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Crossing of energy levels
Physica 40 (1968) 125-138
o North-Holland Pzlblishing Co., Amsterdam
CROSSING
OF ENERGY
LEVELS
W. J. CASPERS Laboratorium VOOY vaste stof fysica, Universiteit van Groningen, Nederland Received 4 April 1968
Level crossing is discussed for a system with a hamiltonian 2 = A + HB, H being an external parameter. A crossing is a degeneracy of two or more energy eigenvalues for a discrete value of H. The number of crossings is related with the degree in H of the basis vectors of the space of operators commuting with &‘: every crossing reduces the degree of one of these basis vectors, if one starts with a basis consisting of the first n powers of 2, n being the dimension of the Hilbert space. It is possible to indicate the maximum number of crossings for a given n. This number being realized, the space of commuting operators is invariant.
1. Introduction. This paper is concerned with the necessary conditions for level crossing of a quantum mechanical system. We consider systems the motion of which is described by a hamiltonian that is a linear function of a parameter H, S@ = A + HB. The eigenvalues of &? are, in general, nonlinear functions of H, b,(H), a = 1, 2, 3,. . . n, PZbeing the dimension of the hamiltonian matrix. The dimension n is considered to be finite in this paper. A level crossing occurs if two eigenvalues are equal for a discrete value of H, Sa(H1) = aii(Hl), tz # ,!?,without being identical for all values of H. At the end of the paper the special case of degenerate levels LTLx(H)3 cd,(H) is shortly discussed. Level crossing was discussed for the first time in relation with the identification
of energy levels of a molecular system for different values of an see Hundi-3). This parameter is for instance the external parameter, distance between the atoms in a two-atomic molecule, or, in an abstract model, the threshold between two minima of the potential energy of a onedimensional system. Hunds) formulated the hypothesis that crossings only occur as a consequence of the symmetry of the system or of a separation of co-ordinates. Intimately related with our problem is the discussion of the change of state of a quantum-mechanical system for an adiabatic or infinitely slow change in an external parameter. Ehrenfest’s adiabatic principle4) states that an atomic system originally in a stationary state will occupy stationary states corresponding with the momentary values of such an external para125
126
W.
J. CASPERS
meter for adiabatic changes of this parameter. The principle originally formulated in terms of the old quantum theory was proved by Born5) for Schrodinger’s wave mechanics. Hund’s “non-crossing rule” implies that a system in its lowest energy state will stay in the ground state during adiabatic changes of an external parameter, except for the exceptional cases for which crossings occur. Von Neumann and Wigners) proved the important theorem that a coincidence of two eigenvalues of a general hermitean matrix only occurs as a consequence of the variation of three parameters. Level crossing is an exception and on the basis of this principle also properties of large systems were discussed. The magnetic response of a macroscopic system for a quasistatic change in the external magnetic field may be evaluated in two different ways. First one may calculate the susceptibility on the basis of Ehrenfest’s principle, which gives the isolated susceptibility. Secondly one has the adiabatic susceptibility as a result of a purely thermodynamical calculation. Perturbation methods applied by Broer7) and Wrights) for large spin systems gave a difference between the two susceptibilities, whereas several authors claimed that the two quantities should be equal (Yamamotoa), Rosenfeldia), Caspers ii)). The apparent controversy should have its origin in the perturbation calculus, which does not converge for these large systems with interaction. The “non-crossing rule” was used as a starting point by the author to prove that the two susceptibilities are equalrl). The contents of this paper are twofold. First a general method is indicated to determine the level crossings for a given hamiltonian. The maximum number of crossings for a given dimension n is shown to be &n(n - 1). The proof is given in section 2 in which also a complete set of diagonal operators is introduced, i.e. operators commuting with the hamiltonian 9. Every diagonal operator, or constant of motion, may be expressed as a linear combination of the operators of this set, which is simply the set of the first n powers of 2. In the third section we show that the given set of diagonal operators may be replaced by a reduced set if a level crossing occurs, i.e. one of the set may be replaced by a polynomial in H with a degree that is one lower. Every crossing reduces the sum of the degrees of the complete set of polynomials by one. In the extreme case of +n(n - 1) crossings all polynomials have degree zero, and the space of diagonal operators is invariant, i.e. independent of H. In section 4 it is shown that the conditions for level crossings as indicated by Hund may be interpreted as special cases of the ones derived in this paper. We finally discuss in the last section some simple cases as an illustration of the theory developed in this paper. A short analysis of the case of degenerate levels is also given.
