represent in H the Cartan-Weyl basis for g, then around a critical point
Uˆ |Mi ∈ P the parameters ξ can be chosen as the real and imaginary parts
0 j
xm, ym of the complex variables zm, m ∈ Σ defined by
|Zi = Uˆz|Mi , Uˆz = Uˆ0e m∈Σ(zmEˆm−zm∗Eˆ−m) . (24)
P
With these parameters, the components of the form ωM in Uˆ |Mi are ωM =
ij 0 xmxn
ωM = 0, and
ymyn
ωxMmyn = −h¯hM|[DzmUˆ,Dz−nUˆ|Mi = −h¯δmnhM|[Eˆm,Eˆ−n]|Mi . (25)
Retaining only the terms linear in z, the equations of motion (23) become
ih¯z˙mhM|[Eˆm,Eˆ−m|Mi = hM|[HˆU0,Eˆ−m|Mi− (26)
hM|[[(znEˆn −zn∗Eˆ−n),HˆU0],Eˆ−m]|Mi .
nX∈Σ
The state |gi = Uˆ |Mi is a critical point if the constant term in (26) vanishes,
0
hM|[HˆU0,Eˆ−m]|Mi = 0 , ∀m ∈ Σ , (27)
which is the static equation used to find Uˆ . However, further will be taken
0
Uˆ ≡ Iˆ, presuming that if the minimum exists, then it represents both the
0
ground state and |Mi ∈ P. Moreover, |gi is supposed to be invariant to the
symmetry groupG ofHˆ. Otherwise, theactionoftheidentity component of
S
G on |gi generates a whole critical manifold for h , related to the collective
S M
modes. Important examples are provided by the anisotropic or superfluid
ground states. These are not invariant to the action of the rotation group
SO(3,R), respectively of the ”gauge” group U(1) generated by the particle
number operator, and the collective behavior shows up in rotational bands.
Let |Mi be a G -invariant minimum, and consider the closed orbits as-
S
sociated to the normal vibration modes. These are specified by the complex
amplitudes {Xm,Ym,m ∈ Σ} and the oscillation period T, of the general
periodic solution
2π
zmω = Xme−iωt +Ymeiωt , m ∈ Σ, ω = . (28)
T
Replacing this expression in (26), a time-independent system is obtained:
hM|[[Hˆ,Bˆ†]−h¯ωBˆ†,Eˆm]|Mi = 0 , ∀m ∈ ∆, (m,M) 6= 0 , (29)
9
where Bˆ† denotes the sum
Bˆ† = (XmEˆm −Ym∗Eˆ−m) . (30)
mX∈Σ
For (28), θ = hM|Uˆ−1∂ Uˆ|Mi from (16) is dominated by the term
U τ
hM|[Eˆm,Eˆ−m]|MiIm(z˙mzm∗) = (31)
mX∈Σ
−ω hM|[Eˆm,Eˆ−m]|Mi(|Xm|2 −|Ym|2) ,
mX∈Σ
and the quantization condition for the amplitudes Xm,Ym becomes
− hM|[Eˆm,Eˆ−m]|Mi(|Xm|2 −|Ym|2) = n, n = 1,2,... (32)
mX∈Σ
The result makes sense only if the quantized amplitudes are within the range
of the linear approximation. Assuming this to be true for n = 1, the au-
toparallel section with period T becomes
|Z˜ωi = eiωtee−iωtBˆ†−eiωtBˆ|Mi (33)
t
and the approximate stationary state given by the time-average is
1
|ωi = |Z˜ωidt ≈ Bˆ†|Mi . (34)
T I t
In the HF case G = SU(n), and the set of operators associated to the roots
m ∈ Σ contains A(n − A) operators {cˆ†cˆ ,A + 1 ≤ i ≤ n,1 ≤ j ≤ A}
i j
expressed in terms of the single-particle creation and annihilation fermion
operators {cˆ†,cˆ,i = 1,n}. In this representation (29) becomes the RPA
i i
equation in the particle-hole channel, and for the first excited state (n = 1)
the quantization condition (32) reduces to the normalization equation for the
RPA operators4, hM|[Bˆ,Bˆ†]|Mi = 1.
Similarly, by choosing G = SO(2n) the RPA equations in the particle-
particle channel5 are obtained.
4This relation between quantization and normalization was obtained using a different
method by S. Levit, J. W. Negele and Z. Paltiel, Phys. Rev. C 21 1603(1980), Eq. 3.39.
5If|Miistheparticlevacuum,form∈Σtherearen(n−1)operators{cˆ†cˆ†,i,j =1,n}.
i j
An example is presented in M. Grigorescu and E. Iancu, Z. Phys. A 337 139 (1990).
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