The maximal kinematical invariance group of
the harmonic oscillator

Autor(en):Niederer, U.

Objekttyp:

ArticleZeitschrift:

Helvetica Physica Acta

Band (Jahr): 46 (1973)
Heft 2

PDF erstellt am:

28.09.2024

Persistenter Link: https://doi.org/10.5169/seals-114478

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http://www.e-periodica.chHelvetica Physica Acta
Vol. 46, 1973.

Birkhäuser Verlag Basel

The Maximal Kinematical Invariance Group of theHarmonie Oscillator
by U. Niederer

Institut für Theoretische Physik der Universität Zürich, Switzerland1)(29. XI. 72)

Abstract. The largest group of coordinate transformations leaving invariant the Schrödinger
equation of the «-dimensional harmonic oscillator is determined and shown to be isomorphic to thecorresponding group of the free particle equation. It can be described as a Galilei group in which thetime-translations have been replaced by the group SL(2, R) of projective transformations. The
relation between the oscillator group and the spectrum generating algebra of the harmonic oscillator
is investigated. The relevance of the oscillator group and the group SL(2, R) for general quantum
systems is discussed.

1.

Introduction
In a recent paper [1] the maximal kinematical invariance group (MKI) of the freeparticle Schrödinger equation, i.e. the largest group of coordinate transformations
leaving invariant this equation, was determined and it was shown that the Schrödinger
group, as it was called, could be described as a Galilei group in which the one-parametergroup of time-translations has been changed to the three-parameter group SL(2,R).
The present paper is devoted to a similar analysis in the case of the Schrödinger equationof the «-dimensional harmonic oscillator
A (t, x) P(t, x)(i d,+ — d2kk
2m

— xk xk) P(t, x)

2

0,

(1.1)

where x (xx,...,xn). The MKI of equation (1.1), denoted by HO(n) and called ndimensional oscillator group, is defined as the set of all coordinate transformations g,(t,x)^g(t,x),(1.2)

with the property that

A[g(t,x)][fJt,x)P(t,x.)]0

for some function f„ and all solutions p of (1.1).
l)Work supported by the Swiss National Foundation.

(1.3)

U. Niederer

192

H. P. A.

At first sight, it might seem that in comparison with the free particle case, thepresence of the potential term in equation (1.1), which fixes an origin of space, vigorously
limits the number of allowed coordinate transformations. This, however, is not
so and the most surprising result of the subsequent analysis is that the oscillator group
HO(n) and the Schrödinger group Schr(n) are isomorphic. In fact, if appropriate coordinates
are used for the harmonic oscillator, the transformations of the two groups are

identical.

In Section 2, we first reduce equation (1.1) by splitting it into two separate sets ofequations for the coordinate transformations g and the companion functions f„
respectively. The g-equations are then solved and the group HO(n) is determined. The/-equations are solved in Section 3 and it is shown that the harmonic oscillator solutions
carry a projective unitary irreducible representation of HO(n). Restricting, for
1 we then produce a plausibility argument for thesimplicity, our attention to the case nexistence of the invariance group HO(l) by connecting its generators to the creation
and annihilation operators of the harmonic oscillator. Section 4 contains a discussionof the relationship between the harmonic oscillator and the free particle which is
intimated by the isomorphism of the two groups H0(1) and Schr(l). It is shown that
there exists a simple formula connecting the solutions of the two systems. In Section 5,
the algebras SL(2,R) and H0(1)2) are established as spectrum generating algebras of
the harmonic oscillator. Finally, in Section 6, we demonstrate that the algebras SL(2, R)
and HO(l) not only occur in connection with the harmonic oscillator, but are actuallyhidden in most quantum systems.
2. Determination of the Group HO(n)To solve equation (1.3) for the unknown coordinate transformations g andcompanion functions/9, we use the same method as in the appendix of [1]. We first define
the derivativesdtdtu(t,x)= — ^kO, ct(t,x)= —,bjt,x)=^,
atdik(t,x)=p7, detdik^0,

(2.1)

dxk

where (t',x') =g(t,x). The differential operator A (t',x') of (1.3) is now expressed as an
operator in (dt, dj) and by comparing different orders of space derivatives of theharmonic oscillator solutions p we obtain from (1.3) the equations
c,

0,

dudkl

(2.2)

u8ik,

2u dt f, + (dlk d, dik + 2imb,) fg

(2.3)

0,

« 9« f9 + (dik 3( dik + 2imbt) dt fg + 2imufg + m2 to2(ux2(2.4)- x'2) /, 0.

(2.5)

Equation (2.3) tells us that the matrix dik can be expressed as
dik
2)

u1'2Rkt,

A group and its Lie algebra are denoted by the same symbol.