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127
2. Determination of level crossings. There are level crossings for those values of H for which the Vandermonde determinant of the eigenvalues 8, equals zero. The Vandermonde differences
E, -
determinant
is given by the product
of all
&,(cx > ,5)
The square of the Vandermonde determinant can be expressed in terms of symmetric functions of the eigenvalues. One has to realize that the determinant Ilbklll,bkl = tf$-‘, k, 1 = 1, . . . n has the same value as Ijatjlj and taking the product of the two determinants one finds
In this expression ?%
s=_
Tr Sm Tr 1
=
is a normalized
trace
LTrZm. n
This normalized trace is a polynomial in H of dregee m or lower. For even m the degree will be m, otherwise the hermitean operator B will be identical to zero, which case is of no physical interest. For the case that the eigenvalues of B are non-degenerate, it is easy to see that for asymptotic values of H we have Vi w constant x Hncnpl) and Vi will be a polynomial in H of degree ~z(n - 1). Because of the fact that Vi is non-negative all real roots of V:(H) = 0 have even multiplicity. From the definition of I”‘n it follows that a twofold root HI corresponds with a single crossing &, - &o = = c(H - HI) + O[(H - HI)~]. For a fourfold root HI one may have two coincident single crossings go1 - 8~ = c(H - HI)+ . . . . 6, - &‘a= c’(H -
--HI) + . . . or two levels that touch for H = HI, 8, - 8~ = c”(H - H1)2 + + O[(H - HI)~]. A sixfold root may indicate three degenerate eigenvalues. One may interpret a point of contact as two or more coincident crossings and in this way one may state that the maximum number of crossings equals $z(n - 1). The degree of Vi is always even: 2s < n(n - l), and the number of crossings is smaller than or equal to s. As a simple illustration we consider the case for which A and B commute. All levels are linear functions of H and for a non-degenerate spectrum of B all pairs of levels will have one and only one intersection point. The number of level pairs equals Qn(n - l), which corresponds with the maximum number of crossings. For the case 2s < n(n - 1) the number of level pairs that are asymptotically parallel equals &z(n - 1) - s. The Vandermonde determinant may be given a geometrical interpretation in the vector space of all operators commuting with 8, as will be done in the following section. If there are no degenerate levels the set I, Z(H), #z(H), . . . SF”-l(H), for a fixed value of H, is a
W. J. CASPERS
128
basis of this vector space. That this series of powers of S is really a basis can be seen in the following way: If we take a representation diagonalizing A? the matrix of the hamiltonian is given by lb,B,,.rl. This representation is unique and a commuting operator C should have the representation jC,&#/. The operator equation C = Cz:t uk#k, giving the components of C for the given basis, corresponds with the set of linear equations C, = xi:; ak &$ fx = 1, . . . PZ,which has a unique solution, because the determinant of this set, which equals the Vandermonde determinant, is unequal to zero. The n dimensional space of diagonal operators undergoes a transformation if the value of H changes. This transformation corresponds with a “rotation” in a space of higher dimension with a basis formed by a set of symmetrized products of A and B. The kth power of A? can be written in the following way Sk
= 2 Hl{Ak--1Bl}, z=o
the braces indicating a symmetrized B. We have, for instance,
product of k - 1 factors A and I factors
{A2B3)=A2B3+ABAB2+APAB+AB3A+BA2B2+ + BABAB
+ BAB2A
+ B2A2B + B2ABA
The number
of terms in a symmetrized
symmetrized space equals
products
n + (n -
are linearly
1) + (n -
product
independent
2) + . . . +
+ B3A2. {Ak-LBl) equals the dimension
1 = *n(~ +
k I . If all
0 of the larger
1).
