hbar expansions in semiclassical theories for systems with smooth potentials and discrete symmetries

0.33 MB·English

by  Holger Cartarius

~
expansions in semiclassical theories for
systems with smooth potentials and discrete
symmetries
6
0 Holger Cartarius, J¨org Main, and Gu¨nter Wunner
0
2
1. Institut fu¨r Theoretische Physik, Universita¨t Stuttgart, 70550 Stuttgart,
n
a Germany
J

0
3

Abstract
]
D
C We extend a theory of first order ~ corrections to Gutzwiller’s trace formula for
. systems with a smooth potential to systems with discrete symmetries and, as an
n
i example, apply the method to the two-dimensional hydrogen atom in a uniform
l
n magnetic field. We exploit the C -symmetry of the system in the calculation of the
4v
[
correctionterms.Thenumericalresultsforthesemiclassical valueswillbecompared

2 with values extracted from exact quantum mechanical calculations. The compari-
v
son shows an excellent agreement and demonstrates the power of the ~ expansion
1
5 method.
0
0
1 Key words: semiclassical theories, Gutzwiller’s trace formula, ~ expansions,
5 discrete symmetries, diamagnetic hydrogen atom
0
PACS: 03.65.Sq, 05.45.Mt
/
n
i
l
n
:
v
i 1 Introduction
X
r
a
Semiclassical theories have become very important for a deeper understanding
of quantum systems, and Gutzwiller’s trace formula [1] has become a powerful
toolforclassicallychaoticsystems.Itprovidesasemiclassicalapproximationof
the quantum level density in terms of classical periodic orbits. In a systematic
expansion of thelevel density in powers of~ it canbeconsidered astheleading
order. Higher orders of this asymptotic expansion have been developed in
several studies [2,3,4], but for a long time were only tested for billiard systems,
i.e., systems with hard walls instead of smooth potentials. By extending an
expansion which was derived by Gaspard et al. [3,4], Gr´emaud [5] developed ~
corrections to Gutzwiller’s trace formula for quantum systems with a smooth
potential. He presented a first-order ~ correction term to Gutzwiller’s trace
Preprint submitted to Elsevier Science 8 February 2008
formula and obtained numerical results for the diamagnetic hydrogen atom.
Comparisons with values extracted from exact quantum calculations showed
that the semiclassical results can be computed with very high accuracy.
However, in the theory presented in [5] some important topics have not yet
been considered. For example, the ~ corrections cannot be calculated for clas-
sical orbits which have a turning point (i.e., the velocity vanishes completely
atthis point).Furthermore, discrete andcontinuous symmetries have not been
included.
If there are discrete symmetries the eigenstates of the quantum system split
up into several subspaces. In these subspaces, classical orbits which are not
periodic without a symmetry transformation contribute to the level density.
Preliminary results for the diamagnetic hydrogen atom, which is an exam-
ple system with discrete symmetries, were published in [6] without explain-
ing which modifications in the numerical calculations the introduction of dis-
crete symmetries entails. Inparticular, an analysis of the differential equations
which have to be solved to obtain the correction terms is not presented. In this
paper we want to give a transparent derivation of the modifications necessary
forthecalculationofthecorrectiontermsinsystems withdiscrete symmetries.
We will show in detail how such symmetries of the Hamiltonian have to be
taken into account. It will be explained in which equations it is necessary to
introduce symmetry operations and which transformations lead to the correct
boundaryconditions oftheclassical Greensfunction, which isanessential part
