Wave packet motions of a single electron in harmonic potentials or a magnetic field are obtained analytically. The phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the electron changes its width and central position periodically. These results are visualized using computer animation techniques

Kobayashi, Takeshi

Tsuru, Hideo

English

NASA Technical Reports Server

N93-27316
SU(2) ACTION-ANGLE VARIABLES
Demosthenes Ellinas
Department of Theoretical Physics, University of Helsinki
Siltavuorenpenger 20, SF-00170 Helsinki, Finland
Abstract
Operator angle-action variables are studied in the frame of the SU(2) algebra, and their
eigenstates and coherent states are discussed. The quantum mechanical addition of action-
angle variables is shown to lead to a novel non commutative Hopf algebra. The group
contraction is used to make the connection with the harmonic oscillator.
1 Introduction
Action-angle variables in quantum mechanics one known to lack, in the operator level, some of
properties of their classical analogues [1,2]. Especially the exponential phase operators for the
harmonic oscillator, occuring in the polar decomposition of the bosonic creation and annihilation
operators (an operator analogon of the polar decomposition of a complex number), lack the unitary
and satify the weaker condition of one side-unitary or isometry operator. Based on the mathemat-
ical fact that, unlike in finite dimensional Hilbert spaces as the Fock space of harmonic oscillator,
in finite spaces an isometry is equivalent to a unitary operator, we have in recent works, suggested
a group theoretical construction of a unitary phase operator by introducing action-angle variables
for the SU(2) algebra and going over to their oscillator counterparts via the InSnfi-Wigner method
of group contraction [3-6]. In this report we will briefly review and then expand this work with
respect to two aspects: first, a set of coherent states will be introduced along the lines of the
displacement operator creating the usual coherent states from the vacuum state and second, we
will show that addition of spins in terms of their action-angles (polar) operators, unlike the usual
addition in terms of the step (cartesian) operators, involves a genuine no commutative, no co-
commutative Hopf algebra structure and relates interestingly the phase operators subject to the
subject of quantum groups.
2 Action-angle Variables and States
Let us start with the SU(2) action-angle operators
J_=e'" J_+J-=__J+e'" (I)
(2)
4_
https://ntrs.nasa.gov/search.jsp?R=19930018126 2020-03-17T06:07:13+00:00Z
where
2j
J+= _--_¢m(2j-m+l)]J;m+l><J;m] , J_--J_ (3)
rn _ O
2j
J3 - y_(m -j)]J;m >< J;ml (4)
and
2j
e_¢ = _ [J;g >< J;g+ 1[ , (5)
t=O
mod(2j + 1), and hh + = h+h = 1 with h m e_¢, h + m e-_'_ the unitary angle operator. Then
from the fact that h, generates the cyclic group Z2j+1 acting as a cyclic permutation in the weight
space of the algebra we can construct phase states
[¢;k >= F[J; k >=
1 2j
v/r23,.._-_ _ wk'_IJ;n > (6)
through the finite Fourier transform FF + = F+F = 1, which maps action eigenstates to angle
eigenstates and conjugates the respective variables, where w = expi(2_r/2j + 1). Indeed, if g :=
w3_+ja then FgF + = h, FhF + = g-] and g(h) acts as step operator in the angle (action) state
basis, i.e,
hlJ;n>=[J;n+l > , h[¢;m>=w_[_;rn> (7)
while
g-'l¢;n>=}¢;n+l> , glJ;m>=w'_lJ;m> (8)
mod(2j + 1) and h a'i+_ = g2j+_ = 1, (notice that the state [d; n > and 14; m > where denoted as
In > and t_,- > respectively, in Refs. 3-6). The noncommutativity between the action and the
angle variables is best expressed by the formula
= h9 (9)
which resembles the exponential form of the Heisenberg canonical commutation relations (CR) as
were originally written by Weyl with the association that here the action operator J3 is a finite
version of the position operator and the angle operator stands for the momentum operator. By