Every crossing and consequently every zero of the Vandermonde determinant corresponds with a linear relation between the first n powers of X, for a given H = HI. This is because there are only n - 1 or less different eigenvalues of S? for this value of the parameter H. For a single crossing, or a single zero of the Vandermonde determinant the coefficient of ~59-1 in this linear relation is unequal to zero. For degeneracies of higher order the coefficients of xn-1 x78-2 ) . . . etc. equal zero. These relations may be multiplied by SP,.YP ..: to find other independent linear relations; for a twofold crossing we find two relations etc. The linear relations between powers of X, for a given H = HI, correspond with linear relations between the symmetrized products {Ak-ZBZ}. The maximum number of these linear relations is &z(n - l), corresponding with the lowest possible dimension of the space of symmetrized products; this lowest possible dimension is n. It could not be smaller because we made the supposition that there are n different eigenvalues for general values of H. If the number of crossings is smaller the corresponding reduction of the dimension of the space of symmetrized products is also smaller, i.e. we find
CROSSING
OF ENERGY
LEVELS
129
a restricted number of linear relations between these operators. There may exist, however, more linear relations, but these cannot be derived from linear relations between powers of &?, for a given H = HI. Formulated in an other way: level crossing is not a necessary condition for the existence of one or more linear relations between the symmetrized products. There may be a reduction of the degree of one of the basis vectors without crossing. In the next section we show that the basis of the space of diagonal operators given by the first n powers of 2’ may be replaced by an other basis, the vectors of which are of lower degree in H, if there are level crossings. Even crossing corresponds with a reduction by 1 of the sum of the degrees of the basis vectors. 3. Red&ion of the degrees of the basis vectors. The space of diagonal operators for a given general value of H, is an n dimensional space for which we may introduce a metric. The inner product of two hermitean operators C and D is defined by Tr CD TrDC CD = ~ =r=
Tr 1
Dc >
and the square of the length of a vector in the operator space is
As was already stated in the previous section every diagonal operator may be expressed as a linear combination of I, Z(H),
. . . X’“-l(H),
which operators give a basis for the space of diagonal operators, except for those values of H for which degeneracies occur. The Vandermonde determinant, already introduced in the previous section may be interpreted as the volume of the hyperparallelepiped defined by the basis vectors et = &+1(H),
i = 1, . . . S.
The square of the Vandermonde determinant is given by
J’i(H) = Ilet
@4II.
That Vn(H) may be interpreted as the volume of a hyperparallelepiped follows from its properties: a. It is a linear function of all the basis vectors. b. It equals zero if there exists a linear relation between the basis vectors. First wezonly consider single crossings. Higher order degeneracies may be considered as limiting’cases for which two or more crossings coincide. For a single crossing there exists a linear relation between the et; the coefficient
W. J, CASPERS
130
of e,(H) may be chosen to be equal to 1. Suppose we have a degeneracy H = HI
for
n-1 e,(Hl)
+
X
afh-#l)
=
0.
k=l
So we can find a linear combination
of the et(H) that has a single zero for
H = HI n-1 eA(H)
=
e,(H)
+
C k=l
ai”e,_k(H)
= (H -
HI) e:‘(H).