of the correction terms. Furthermore, we want to give a deeper understanding
of the significance of the symmetry of the system for the quantum spectrum
and the connection to periodic orbits.
We will apply the method to the diamagnetic hydrogen atom. The three-
dimensional hydrogen atom in a uniform magnetic field has a continuous
symmetry, namely the rotational invariance around the magnetic field axis.
Continuous symmetries have a substantial influence on the correction terms.
Actually one cannot obtain the correct results for the first-order ~ correction
for the three-dimensional hydrogen atom with the formulas presented because
the rotational invariance leads to additional contributions to the first-order
corrections. Therefore, we will consider the hydrogen atom as a pure two-
dimensional system as it was done before in [5]. If one uses semiparabolic
coordinates, the potential of the two-dimensional diamagnetic hydrogen atom
exhibits a discrete C -symmetry. The ~ corrections will be calculated for
4v
selected periodic orbits taking into account that discrete symmetry. The semi-
classical results will be compared with the analysis of exact quantum calcu-
lations. The agreement between the results of both methods turns out to be
very good.
The outline of the paper is as follows. In section 2 we will first give a summary
2
of the derivation of the ~ corrections without taking into account discrete
symmetries. In section 3, we will introduce the hydrogen atom in a uniform
magnetic field with all aspects relevant for the calculation of the correction
terms. We will also discuss the influence of the rotational invariance of the
three-dimensional hydrogen atom. Then we will extend the ~ corrections to
discrete symmetries, calculate results for the two-dimensional hydrogen atom
and compare them with exact quantum mechanical calculations in section 4.
2 Semiclassical approximation of the quantum level density
As mentioned in the introduction the ~ corrections to Gutzwiller’s trace for-
mula for systems with a smooth potential were derived by Gr´emaud [5] based
on earlier work by Gaspard et al. [3,4]. In this section, for the reader’s conve-
nience we will briefly summarize all major steps of this derivation by following
the way described in [5] in order to be able to include discrete symmetries in
section 4.
2.1 The trace of the propagator and the first part of the ~ correction
The starting point for the derivation of the semiclassical level density which
is used in this theory is a discrete version of Feynman’s path integral repre-
sentation of the propagator
1
K(~q,~q ,T) = d~q d~q d~q
0 (2πi~∆t)Nf 1 2··· N−1
2 Z
i N−1 ~q ~q
n+1 n
exp L − ,~q ∆t+O(∆t) , (1)
× "~ ∆t n! #
n=0
X
for a system with f degrees of freedom and a time independent Hamiltonian:
p~2
H = +V(~q). (2)
2
K(~q,~q ,T) represents the propagation of a particle going from ~q to ~q = ~q
0 0 N
in time T with ∆t = T/N. L(~q˙,~q) denotes the classical Lagrangian of the
system.
For the calculation of the level density one has to know the trace of the prop-
agator, which can be obtained by setting ~q = ~q in (1) and by integrating over
0
3
this variable:
1
K(T) = K(~q ,~q ,T) d~q = d~q d~q d~q d~q
0 0 0 (2πi~∆t)Nf 0 1 2··· N−1
Z 2 Z
i N−1 ~q ~q
n+1 n
exp L − ,~q ∆t+O(∆t) . (3)
× "~ ∆t n! #
n=0
X
After expanding the exponential function and including all terms which pro-
duce contributions to the leading order in ~ and to the first-order correction,
the integral is evaluated as described in [4]. The result in the limit ∆t 0 is
→
given by:
K(T) = K(0)(T) 1+ i~ Tp dt C (T,t )+O ~2 . (4)
Xℓ ℓ ( Tp Z0 0 1ℓ 0 (cid:16) (cid:17))
In this formula the sum runs over all periodic orbits of the classical system