virtue of this analogy we may interpret eqs. (7-8) ks the translations along the two different
directions of the phase space of our problem, which due to the module condition is a lattice
torus, pararnetrized by the discrete action and angle values. Also eq. (9), exhibits the unusual
noncommutative character of two succesive translations along different directions. Moreover, the
effect of group contraction which is discussed below, is to increase the density of the lattice points
until the continous limit j --. oc. Furthermore this association to position and momentum suggests
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that we should look for the "number states" IN; rn >, m = 0, 1, ..., 2j in our finite system. Indeed
by diagonalizing the finite Fourier transform FIN; m >= if"IN; m >, we find the number states
IN; m >, related e.g. with the orthonorma] action states as:
2j
Ig;k >= _ [J;m >< J;m[g;k > , (i0)
m=O
with expansion coefficients given in terms of the Hermite polynomial, Hk with discrete argument,
< J;mlN; k >= _ e- ,_'r+_(P(2i+_)+_)2Hk 27r .(p(2 + 1)+,.) (11)
This situation is akin to that of the harmonic oscillator numberlrt:ates which are similarly eigen-
states of the usual Fourier t_ransform operator which conjugates position and momentum operators,
a fact that stems from the property of the oscillator eigenstates exp(- 1 x2)Hk(z), to be their own
Fourier transforms. Especially the vacum or lowest number state is,
_j
IN;0 >= _ w]"203(im]i(2j + 1))lJ;rn > (12)
m=O
where 0a is the theta-Jacobi function [7]:
oo
Oa(zlr)= _ e"'2+°+'''°2 (13)
8 OQ
Having the action [J;m >, the angle [_;n > and the number states [N;k > as were given
above, we can further built, as have been outlined in Ref. 4, the quantum theory of action-
angle variables by introducing the corresponding coherent states acting on the vacum IN; 0 >,
with a displacement operator. Such an operator is furnished by the unitary traceless elements
J,_,,,2 "- w"_2/2g "_h''2' where J+ = J-,-_,--;2 = J2j+t-,,_,2j+l-,,2, with (ma,m2) pairs
• -- ml ,frl2
belonging to the square index-lattice 0 < rn],m2 <_ 2j with boundary conditions and the (0,0)
pair excluded.
The following interesting properties of these operators suggest them as the Glauber displace-
ment operator of our case; first they constitute an orthonormal set of (2j + 1)2 _ 1 elements obeying
the relation
< J_,Jn >:= Tr J,_Jn = (2j + 1)_,t+_,6 ,
where e.g. Jm = J,m=2, and further,
J,_J_ = w-½"ax_ J,,a+_
and
=
and finally
[J_a, J_] = -2i sm 2--_-_mj+
(14)
(15)
(16)
(17)
43
mod(2j + 1), while rfi x 6 = mln2 - m_nl. With the aid of these operators we now introduce
coherent states ]l'>, for the action-angle system by acting on the vacum:
_j
I[>:= J_N;O >=w] tzt_ _ wtz_+_O3(imli(2 j + 1))lJ;m + g_ >
m,_O
(18)
These are now coherent states defined on the lattice phase space which is the appropriate phase
space of the quantum action-angle variables. They involve the Jacobi thets functions which are
also appearing in the case of the ordinary coherent states when, looking for a complete subset out
of the over complete set of coherent states we lattice the phase space. Elsewhere, the normalization
and minimum uncertainty properties of the states will be studied in detail.
3 Quantum Angles Addition
Let us now turn to the case where there are several action-angle degrees of freedom and search
for the way we combine them quantum mechanically. The similar problem for the "cartesian"
generators di, with [Ji, Jj] = 2ieqs, Jj, is the fundamental theme of addition of spins and customanily
is solved by tensoring the generators,
AJ_:=&®I+ l®J_ (19)
which again satisfy the commutation relations, [md_, AJi] = 2ieokAd k. In our case, for the "polar"
generators g = w (A+il) and h = w v{J3+il)F* with wgh = hg we must find an appropriate tensoring
(coproduct in the jargon of Hopf algebras), which provides such Ag and Ah that wag = Ah. Two