It cannot be a zero of higher order because V:(H) has a double zero for H = = HI, as we only consider single crossings. From the determinant properties it immediately follows that in terms of a new basis we may write V:(H)
= (H -
H1)2 /leil)(H) eil)(H)ll, ell)(H) = es(H), e:‘(H)
= &H-
i =
1, . . . n-
1,
G(H). 1
The new basis vector e:‘(H) has a degree n - 1 in H. Now we consider a second value of H for which there is a crossing. Also for this value H = H2 there exists a linear relation between the vectors of the original basis n-1
e,(Hz)
+ C Z~2’e,_~(Hz) = 0. k=l
This linear relation may be translated into a relation between the vectors of the set eb”; the coefficient of ec’ may be chosen to be equal to 1. Again we have a single zero n-1
ei”(H)
+ x ai2’ec?l,(H)
= (H -
Hz) e:‘)(H),
k=l
and we may write Vi(H)
= (H -
H1)2 (H -
ei2’(H) = e;‘)(H),
Hz)2 llej2)(H) ei2’(H)ll, i=
1, . ..n-
1.
The vector ei2) is defined above. Now we have basis vectors ek2)(H) of degree i - 1 in H for i = 1, . . . n - 1, and the vector e:)(H) has the degree 1z - 2. For the first zero the vector e,(H) was replaced by e:)(H) and the degree of the nth basis vector was reduced by one; for the second zero e:)(H) was replaced by e:)(H) and again the degree of the nth vector was reduced by one. The ejl) as well as the ej2) form a basis for the space of all diagonal operators for those values of H for which no degeneracies occur. This procedure can be continued until V:(H) is factorized in the following way Vi(H)
=lcr
(H -
Hz)~
Ile$“)(H)ej”)(H)Il,
CROSSING
OF ENERGY
LEVELS
m being the total number of zero’s of Vn, corresponding of crossings. This factorization
is performed
131
with the total number
in m steps, each of these steps
corresponding with a linear transformation of the basis vectors. In this linear transformation one of the basis vectors is replaced by an other, the degree of which in H is one lower. After the kth step the expression for V:(H)
reads
v;(H)= ,bl (H - Hd2 Ileik’(H)ebk)(H) 11,
k=
1,2,...m.
The k + Ith step (k + 1 < m) is made in the following way. First we order the ekk)(H) according to climbing powers of H and perform an orthogonalization procedure for the @)(Hk+l). i.e. from every ei”)(Hk+l) we subtract components parallel with ej”)(Hk+l), i < i. We define the set @+l)(H) in the following way
t$k+l)(H) = ejk)(H) _ C i
ei”‘(Hk+l)e,!k)(Hk+l) e:“)(H), ey)(Hk+l)
ey)(Hk+l)’
1, 2, 3, . . .)
-(kfl)(H) until we arrive at a vector eP A new basis is defined by
with a vanishing length for H = Hk+l.
eik+l)(H)
= ei”‘(H)(i < p), e:+“(H)
eikfl)(H)
= eik’(H) (i > f~),
and the corresponding jleik’(H) er)(H)lj
determinants
= (H -
=
1 H _ Hk+l
@+“(H),
are related in the following ..Hk+1)2 Ileik+l)(H) t~i’+~‘(H)Il.
way
The set eikfl) could not be degenerate for H = H,++I because V:(H) only has a double zero for this value of the external parameter. Continuing the procedure for all zero’s of V:(H) we finally arrive at the factorized form given above and for a total number of crossings m the sum of the degrees of the basis vectors equals n-1
d = x I I=0
m = &Z(TZ-
1) -
m.
The number d equals 0 for the case of maximum number of crossings m = = $n(KZ- 1). The reduction procedure is not unique but in our scheme we have, for every zero of V:(H), a reduction of a basis vector of lowest possible degree. Still there is some ambiguity because there may be more than one basis vector of this degree in the linear combination that vanishes for the corresponding value of H.