(2.6)

The Maximal Kinematical Invariance Group of the Harmonic Oscillator

Vol. 46, 1973

193

where ReO(n) is an «-dimensional rotation. The integrability conditions for (2.4),(2.5) and the derivatives (2.1) then imply the relations

u

u(t),

2ubi

Rik

constant,

9, bk

(2.7)

AJt 8tk,- ùbt + 2co2(u2xi - u1/2x'k Rki) 0,(2.8)

and as an intermediate result we obtain, after some rearrangement, the equations

t'=\ dtu-^t), x'
y + co2ydtf0

u~1/2[Rx + y(t)],uü-\U2 + 2co2(u2-l)=0,0,

ù

f, im[R-i y - i u- (x + R-' y)], fg,

im17-2 K**2

ZU

-

(2.9)

œ2) iRx Ay)2 + co2u2x2 + u2 y2-nùuù(Rx + y) y] /, + —fg,•

4U

where the more convenient vector y has replaced the vector b. The set (2.9) of equations
has the advantage that it decomposes into equations for the coordinate transformations
g and, for known g, into equations for the companion functions fg. We first solve
the g-equations ; the solution of the /-equations is given in the next section.

Solution of the g-equations
The solution of the equation for u(f) is(1+ n2)-l[(ctn + ß)2 + (yi) + 8)2],u(t)(2.10)

where v tantôt and «8 — ßy 1. If (2.10) is used in the expressions for (t',x') we
obtain the result that the group HO(n) is the set of all coordinate transformationsof the form

g=(S,a,v,R),
u x)
g(t,

(l— arctan ettancot + ß—,^\toy tan cot + 8+ tan2 tot
ct tan cot + ß)2 + (y tan tot + 8)2
1

x [Rx + vsina^ + acostof]I,

1/2(2.11)

where5|a P)eSL(2,R);a,ve

ReO(n).

Note that the time-translations, a symmetry which is always present for time-independent
potentials, are contained in (2.11) if S is a two-dimensional rotation.
HO(n) in oscillator coordinates
The transformations (2.11) take on a much simpler form if a new set of coordinates,
called oscillator coordinates, is used, namely
7]

tan tot,|(1 + tan2 tot)1'2 x(cos tot)-1 x.

(2.12)

U. Niederer

194

H. P. A.

In these coordinates the HO(n) transformations are(ct7i + ß

gfofl=JLIT'\yrj +
O

Ri+vri + a\+s J '
yrf +(2-13)

O

and the group HO(n) appears as an enlarged Galilei group in which the one-parameter
subgroup of time-translations has been replaced by the three-parameter group SL(2, R)of projective transformations. The product and the inverse in HO(n) can easily beworked out from (2.13) with the result

g3=g2gi-

S3=S2SX,

R3

R2RX,

a3

R2a, + ß,v2+8,a2,

v3

i?2v1 + a1v2 + y1a2,(2.14)

g'^g-1: S'=S-\ R' R~l,a' -R-l(eta-ß\),y' -R-1(8v-ya),(2.15)

where S and R are multiplied as matrices in IR2 and R" respectively.
The most surprising fact about the group HO(n) is the finding that it coincides with
the Schrödinger group Schr(n), i.e. the MKI of the «-dimensional free particle [l]3).
More precisely, the group HO(n) acts on the oscillator coordinates (v,$) exactly as thegroup Schr(n) acts on the Cartesian coordinates (t, x) of the free particle. This relationshipbetween harmonic oscillator and free particle is discussed further in Section 4.
3. The Oscillator Representation of HO(n)The general form (2.11) of the coordinate transformations g only solves part of theequations (2.9) and it now remains to determine the functions/, accompanying a given
transformation g. The two differential equations for/, in (2.9) are easily solved with theresultfjt,x)=ujtyi*exp

Aujf)

hJt.x)(3.1)

hjt, x) ÙJt) [Rx + yjt)]2 - 2ujt) yjt) [2Rx + yjt)],
where ujf) is given by (2.10), andyjt)(1+ v2)-1/2(\v + a)

vsin cot + a cos cot.

(3.2)

The functions fg satisfy the important relation

fg2[gi(t,*)]fSlit.x) =<*(g2.gi)f.iai(t,x).

«jfe.gi)

exp

(3-3)

[- \imco(-ct, v2 R2 ax + ß, v2 R2 v, - y. a2 R2 a,+ 8xa2-R2vx)].

