Nuclear Physics 4 (1957) 289---294; (~) North-Holla~:d Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
T H E C L A S S I F I C A T I O N OF S T A T E S OF S U R F A C E V I B R A T I O N S G. R A K A V Y C E R N Theoretical Study Division at the Institute for Theoretical Physics,
University o/ Copenhagen ? R e c e i v e d 13 April 1957 A b s t r a c t : T h e classification of s u r f a c e oscillations is i m p r o v e d b y i n t r o d u c i n g a n a d d i t i o n a l q u a n t u m n u m b e r w h i c h is specially s u i t a b l e for t h e t r e a t m e n t of " v - u n s t a b l e oscillations". A selection rule i m p l i e d b y this q u a n t u m n u m b e r is discussed.
1. I n t r o d u c t i o n In the theory of surface vibrations of order 2 1) the problem arises of the classification of the states of a (22+ 1)-dimensional oscillator according to angular momenta. In the most interesting case of surface oscillations of order two the first few low-lying states m a y be uniquely classified by means of the total excitation N, the angular momentum I, and its z-component M. For an excitation N = 4 or higher several states with the same I and M appear, and additional quantum numbers are needed to distinguish among them. In this note elementary group theory is used to construct an additional quantum number, which turns out to be suitable for the classification of the states also in the general case of "y-unstable oscillations" ~).
2. T h e M u l t i d i m e n s i o n a l
Oscillator
We define a (22+l)-dimensional space with coordinates xm (m : 2, 2--1 . . . . . 0 . . . . . --2). A set of coordinates x m corresponds to a given set of coordinates X Y Z in ordinary 3-dimensional space; if we rotate the X Y Z system by Eulerian angles afly, the coordinates x m transform to new coordinates x'~ by means of the matrix Da(~fly). We furthermore impose on the (complex) coordinates x~ the conditions
xm*=
(1)
It m a y sometimes be convenient, instead of the complex coordinates xm, to use real coordinates q~ defined by ? P e r m a n e n t address: H e b r e w U n i v e r s i t y , J e r u s a l e m , Israel. 289 September 1957
290
~. m~KAVY
zm = ~
1
(qm+ iq,+~)
zo = qo (-1)~ x-,,,
--
-
V~
(m > 0)
(3)
(qm--iq~+,.).
The Hamiltonian of the system is defined as follows:
H =--½ X (--)" .__~
~-½ X (--)"*xmx_,,, 0 x . Ox_.
P.A ~ l
=-½X
m--~
(3)
2,1
J+iX ,..o Oq fl
q:-
ZH,
~-o
and the scalar product of states ~x(x) and ~on(x) is defined to be
(~,,(~), ,p~(x)) =
oo +oo.
c f ..y
~,~*
(x)v,~ (x) H dq,.
¢,--oo',d --o0
°
(4)
0
We now proceed to write down the eigenstates of the Hamiltonian (3) with the aid of the creation and annihilation operators defined as follows:
'(
am=-~-~ (--)'~m
a~ (5)
am
=
1 "~--~fX_m+ (-) ,~ 0
These operators satisfy the usual commutation rules
[a~, ak,]---[ak, a~,] = 0, [a,, a~,] = ~,,,
(6)
and transform with the matrices Da(Rn) and D -1 = D*a(Rs), respectively. The "reality conditions" (1) are satisfied if d,~ is represented b y the hermitian conjugate matrix of a~, and consequently we m a y use the usual representation of these operators. The set of orthonormal states
io >, where iO ) = ~-a-i exp {--½ • (--)'~XmX_,~ } = z~-a-t exp X -- ½q,' and C = [II(N=/)]-~ m
(Nk = number of times the index k appears in the row the space of eigenstates of excitation N.
mare ~ • • • m,,),
span
STATES OF SURFACE VIBRATIONS
291
3. Classification of the States To linear transformations of the operators din: x,,,
-
Z c,,.d.,
(8)
correspond linear transformations of the states (7), in fact, they transform like symmetrical tensors of rank N. The space of symmetrical tensors is irreducible under the linear group t, but if we restrict the transformations (5) to orthogonal transformations, the tensor space m a y be reduced into subspaces b y the process of contraction a). The space of symmetrical tensors of rank N breaks up into a traceless space and trace spaces of orders v = (N--2); v = (N--4) etc. A trace space of order v transforms like the traceless part of the space of symmetrical tensors of rank v (occasionally we shall refer to v as "seniority"). We define our orthogonal group R2~+1 as the group of all transformations (8) leaving invariant the quadratic form
Z
(9)
This form is kept invariant b y the transformations De(R3), and consequently the group D~(Ra) is contained in R2a+l. It should also be remarked that the transformations belonging to the orthogonal group R~a+l are in fact simple coordinate transformations in the x~ space while the general linear transformations (8) are not. Consequently the transformations of R~+ 1 leave invariant also the form ,.3 = Z
(lo)
Finally we reduce each subspace (v) into its irreducible invariant parts with respect to the group of transformations DI(R3). Each is a basis for some representation/fl(Ra) and will be designated b y I. We m a y now classify the states b y the row of quantum numbers N, V, I, M describing the row of subspaces with respect to R2a+l, R3, and R 2 in which the state lies. This classification is not complete because a certain I m a y appear more than once in a given subspace (v), however, this does not occur for small values of the seniority (for example: for 2 = 2 the first case occurs at v = 6 when I = 6 appears twice). The reduction of the space (v) into its subspaces I m a y be achieved directly b y using methods of group theory, but it is also possible to achieve it b y an elementary procedure: First one breaks up the total space of states of a given excitation into the subspaces I. This is achieved simply b y counting t T h e state of a group of particles in a n o s c i l l a t o r wel l m a y be w r i t t e n i n a w a y s i m i l a r t o eq. (7). I n t h i s case some of the creation operators refer t o p a r t i c l e no. 1, s o m e t o particle no. 2 etc. T h e s t a t e s are n o t c o m p l e t e l y s y m m e t r i c a l , a n d t h e t e n s o r space m a y be s p l i t u p i n t o i n v a r i a n t subspaces under t h e l i n e a r group. T h i s procedure m a y provide us e ful q u a n t u m numbers.