1%‘. J. CASPERS
132
The basis vectors
e:“)(H)
form a complete
set of diagonal
operators
lowest possible degrees. As was stated before the linear relations powers of Z(H) for a fixed value of H, correspond with linear
with
between relations
between the symmetrized products of A and B. Every crossing reduces the dimension of the space of these products. The vectors elk’(H) undergo a transformation in this space if the value of H is changed. The vector space defined by this basis is also transformed if H changes, except for the case that m = &(PZ - 1) for which the space of diagonal operators is invariant, all basis vectors ep’ being independent of H. It may be possible that the degrees of the basis vectors may be further reduced, without this reduction being derived from zero’s of V:(H). For this case V:(H) will contain a factor that is the square of a polynomial, which polynomial has no real zero’s. This will be shortly discussed in relation with the example of section 5.1, i.e. the spin hamiltonian for S = 1. Multiple crossings for which more than two levels coincide for a given value of H or for which two pairs of levels cross for the same value of the parameter etc., can be discussed as limiting cases of the foregoing analysis. These cases correspond with fourfold, sixfold etc. roots of V:(H). 4. Hund’s colzditions for level crossing. Hunds) has discussed several cases for which crossings may occur. First of all he stated that for a sufficiently general mechanical problem the coincidence of two levels cannot be realized by varying one external parameter. Von Neumann and Wigners) showed that for such a case only the correct choice of three (external) parameters results in a coincidence or level crossing. The first case mentioned by Hund for which a crossing appears for one external parameter is the case of a separable co-ordinate, e.g. the component of the total angular momentum in a direction of axial symmetry. One of the basis vectors e:“)(H) may now correspond with this component of the momentum, being the operator for infinitesimal rotations about the symmetry axis. This operator is independent of H and our reduction in this case may result in a eim) # I that is also independent of H. Another possibility is the invariance of the hamiltonian for the interchange of two identical particles; also in this case one of the e:@(H) # I may be independent of H. These two cases may be characterized by the word symmetry. Another important case, however, not mentioned explicitly by Hund, is the one of two (or more) systems without interaction. Both systems are supposed to be coupled with the external field H, and the hamiltonian takes the form
9
= Xr
+ X2,
%r
= A(1) + B(1) H, 2’2 = A(2) + B(2) H,
in which the operators A(i) and B(i) only contain variables of the ith system. Here we also have separation of co-ordinates but, in general, there is not one single co-ordinate that commutes with 8 and the system does not show
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LEVELS
133
a symmetry. A discussion of this last case of separation of co-ordinates is given in section 5 as an example of the theory developed in the preceding sections. We also discuss in this last section another case of level crossing not based on symmetry. First of all the relatively simple case of the spin hamiltonian for S = 1 is analysed. 5. Discussion
of
special cases.
5.1. Spin hamiltonian for S = 1. In the description of paramagnetic resonance spectra of salts of the iron group use is made of the concept of spin hamiltonian, which gives the spin splitting of the lowest orbital state. Effects of spin-orbit coupling are included in this effective hamiltonian, For S = 1 the most general which only depends on the spin variables 12-9. spin hamiltonian in the case of an isotropic g-tensor is given is) by X,s
= D[S2, -
$S(S + l)] + E(S;
-
S;) + g,!$,H.S,
in which /?ois the Bohr magneton. The X, y and z axes correspond with the principal directions of the spin splitting, which can be described by a tensor of the second rank if S = 1. So far the magnetic field H has an arbitrary direction with respect to the principal axes. If the polar angles of H are defined in the usual way (6 being the angle between H and the z axis and q~the angle between the projection of H on the xy-plane and the x axis) the secular equation of Xs reads g3 +
P=
PS + 4 = 0, -(&D2 + E2 + hz),
h = gBo IHI,
q = &Da - jDE2 - QDh2(3 cos 26 + 1) -$Eh2(1
- cos 26)cos 2~.