•

•

¦

(3.4)

In [1] the subgroup SL(2,R) has been decomposed into the three one-parameter groups oftime-translations, dilations and expansions; but the form (2.13) where the group SL(2, R) isleft as a whole is easier to handle. The generalization of Schr(3) to arbitrary dimension is
obvious.

The Maximal Kinematical Invariance Group of the Harmonic Oscillator

Vol. 46, 1973

195

The verification is straightforward but tedious. We can now define a projective
representation g -> Tg of HO(n) on the solutions
p of the oscillator equation (1.1) by simply

putting(Tgp) (t,x) =fg[g-\t,x)] p[g~\t,x)],(3.5)

where, by the definition (1.3) of the MKI, the new functions Tg p are again solutions ofequation (1.1). The map g -> Ta satisfies, as can easily be proved from (3.3), the relation

T,2Tgt(3.6)

to(g2,gx)Tg29l

and hence defines a projective representation of HO(n). Furthermore, the representation(3.5) is unitary with respect to the inner product\d"xp*(t,x)p2(t,x),(Px,P2)(3.7)

because the factor ug,A in (3.1) exactly corrects for the non-invariance of the measured"x. Finally, the irreducibility of the oscillator representation follows from the fact thatin oscillator coordinates, it coincides with the Schrödinger representation [1] of Schr(n)which is irreducible.
The equations (2.11) for the transformations g and (3.1) for the companion functions
fg represent the complete solution of (2.9) and hence of the condition (1.3) for the
MKI of the «-dimensional harmonic oscillator. In the remainder of the paper we may,for simplicity, confine ourselves to the case « 1 because the influence of the dimension« is comparatively trivial.
The Lie algebra H0(1)The generators of H0(1) are defined byl-is-l- iaP + ivK + o(2),77(s, a,v)

where the parameters sS(s)s(sx,s2,s3) of SL(2, R) are chosen according to

Si

S

2

2\s2-s3 -sx

cosh- + -sinhs

(3.8)

(s,x

Sn

+ S3\

j

(3-9)

+(s2 + s2 + s2y>2.

lit

In the oscillator representation (3.5) the generators are found to be
*

11

2co

In

2co

h

i

2co

sin 2tot d.

*'2

cos 2cotx 9,x*A

sin 2cot,cos 2cot + Vrncox2
2

cos 2cot d. + -sin 2cotx dr + -sin 2cot + Amtox2 cos 2cot,
A
2

d„

P

—i cos cot dx + mcox sin cot,Ki sin cot dx + mcox cos cot,

(3.10)

U. Niederer

196

H. P. A.

and the corresponding commutators are%-P,[h, K]--K,[I2,I3]=-iIi, [I2,P] l-K,

[I2,K]

l-P,[Ix, I2]

il3,[Ix, P](3n)[I3,K] l-P,
[I3,h]=-il2, [I3,P]=--K,
2iZ[P, K]-imco.

Note that in a true representation the generators P and K commute and that the nonvanishing of the commutator [P,K] in (3.11) is due to the projective nature of theoscillator representation.
The oscillator representation on energy states

We obtain a discrete description of the oscillator representation of H0(1) if
instead of arbitrary solutions of the harmonic oscillator equation we use eigenstatesof the energy operator id,. An orthonormal energy basis is given by

PJt,x)

t rrtrw l(mco\(nl)-l/2\

l

î

l

i

exp[-ito(n + \)t-\mcox2]HJN2mcox),(3.12)

where « 0,1,2,... and H„ are the Hermite polynomials. On these states the generatorsof HO(l) are the ladder operators/+ 9« » (h + u2) 9n=- V(«+ !)(« +2) pn,

I-

<A„

h<Pn

('1

- ih) ^n -- Vn(n-l) p„-2,
2

P+ 9«=iP + iK) 9n

P_ pn(3.13)-2(n + i)9n,(P

iV2mco(n+l)P„+x,- iK) p„ -i \/2mcon pn-i.

The form (3.13) of the oscillator representation suggests an argument to makeplausible the existence of the invariance group HO(l) by relating it to the existence ofthe creation and annihilation operators (a1,a) of the harmonic oscillator. Indeed, we
haveP+

iV2mcoe-ic"a7

I+=-e-2"°'a^2, I_P_-iV2mcoei0" a,--e2i°"a2, I3- i(aa* + aU).