such coproducts we have found,
_g=g®g , _h=h@l+g®h (20)
and
Ag=g®g , Ah=h®g+g-l®h (21)
which both have the remarkable property of not been the same under permutation of their com-
ponents involved in the tensor products. This is distingly different to the usual addition of spins,
where there is no sence of order in the tensoring the spins. Technically speaking we have here
a natural case of no co-commutativity unlike in eq. (19), where the product is co-commutative
[8-11]. We end here this discussion, as we intent to expand it elsewhere, by saying that it is also
possible to show the Hopf and quasi triangular Hopf algebra structure of the above tensoring and
then to find the R-matrix and to verify the Yang-Baxter equation.
4 Contraction to the Oscillator
Before we came to conclusions let us mention that as was shown in Ref. 3 via the group contraction
that the SU(2) action-angle variables can be contructed to those of the oscillator and the dynamical
aspects of this procces could be exemplified by studing the Jaynes-Cummings model. We illustrate
44
now thisideabe contractingthe SU(2) generatorsto the oscillatorgeneratorsin the Bargmann
analyticrealization.In the spaceofanalyticpolynomialsofdegree2j the SU(2) algebraisrealized
a.S_
2d
J+=-z _zz ÷z2j
d d
J- = d--; ,/3 = ZTz - j (22)
where z is the complex label of the spin coherent states, and geometrically stands for the projective
coordinate of the coset sphere SU(2)/U(1) ,._ S 1. Transforming now the generators like J± -_
J±/v/_ "] and J3 "* J3 +jl we find in the large j limit, the oscillator generators in their Bargmann
form as follows:
"-_ = 2j d(.cr_'yz) + _ = a+ (23)
J_ d d
-- = a (24)
and
d dr,,=-
J3 + j = d(v z) oT = g (25)ao_
where v_Z ,,_a isthe complex variableof the Glauber coherentstateswhich isnow becoming
the coordinateof the tangent phase planeof the harmonic oscillator.One can furthershow that
the overlap,the completeness relationand allother notionsof the spin coherent statescan be
contractedto theirrespectiveoscillatorcounterparts.Moreover in Ref. 5 has been shown how a
q-deformed oscillatorwith q deformationparameter to be rootof unitycan be employed to define
action-anglesvariablesin a finiteFock Hilbertspace and a number of theirpropertieshave been
worked out. In such an approach we have shown [5],thatthe contractionmethod issubstituted
by the limitprocedureof undeforming the q-oscillatorto the usual ocillators.
5 Conclusion
In conclusion, we have shown that the quantization of action-angle classical variables can be
developed in the framework of the SU(2) algebra in a manner which allows for the classical
properties of these variables to find well defined operator analogues. Interesting relations to
the quantum groups and Hopf algebras axe naturally emerge from the present method of angle
quantization which will be pursued further, together with the introduction of the Wigner function
for the action-angles variables and the star and Moyal product defined between functions of the
phase space of our problem.
References
[l]P.A.M. Dirac,Proc. R. Soc.Lond. A, 114, 243 (1927).
45
[2] P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40,411 (1968).
[3] D. Ellinas, J. Math. Phys. 32, 135 (1991).
[4] D. Ellinas, J. Mod. Optics. 38, 2393 (1991).
[5] D. Ellinas, Phys. Rev. A. 45, 3358 (1992).
[6] M. Chaichian and D. Ellinas, J. Phys. A 23, L291 (1990).
[7] D. Mumford, Tara Lectures on Theta (Birkhauser, Boston, 1984).
[8] M.E. Sweedler, Hopf algebras, (W.A. Benjamin, Inc. New York 1969).
[9] V.G. Drienfeld, Sov. Math. Pokl. 32, 254 (1985).
[10] M. Jimbo, Left. Math. Phys. 10, 63 (1985); Commun. Math. Phys. 102, 537 (1986).
[11] N. Yu. Reshetikhin, L.A. Takhtajan and L.D. Faddeev, Leningrad Math. J. 1, 193 (1990).
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Wave packet motion in harmonic potential and computer visualization

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