It is well-known from the theory of the algebraic equations of the third degree that a necessary and sufficient condition for a twofold root is
&P3 + q2= 0, which is equivalent with the criterion &#‘!‘! ‘+i-2]/ = 0 of section 2, because the value of this determinant is : - (&fi” + 42). This quantity is non-negative and it is only zero for those values of h, 6 and q~for which there is a degeneracy. The easiest way to locate these degeneracies is to differentiate with respect to the three variables h, cos 26 and cos 29. Differentiation with respect to cos 26 and cos 2g, gives
2q(+Dh2 - $Eh2 cos 297) = 0, 2q.$Eh2(1 - cos26)
= 0.
The case q = 0 is almost trivial levelcrossing. Thisconditiongives
because it implies p = 0 as a condition for D = E = 0, with a crossing for h = 0. (I)
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134
The other solutions
E = 0;
J. CASPERS
of the second equation
h = 0;
cos26
= 1.
For the first case we find as a nontrivial with a zero of Vandermonde’s crossing is
are
determinant,
solution
h = 0, corresponding
so the second possibility
of level
DfO,E=h=O.
(II)
For k = 0 there is another possibility This case is equivalent with (II) apart Finally the most interesting case is for redundant for this case because q doesnot -A$”
-
q2 zz &(+D2
+ E2 +
@)3
of level crossing given by E = fD. from an interchange of two axes. cos 26 = 1. The condition for v is depend on v. There is a crossing for -
(&D3
-
gDE2
-
tDj$)2
zz
0,
or
(E2 + h2)(02 - E2 - h2)2 = 0, which gives a level crossing for ks = 02 -
IDI > /El
if
cos28
=
E2,
1.
(III)
For the trivial case (I) we may choose for the z axis the fixed direction of H. The energy eigenvalues are h, 0, -h, and for h = 0 there is a threefold degeneracy, or a coincidence of three single crossings. The sum of the degrees of the basis vectors of the space of commuting operators is 3 for the general case (3 = &Z(PZ- 1)). Case (I) shows a reduction of 3, corresponding with the threefold degeneracy for h = 0, giving a simple illustration of the general theory of sections 2 and 3. Level crossing here is a consequence of the high symmetry of the system, which is invariant for all rotations about the z axis. Commuting operators are E, S, and Sz. In the second case it is quite clear that the degeneracy for h = 0 is independent of the direction of the field; H need not be parallel with a symmetry element of the system. The energy eigenvalues for lz = 0 are QD and -$0, the first one is twofold. The hamiltonian for h = 0 obeys the equation &?s + +DX - $02 = 0, and a nontrivial commuting operator is given by
e3 =
X;(h)
+ QDXs(h) h
- $D2 ,
and a complete set is ei = I, e2 = A?(h), es, the sum of the degrees is 2. In case (III) it is again the symmetry of the hamiltonian that leads to level crossings. For cos 26 = 1 (6 = 0 or 7~) the eigenvalues of Xs are pi = -go,
b2 = QD + JEZ + h2,&3 = &D -
JEW + h?
For E # 0 &2 and 8s do not cross, but 4’1 and 83 have intersection
points
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LEVELS
135
for 02 = E2 + h2, h = *dD2
-
E2
if
D > IEl.
In this case br and 8s have no crossing. For -D > IE) 61 and &s have intersections for two values of 12. The maximum number of crossings is two, corresponding with a nontrivial constant of motion es = Si, and a basis is ei = I, es, es = 2; the sum of the degrees is 1. For IDI < IE 1 we have the same basis and there are no crossings. But it was already stated that the existence of coincidencies is not a necessary condition for the reduction of the degrees of the basis vectors. For E = 0 the system shows maximum symmetry, i.e. axial symmetry and a complete set of commuting operators is ei = I,
e2 = S,
and
es = Si,
all of degree zero. There is a reduction of 3 corresponding with level crossings for h = 0, 12=-&D. In the trivial case we have the same set of commuting operators. 5.2. S$Gn hamiltonian for higher values of S. We do not have the intention to discuss in detail the general case for these higher spin values. We only want to draw the attention to two special cases the first of which is rather wellknown. For half-odd spin values there are always degeneracies for h = 0 because of Kramers’ theorem. All levels have even degeneracy if there is no external field, and this degeneracy is not related with geometrical properties of the system, but has its basis in the time reversal symmetry. For S = 2 one can find examples with degeneracies that are not a consequence of symmetry or of a general physical principle like Kramers’ theorem. The general spin hamiltonian for this case takes the form
*s =M=--2 ; B;M TsM(S)
+ ;
BzM TIM(S) + h-S>
M=-4
the z axis corresponding with the direction of the external field. The TIM and TJM are the wellknown tensor operators for the spinls), and BUM and BUM are complex constants obeying the symmetry relations BLM = (---I"B~-~,
L = 2, 4.