(3.14)(3.15)The operators P± defined in (3.14) act on solutions of equation (1.1) in the same way as(<zt, a) act on solutions of the time-independent equation, hence they generate asymmetry and the only question is whether it is a kinematical symmetry, as opposed toVol. 46, 1973

The Maximal Kinematical Invariance Group of the Harmonic Oscillator

197

internal symmetries which are not considered here. The symmetry is of the kinematical
type, i.e. is a coordinate transformation, if the operators P+ can be represented as
first-order differential operators in the coordinates (t,x), because the corresponding
coordinate transformations are then obtained from the Lie differential equations [2].
Since (a7,a) are constructed linearly from dx and x the operators P± do generate akinematical symmetry. Furthermore, the second-order operators I of (3.15) again
generate a kinematical symmetry because the wave equation (1.1) serves to convert thesecond-order operator d2 into the first-order operator dt. The fourth independent
expression quadratic in (aT,a), namely [a,a^], is a c-number; it does not generate asymmetry but is responsible for the oscillator representation to be a projective
representation. Neither do higher order expressions in (a7,a) generate kinematicalsymmetries because they
definitely are of at least second order in dx and dt.
4. Harmonic Oscillator Versus Free Particle

Having seen in Section 2 that the MKI of the harmonic oscillator is the same groupas the MKI of the free particle we now proceed to analyze the connection between these
two systems. We first note that an oscillator motion in Cartesian coordinates implies afree motion in oscillator coordinates and vice versa :

d2î

0o—^
dr]

x + to2x0.

(AA)Thus, apart from the fact that the oscillator coordinates only describe half a period of
the oscillator motion, the two systems are, on the level of classical mechanics,
transformed into each otherby a change of the coordinate system. The implication (4.1)
also holds in the framework of quantum mechanics if the coordinates x and £ arereplaced by the expectation values

dxxp*(t,x)p(t,x),(p,xp)(9, Çcp)jdi(l + 7,2)-ll2Ç9*(V, fl 9(7), £),(4.2)

respectively, where p is a solution of (1.1) and 9 is a solution of the corresponding wave
equation in oscillator coordinates, namely(•a'+^+ï^^-^(n^*1)^'fl-0-(4-3)

How closely related the two systems indeed are is shown by the existence of a simple
formula connecting the harmonic oscillator solutions to the free particle solutions inoscillator coordinates. Let xiv,£) De a solution of the free particle equation with mass
mco,

2mco

y

92JX(^)=0,
J

(4.4)

normalizable with respect to the inner product(Xi-Xi)\dëx*iv>£)x(v.i)-(4.5)

U. Niederer

198

H. P. A.

Then to each of these solutions v there corresponds a normalizable solution p of theharmonic oscillator equation (1.1), given by
P(t, X)(l+7]2y* exp I - \imco ^~2 A xfo. &

¦

(4-6)

Note that the one-to-one correspondence (4.6) between the solutions p and x is local
and does not depend on the solutions itself. It can also be shown that (4.6) is invariant
under HO(l), i.e. that the mapping p+->x commutes with the transformations Tg of
the oscillator and the Schrödinger [1] representation.
What is different for the two systems is, of course, the form of the energy operator
which for the system (4.4), considered as a harmonic oscillator, is not id„ as it would be
for a free particle, butito[(l + r,2) dn + r)Çdl, + i71- \imcoi2].

(4-7)

It can be checked that the solutions of (4.4) which are eigenstates of the operator (4.7)
lead back, with (4.6), directly to the solutions (3.12) of the harmonic oscillator.
5.

HO(l) and the Spectrum Generating Algebra

It is well known [3] that SL(2,R) is the spectrum generating algebra of the onedimensional harmonic oscillator in the following sense : Consider the three Hermitean
operators

Lx=--(2xdx+l),
A
L2
L3

(d2x

Amco

+ m2to2x2),(5.1)-^—(d2x-m2co2x2)=—-H,
Amco

2co

which are obtained by closing with respect to commutation the two operators d2 and x2
appearing in the Hamiltonian H of the harmonic oscillator. They form the Lie algebraSL(2, R) ~ SO(2,1) ~ SU(1,1) and the corresponding Casimir operator is a c-number,
namely-L\ -L\ + L\

-3/16.

(5.2)

Hence the Hilbertspace of the harmonic oscillator is composed of one or more of thespaces of those unitary irreducible representations of SL(2,R) which belong to theCasimir value 3/16 and whose L3-spectrum is bounded from above (to guarantee alower bound for the Hamiltonian). There are only two representations satisfying thesecriteria [4] and if both are combined they reproduce the correct spectrum of H. For thisreason SL(2, R) is called the spectrum generating algebra of the harmonic oscillator.
Now the question naturally arises whether the subalgebra SL(2,R) czHO(l) is
related to the spectrum generating SL(2,R). The Casimir operator of the former
SL(2, R) is obtained from (3.10) and is given by--I2-Ii + I2 imx2A(t,x)-3jl6,(5.3)