described by the Hamiltonian (2) with period T. T represents the time period
p
of the primitive periodic orbit, i.e., if the orbit consists of multiple repetitions
(0)
of a periodic orbit, one considers only the basic traversal in this case. K (T)
ℓ
stands for the leading order of the semiclassical approximation. It is identical
with Gutzwiller’s [1] result, and is given by:
1 T
(0) p
K (T) =
ℓ √2π~ ∂ T det(m (T) 1)
E ℓ
| − |
q i iπ iπ
(cl)
exp W (T) µ + sign(∂ T) . (5)
× ~ ℓ − 2 ℓ 4 E
(cid:20) (cid:21)
In this expression one can find the (2f 2) (2f 2)-dimensional stability
− × −
matrix m, which represents the stability properties of the periodic orbit. The
determinantdet(m (T) 1)isrelatedtothe2f 2f-dimensionalmonodromy
ℓ
− ×
matrix. This relation is shown in section 2.4.2. A further term is the classical
action
T
W(cl)(T) = L(~q,~q˙) dt, (6)
ℓ
Z0
µ is the Maslov index, and ∂ T is the derivative of the time period with
ℓ E
respect to the energy E. In equation (4) C is the abbreviation of the first-
1ℓ
order correction in ~. As one can see in (4) one has to use the average of
C (T,t ), where t parametrizes the periodic orbit. The integral over t runs
1ℓ 0 0 0
from 0 to T , i.e., one has to integrate over the whole primitive periodic orbit.
p
The explicit expression of C (T,t ) needs lengthy calculations, which are
1ℓ 0
4
discussed in [3,4,5] in detail. The result reads:
1 T
(cl)
C (T,t ) = dtV ~q (t) (t,t;t ) (t,t;t )
1ℓ 0 8 ,i1i2i3i4 ℓ Gi1i2 0 Gi3i4 0
1 T Z0 T (cid:16) (cid:17)
+ dt dt′V ~q(cl)(t) V ~q(cl)(t′) 3 (t,t;t )
24 ,i1i2i3 ℓ ,j1j2j3 ℓ Gi1i2 0
Z0 Z0 (cid:16) (cid:17) (cid:16) (cid:17)(cid:16)
(t,t′;t ) (t′,t′;t )+2 (t,t′;t ) (t,t′;t ) (t,t′;t )
×Gi3j1 0 Gj2j3 0 Gi1j1 0 Gi2j2 0 Gi3j3 0
V (t ) T (cid:17)
,j 0 (cl)
+ dtV ~q (t) (0,t;t ) (t,t;t ), (7)
2 q~˙ℓ(cl)(t0) 2 Z0 ,i1i2i3 (cid:16) ℓ (cid:17)Gji1 0 Gi2i3 0
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
∂nV ~q(cl)(t)
(cl) ℓ
V ~q (t) = (8)
,i1...in ℓ ∂q (cid:16)...∂q (cid:17)
(cid:16) (cid:17) i1 in
(cl)
are derivatives of the potential evaluated at the point ~q (t) on the classical
ℓ
orbit. are the components of the classical Green’s function, which is a
ij
G
solution of the linearized equation of motion:
d2 ∂2V
1 ~q(cl)(t) (t,t′) = 1δ(t t′). (9)
− dt2 − ∂~q∂~q !G −
(cid:16) (cid:17)
Due to the factor 1/ ~q˙(cl) 2 in equation (7), the correction term is singular for
| |
vanishing velocities, and because of the integral over t in equation (4), the ~
0
correction term diverges for orbits with a turning point.
2.2 The classical Green’s function for the trace of the propagator
Theclassical Green’sfunctionisanessential partofthefirst-order~correction
to the trace of the propagator. In order to be able to look at the symmetry
transformations of C in equation (7), one has to know its structure. In this
1ℓ
section we discuss all important parts of this structure.
From the derivation of the correction term C , it follows that the classical
1ℓ
Green’s function has to fulfil the boundary conditions [5]
(0,t′) = (T,t′),
G G
(0,t′) = (T,t′) = 0, (10)
Pt0G Pt0G
˙(0,t′) = ˙(T,t′),
Qt0G Qt0G
with the projection operator along the direction of the classical orbit at
Pt0
time t , which has the form
0
~q˙(t ) ~q˙(t ) q˙ (t )q˙ (t )
0 0 i 0 j 0
( ) = ⊗ = ; (11)
Pt0 ij  ~q˙(t ) 2  ~q˙(t ) 2
0 0
 ij
 (cid:12) (cid:12)  (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
5
furthermore, = 1 , with the f-dimensional unity matrix 1. If we use
Qt0 −Pt0
the notation
(t,t′) = (t,t′) for 0 t t′,
−
G G ≤ ≤ (12)
(t,t′) = (t,t′) for t′ t T,
+
G G ≤ ≤
it is possible to write the classical Green’s function as a product [5]
(t,t′) A (t′)
G± = M(t) ± (13)
   