The BLM as well as the TLM transform like the spherical harmonics YLM, as far as the rotations are concerned. The TLM may be normalized in such a
136
way
W.
J. CASPERS
that the matrix elements are given by <~M~ITLMI
2M2)
=
<2MzLMI2Ml>,
the symbol in the right member being a Clebsch-Gordan coefficientle). For a suitable choice of the constants BLM the Hilbert space falls apart into two subspaces corresponding not connected by Zs.
with M = 2, -2
and M = 1, 0, - 1, which are
Taking B4s = B44 = 0 we find that there are no matrix elements connecting 122) with 12 - 2) and 12 - l), and 12 - 2> with 12 2> and 12 1). The matrix elements (2 2 I&‘sl 2 1) and (2 2 IL%?s~ 2 0) are zero if B;,<2121
/22)+B;,<2141
122>=0,
B;,<2022~22)+Bf,(2042~22)=0, and the same equations guarantee that there are no matrix elements connecting the states 12, -2), 12, -1) and 12, -2>, 12, 0). The four ClebschGordan coefficients in the equations are non-zero and for an arbitrary choice of the BUM we can always find B~M(M = j= 1, *2), in such a way that the Hilbert space is separated in two subspaces and the eigenstates of X’s are vectors in these subspaces. The eigenvalues for the subspace (12, -2>, 12, 2)) are linear functions of h given by Bso(2 2 2 012 2) + B40<2 2 4 012 2) f
2k,
which have a crossing for h = 0. These levels will certainly have an intersection with all triplet states, because the asymptotic slopes of these states is 0, 1 or - 1. So we have a total of 7 intersections, which cannot be interpreted in terms of symmetry or of separation of coordinates. 5.3. Systems &thoztt interaction. If we have two systems without interaction we may expect, in general, a number of crossings, which disappear, at least partially, if an interaction between the systems is introduced. We suppose that the levels of the individual systems do not cross, to make-the discussion as simple as possible. The dimensions of the Hilbert spaces of the two, uncoupled, systems are supposed to be nl and ns. If the total system with interaction has no special symmetries, resulting in level crossings, the sum of the degrees of the basic commuting operators is +1n2@v2-
I),
corresponding with the first nias powers of the total hamiltonian. For the composite system without interaction we have the following basic set &$Z~,
k = 0, ...n1 - 1, I = 0, . . . ns -
1,
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137
21 and 9s being the hamiltonians of the two individual systems. This basic set cannot be reduced and the sum of the degrees in this case equals m-1 c s=o
m-1 z (s + t=o
q
and the decoupling @r?zs(n@s
-
=
~W2(~1
+
n2 -
4,
of the systems results in a maximum 1) -
~12&+zr + ns -
number of crossings
2) = 913&+21122 -
921-
ns +
1).
This reduction will be illustrated with a simple case. We consider two spins 8 with different g values, in one and the same external magnetic field h superposed on constant local fields different for the two spins. X1 and 2s may be put in the general form #1=
g1ho1*&+
g1h.L
22
=
gzhoz*Sz
+
g2hSzz,
for a suitable choice for the unit of the magnetic field. The individual energy eigenvalues are G
= zttg&w&,,
+
%,,
+
(ho,,, +
W2)"*
8:
=
+
G2.y
+
@02,2 +
42)i.
f&2(%2,,
We take gl and gs > 0. There are two energy levels of the system of the two spins, 8: + 6; and 8; + 6’ i, that will have two intersection points for a certain range of values for the components of her and has. The trivial case her = ho2 = 0 will not be considered. The two levels intersect for
which has real solutions for (&or,,
-
&o,,J2
-
(g; -
gMh,2,
-
&%)
2 0.