The Maximal Kinematical Invariance Group of the Harmonic Oscillator

Vol. 46, 1973199

where A (t, x) is the differential operator of the harmonic oscillator equation. Hence theoperators I, when acting on oscillator solutions, also generate the spectrum. The bases
I and L of the two spectrum generating algebras are related by a rotation in the 1-2plane :Lxcos2cotlt — sin2eotI2,

L2

sin 2cotIx + cos 2cotI2,

L3

I3,

(5.4)

where in I the operator id, has to be replaced by — (d2 — m2co2x2)j2m.
Thus we have obtained the interesting result that the MKI of the harmonic oscillator
incorporates the spectrum generating algebra as a subalgebra. Let us finallymention an important difference in the spectrum generating power of SL(2,R) andH0(1). To cover the full energy spectrum of the harmonic oscillator, we need two
irreducible representations of SL(2,R), corresponding to the fact that the generators
I of (3.13) are two-step ladder operators, whereas in the case of H0(1) one irreducible
representation is sufficient because the additional generators P+ are one-step ladder
operators. In other words, H0(1) may with even better reason be called the spectrum
generating algebra of the harmonic oscillator.
6. The Relevance of SL(2,R) and HO(l) for Quantum Systems

In the preceding sections the groups SL(2,R) and H0(1) were established as
invariance groups of a wave equation describing a special quantum system, namely the
one-dimensional harmonic oscillator. In the present section, we want to point out that,independent of a special system and its wave equation, the two groups occur in anatural way, 1) in all one-dimensional non-relativistic quantum systems, and 2) in all
Boson systems. In both cases the pattern is the same there exists a given algebra of
two basic quantities and SL(2, R) is the algebra of all expressions which are of second
order in these quantities (hence called the strictly quadratic algebra), whereas H0(1)is the algebra of all expressions of first and second order (called the quadratic algebra).
More precisely
:

:

(1)

SL(2,R) is the strictly quadratic algebra of the Heisenberg algebra [Q,P]=i,i.e. the three Hermitean operatorsix

i(QP + PQ), i2i(Q2-P2),h-i(Q2 + P2),

(6-1)

satisfy the commutation relations of SL(2, R).
(V) The algebra (6.1) can be completed to the algebra H0(1) by inclusion of the
Heisenberg algebra itself because the five operators (I, P, K Q) satisfy thecommutation relations (3.11) for mco 1. Thus H0(1) is the quadratic algebraof the Heisenberg algebra.
(2) SL(2,R) is the strictly quadratic algebra of the pair (a7,a) of Boson creation andannihilation operators, i.e. the three Hermitean operators

Ix=-l-(a2-aV),

I2\(a2 + aA2),

I3-\(aa^ + aAa),(6.2)

satisfy the commutation relations of SL(2, R). A similar relation was exploited in(3.15).

U. Niederer

200

(2')

H. P. A.

HO(l) is the quadratic algebra of the Boson operators (a1,a), the additional
generators being defined byP--}=(aï-a), K-^=(ai + a).

(6.3)

There is a different interpretation of SL(2, R) and HO(l) in connection with Boson
systems, namely, SL(2, R) and HO(l) are both spectrum generating algebras of the
Boson number operator aA a. This is clear from (6.2) where aA a —\I3 — andfrom the fact that the Casimir operator of the algebra (6.2) has the value —3/16.
The two irreducible representations of SL(2,R) with this value of the Casimirand /3-spectrum bounded from above yield the correct spectrum of aJ a. In thecase of the larger algebra H0(1), these two representations together form a single
irreducible representation.

i

Note that in both the cases (1) and (2) there exists besides the operators I a fourthlinearly independent non-vanishing quadratic expression, namely the commutator of
the basic quantities. This commutator is a c-number hence, strictly speaking, it is not
HO(l) itself but a projective representation (or, in different terms, a central extension)of HO(l) which is the quadratic algebra of the basic quantities.
We conclude these remarks by pointing out that in the case of the harmonic
oscillator the two statements (1) and (2) collapse into one because the operators (aA,a)are linear expressions of the operators (Q,P),
REFERENCES[1] U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv.
Phys. Acta 45, 802 (1972).
[2] L. S. Pontrjagin, Topologische Gruppen (Teubner, Leipzig 1957).
[3] S. S. Sannikov, JETP 22, 1306 (1966); H. Bacry and J. L. Richard, J. Math. Phys. 8, 2230
(1967) ; P. Cordero and G. C. Ghirardi, Fortschr. Physik 20, 105 (1972) and referencescontained[4]

therein.
S. Komy and L. O'Raifeartaigh, J. Math. Phys. 9, 738 (1968).