˙ (t,t′) B (t′)
± ±
G   
   
with the2f 2f monodromymatrix M(t) andthefourf f matrices A and
±
× ×
B . The monodromy matrix is a symplectic matrix, which can be obtained
±
by solving the linearized Hamiltonian equations of motion
∂2H
M˙ (t,T) = Σ M(t,T). (14)
~ ~
∂X∂X
Here, Σ is the matrix
0 1
Σ =
 
1 0
− 
 
and 1 is the f f unity matrix. For a Hamiltonian of the form (2) it has the
×
structure:
J (t) J (t)
M(t) = 2 1 , (15)
 
J˙ (t) J˙ (t)
2 1
 
where J and J are f f matrices. The boundary conditions at time t = t′
1 2
×
are
(t′,t′) (t′,t′) 0
− +
G = G + . (16)
     
˙ (t′,t′) ˙ (t′,t′) 1
− +
G  G   
     
Exploiting the formulation (13) of the classical Green’s function and the sym-
plecticity of the monodromy matrix M(t), one can formulate the boundary
condition (16) as follows:
A (t′) A (t′) 0 A (t′) J (t′)T
+ = − M(t′)−1 = − − 1 . (17)
         
B (t′) B (t′) − 1 B (t′) − J (t′)T
+ − − 2
         
         
If this expression is combined with the condition
1 0 A (t′) 1 0 A (t′)
− = M(T) + , (18)
      
0 B (t′) 0 B (t′)
 Qt0  −   Qt0   + 
      
6
which is identical with the first and third condition of (10), it leads to the
matrix equation:
1 0 A (t′) 1 0 J (t′)T
[M(T) 1] − = M(T) − 1 . (19)
       
0 − B (t′) 0 J (t′)T
 Qt0   −   Qt0   2 
       
This equation has to be solved in order to obtain the matrices A and B .
− −
When A and B are known one has the solution for the classical Green’s
− −
function (13). Unfortunately, the matrix
1 0
[M(T) 1]
 