292
G. RAKAVY
the M values: let km be the number of times the factor a m occurs in (7), then, corresponding to a rotation 9 around the Z-axis the state (7) is multiplied b y exp iMp, where +~1
M:
+~
~. ink., tr~---~
( X km~-N).
(11)
m~--a
Let us denote the number of solutions of equations (11) (i.e. the number of states with the given M) b y bN(M ). The number of times a subspace of given I occurs in the space of symmetrical tensors of rank N is
c2v(I) = bN(M = I)--b~(M = I + 1 ) ,
(12)
and finally we obtain the number of times I appears in (v):
d,(I) = c,,(I)--c,_z(I).
(13)
In the following table the reduction of the first few subspaces of lowest seniority is given for vibrations of order 2 = 2: TABLE 1
dv(I ) (~= v
~
0
1
2
3
4
2)
5
6
7
8
9
10
1 1
1 1 1 2
1 1
1 1 l
1
1 1
11
12
1
1 1 1
1 1 1
1
1
1 1 1 1 1
1
If we can find a subgroup of tL2a+xwhich contains Da(R3), we m a y perform the reduction of the spaces (v) into I in two steps. First we reduce (v) into invariant irreducible subspaces with respect to this additional group (let us classify them b y some symbol (u)), and then proceed to reduce each space (u) into the subspaces I. This procedure yields an additional quantum number (u), which m a y serve to distinguish among states with the same I appearing in the same space (v). An example of such an additional group occurs for vibrations of order ;t = 3. The group R~ has a special subgroup denoted G~. which in turn contains Ds(R3) as subgroup. There exists an alternative starting point for our discussion: Instead of using the fact that the space of states (7) is invariant (and irreducible) under the general linear transformations (8), we could base our discussion on the fact that the Hamiltonian (3), which can also be written in the form: {am am}, is invariant under the linear transformations (8), provided we define the transformations of the am's to be contragredient to those of the ara'S.
STATES
OF
SURFACE
VIBRATIONS
4. M o d i f i c a t i o n o f t h e H a m i l t o n i a n b y T e r m s D e p e n d i n g ("y-Unstable Oscillations")
293
on r2
If the Hamiltonian (3) is modified by a function of r 2 only, the eigenstates of the modified Hamiltonian can no longer, generally, be written in the form (7), and N is no longer a good quantum number. Yet, as r ~ is invariant under the orthogonal transformations (Rea+l), the seniority (v) remains a good quantum number (the same applies obviously to ! and M). This means that the degeneracy between states lying in the same irreducible space (v) is not lifted when such a modification is introduced into the Hamiltonian, and that selection rules implied by the seniority quantum number remain valid. A simple example of a selection rule is the one for electric 2a-pole radiation: The operator in question is proportional to x~ 4). The coordinate x,~ has matrix elements only between states with a seniority difference of i 1. This is easily seen in the following way: B y multiplying traceless tensors of the rank v by x's, we get tensors of rank v + 1. These tensors m a y be split into traceless parts (i.e. states of seniority v + l ) and trace parts which are obtained by a single contraction. Further contractions are impossible as these will include two indices of the original traceless tensors. Thus the trace space is irreducible and is equivalent to the space of traceless tensors of rank (v--l) (i.e. contains only states of seniority v - - l ) . The selection rule discussed in ref. *) is a special case of our result. Finally we remark that the Hamiltonian m a y be separated into a part depending on r 2 only, and another part depending on other (22) coordinates, and t h a t the separation constant is simply related to our seniority q u a n t u m number v. Introducing, besides r, 22 coordinates ~, which are homogeneous functions of order zero of the xm's, the Hamiltonian (3) can be written in the form a2 a S = - - ½ {1 ~y~r2r
A+)l(~--l).}r2 + ½ r '
(14)
(with possibly some function of r added to it), where 0
--A = X Pk
02
+ X Q,,
a j'
pk = r~ X a~ax_--~ ~ ,~ (-)'~ Oxm '
Qij -- r2 ~,~ ( - ) m axm a~, Ox_ 0~ m '.
Pk and Q~j are homogeneous functions of order zero in the x's and consequently depend only on the ~'s. The wave functions m a y be factored in the form ~ ---- r-a~p(r)y,(~). The equation for ~(r) is independent of the number of dimensions; hence we m a y use the known results from the 3-dimensional case and determine for a given energy the possible values of A. We also know t h a t the states of excitation N have seniorities v = N; N - - 2 . . . ; hence we
294
G. RAKAVY
m a y easily determine the following connection between A and v: A = v(vq-22--1). References
1) 2) 3) 4)
A. L. H. A.
Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 2b (1952) no. 14 Wilets a n d M. Jean, Phys. Rev. 102 (1956) 788 Weyl, The Classical Groups (Princeton U. P., 1939) pp. 150--151 Bohr a n d B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 16
(15)