−
0
 Qt0 
 
is singular but, as was shown in [5], there is a solution for classical trajectories
which have no turning point (i.e. the velocity q~˙(t) never vanishes completely
along the orbit), viz.
A (t′;t ) 1 q~˙(t ) q~˙(t ) 0
− 0 0 0
= = + ⊗ , (20)
B (t′;t ) X X0 ~q˙(t ) 2 p~˙(t ) ~q˙(t ) 0X0
− 0 0 0 0
   ⊗ 
  (cid:12) (cid:12)  
(cid:12) (cid:12)
where represents a particular soluti(cid:12)on of(cid:12)equation (19), which can be deter-
0
X
mined by a singular value decomposition (see e.g. [7]). The second condition
of equation (10) has been used to derive this result.
In summary, it is possible to obtain the classical Green’s function by solving
the linearized equation of motion (14) of the monodromy matrix and by cal-
culating A and B from equation (20). Then the correction term C (T,t )
− − 1ℓ 0
follows from equation (7). In equation (20), the factor 1/ ~q˙(cl) 2 appears in
| |
the construction of C (T,t ) a second time. The reason for this lies in the
1ℓ 0
application of the projection operator in the second condition of (10).
Pt0
The integrals and double integrals in equation (7) can be transformed into a
set of ordinary differential equations, which can be computed effectively. The
transformation of the integrals is discussed in [5] in detail.
2.3 The trace of the quantum Green’s function and the second part of the ~
corrections
The next step on the way to the level density is the trace of the quantum
Green’s function, which can be obtained from the trace of the propagator via
7
the semi-sided Fourier transform
1 ∞ i
G(E) = G (E) = dT exp ET K (T). (21)
ℓ i~ ~ ℓ
ℓ ℓ Z0 (cid:20) (cid:21)
X X
One has to include all contributions to the leading order as well as the first-
order~correctioninthesemiclassicalapproximationoftheintegral.Themeth-
ods which are used to evaluate this integral are discussed in [4]. We do not
want to repeat the calculation but present the results obtained in [4] and [5],
which can be summarized as follows:
1 T i iπ
0p (cl)
G (E) = exp S (T ) µ
ℓ i~ det(mℓ(T0) 1) (cid:20)~ ℓ 0 − 2 ℓ(cid:21)
| − |
q 1+i~ C (T )+CT→E(T ) +O ~2 , (22)
× 1ℓ 0 1ℓ 0
n (cid:16) (cid:17) (cid:16) (cid:17)o
where the leading order, belonging to the 1 in the curly brackets, is known
from Gutzwiller’s trace formula [1]. The reduced action
(cl) (cl)
S (T ) = W (T )+ET (23)
ℓ 0 ℓ 0 0
is used in this expression, and µ represents the Maslov index. The first-order
ℓ
~ correction consists of two terms. The first,
1 T0p
C (T ) = C (T ,t ) dt , (24)
1ℓ 0 1ℓ 0 0 0
T
0p Z0
is the same as in equation (7) but now with fixed period T , and the second
0
is the contribution from integral (21) to the ~ correction, and is given by:
1
CT→E(T ) = C(1)(T )2 +C(2)(T )
1ℓ 0 (2) 0ℓ 0 0ℓ 0
2W (T )
ℓ 0 (cid:16) (cid:17)
1W(3)(T )C(1)(T ) 1 W(4)(T ) 5 W(3)(T )2
ℓ 0 0ℓ 0 ℓ 0 + ℓ 0 . (25)
− 2 W(2)(T )2 − 8W(2)(T )2 24W(2)(T )3
ℓ 0 ℓ 0 ℓ 0
(n) (n)
W and C are the derivatives
ℓ 0ℓ
∂nW(cl)(T) ∂nC(cl)(T)
W(n)(T) = ℓ , C(n)(T) = 0ℓ
ℓ ∂Tn 0ℓ ∂Tn
(cl)
of the action W (T) and of the logarithm of the amplitude of the trace of
ℓ
the propagator
T
p
C (T) = ln , (26)
0ℓ
 ∂ T det(m (T) 1) 
E ℓ
| − |
q 
respectively.
8
2.4 Contributions to the correction terms of the trace of the semiclassical
Green’s function
Inthissectionwewanttosummarizethecontributionstothesecondcorrection
term CT→E which are important for the understanding of the symmetry prop-
t
erties of this term. As we will see, all parts of equation (25) can be obtained
as numerical solutions of differential equations of some “new” coordinates.
2.4.1 Derivatives of W(cl)(T)
ℓ
Because of the relation
(cl)
∂W (T)
W(1)(T ) = ℓ = E(T ) = H(X~(t,T )) (27)
ℓ 0 ∂T (cid:12) − 0 − 0
(cid:12)
(cid:12)T=T0
(cid:12)
(cid:12)
(cid:12)
(cl)
all higher derivatives of the action W (T ) with respect to the time period of
ℓ 0
the orbit can be expressed as derivatives of the classical Hamiltonian, where
X~(t,T ) = (~q(t,T ),~p(t,T )) represents the phase space vector of the orbit.
0 0 0
Introducing the derivatives
∂nX~(t,T)
X~(n)(t,T ) = , (28)
0 ∂Tn (cid:12)
(cid:12)
(cid:12)T=T0
(cid:12)
(cid:12)
(cid:12)
which appear as coefficients of the Taylor expansion
∞ 1
X~(t,T) = X~(t,T +δT) = X~(n)(t,T )(δT)n (29)
0 0
n!
n=0
X
oftheorbitX~(t,T)aroundareferenceorbitX~(t,T )withperiodT ,thehigher
0 0
(cl)
derivatives of W (T ) are:
ℓ 0
W(2)(T ) = H (X~(t,T ))X(1)(t,T ),
ℓ 0 − ,i 0 i 0
(3) ~ (1) (1) ~ (2)
W (T ) = H (X(t,T ))X (t,T )X (t,T ) H (X(t,T ))X (t,T ),
ℓ 0 − ,ij 0 i 0 j 0 − ,i 0 i 0
W(4)(T ) = H (X~(t,T ))X(1)(t,T )X(1)(t,T )X(1)(t,T )
ℓ 0 − ,ijk 0 i 0 j 0 k 0
3H (X~(t,T ))X(1)(t,T )X(2)(t,T ) H (X~(t,T ))X(3)(t,T ).
− ,ij 0 i 0 j 0 − ,i 0 i 0
(30)
From Hamilton’s equations of motion for the phase space coordinates
X~˙(t,T ) = Σ∂H(X~(t,T0)),
0 ∂X~
9
onecaninfertheequationswhichgovernthemotionofthederivativesX~(n)(t,T ):
0
X˙(1)(t,T ) = Σ H (X~(t,T ))X(1)(t,T ),
i 0 ij ,jk 0 k 0
X˙(2)(t,T ) = Σ H (X~(t,T ))X(1)(t,T )X(1)(t,T )
i 0 ij ,jkl 0 k 0 l 0
~ (2)
+Σ H (X(t,T ))X (t,T ),
ij ,jk 0 k 0 (31)
X˙(3)(t,T ) = Σ H (X~(t,T ))X(1)(t,T )X(1)(t,T )X(1)(t,T )
i 0 ij ,jklm 0 k 0 l 0 m 0
+3Σ H (X~(t,T ))X(1)(t,T )X(2)(t,T )
ij ,jkl 0 k 0 l 0
+Σ H (X~(t,T ))X(3)(t,T ).
ij ,jk 0 k 0
These equations are inhomogeneous differential equations. Their solutions are
of the type
X~(1)(t,T ) = M(t,T )X~(1)(0,T ),
0 0 0
X~(2)(t,T ) = M(t,T )X~(2)(0,T )+F~(2)(t,T ), (32)
0 0 0 0
X~(3)(t,T ) = M(t,T )X~(3)(0,T )+F~(3)(t,T ),
0 0 0 0
where M(t,T ) is the monodromy matrix, which was introduced in section
0
2.2. Together with the condition
~ ~
X(t = 0,T ) = X(t = T ,T ) (33)
0 0 0
for periodic orbits, the solutions (32) lead to the following equations for the
initial values:
[1 M(T ,T )]X~(1)(0,T ) = X~˙(T ,T ),
0 0 0 0 0
−
[1 M(T ,T )]X~(2)(0,T ) = 2X~˙(1)(T ,T )+X~¨(T ,T )+F~(2)(T ,T ),
− 0 0 0 0 0 0 0 0 0 (34)
[1 M(T ,T )]X~(3)(0,T ) = 3X~˙(2)(T ,T )+3X~¨(1)(T ,T )+X.~..(T ,T )
0 0 0 0 0 0 0 0 0
−
+F~(3)(T ,T ).
0 0
The matrix [1 M(T ,T )] is singular. A particular solution can be found
0 0
−
via a singular value decomposition. For a general unstable orbit the kernel is
one-dimensional, i.e., the solution space is in general one-dimensional (see [5])
(2)
but, as also shown in [5], W (T ) can directly be computed if a particular
ℓ 0
solution X~(1)(0,T ) from (34) is known:
0 0
W(2)(T ) = ~ H(X~(T ,T )) X~(1)(0,T ). (35)
ℓ 0 −∇ 0 0 · 0 0
(3) (4)
Similar relations can be found for W (T ) and W (T ), in which also only
ℓ 0 ℓ 0
a particular solution of equation (34) is required.
10