The general conditions for which this inequality is obeyed are rather complicated but it is easy to show that for h 02(h 01) sufficiently small and gs > > gr(gr > gs) there are two crossings. This agrees with the predicted value #nrns(7zr~zs- 1~1- ns + 1) = +.2*2(4 - 2 - 2 + 1) = 2. 5.4 Degenerate levels. For the case of levels identical for all values of the external parameter H the situation is essentially different from the one in which the levels intersect for a finite number of values of the external field. The Vandermonde determinant is equal to zero for all values of H and the operator space corresponding with the basis IA? . . . Xn-1 is only n - 1 dimensional, except for those values of H for which additional intersections occur, resulting in a dimension still smaller. From a study of a diagonal representation of X one may conclude that the space of hermitean operators
CROSSING
138
OF
ENERGY
LEVELS
commuting with X is 1z + 2 dimensional, for all values of H. For the case of more degenerate levels this dimension is correspondingly increased. A systematic study of the algebraic properties of this space of larger dimension has not been performed so far. Con&.&on. An analysis has been made of the properties of the vector space of all hermitean operators commuting with a given hamiltonian Z@ = = A + BH, H being an external parameter. This analysis was made to find conditions for level crossing. The vector space is 1zdimensional, n being the dimension of the hamiltonian. A basis is formed by the first n powers of S : I, 2, 82 . . . Zfi-1. The basis vectors are polynomials in the parameter H of degree 0, 1, 2, . . . n - 1. For every crossing of levels one of the basis vectors may be replaced by an other polynomial, with a degree that is one lower. The maximum number of intersection points is $n(n - 1) corresponding with a basis independent of H, or with an invariant vector space. The study of the non-crossing rule is made especially as a basis for a future analysis of the properties of the energy spectrum of large systems of particles with interaction. For these systems the response in slowly varying external fields has our special interest ii). Acknowledgements. The author is greatly indebted to Drs. H. P. van de Braak, Ir. J. Hennephof and Drs. G. Vertogen for critical reading of the manuscript and for stimulating discussions.
REFERENCES 1) Hund,
F., 2. Phys.
40 (1927)
2)
Hund,
F., 2. Phys.
42 (1927)
3) 4)
Hund, F., Z. Phys. 52 (1929) Ehrenfest, P., Proc. K. Ned.
601. Akad.
5)
Born,
167.
6)
Von Neumann,
7)
Broer,
8) 9)
Wright, A., Phys. Rev. 76 (1949) 1826. Yamamoto, T., Phys. Rev. 119 (1960) 701.
M., Z. Phys.
40 (1927)
742. 93.
J. and Wigner,
L. J. F., Physica
E., Physik.
17 (1951)
Rosenfeld,
Caspers, W. J., Physica 25 (1959) 645. Pryce, M. H. L., Proc. Phys. Sot. (London) Abragam, A. and Pryce, M. H. L., Proc. Roy.
14)
Bleany,
B. and Stevens,
67.
15)
Caspers,
W. J., Theory
16)
p. 41; 143 ff. Edmonds, A. R., Angular
K. W. H., Rep. of spin
relaxation,
momentumin
1957) p. 37 ff.
591.
Z. 30 (1929) 467.
11) 12) 13)
(Princeton,
27 (1961)
16 (1914)
531.
10)
Press
L., Physica
Wetensch.
A63 (1950) 25. Sot. (London) A205 (1951)
Progr.
Phys.
Interscience
quantum
16 (1953) Publ.
mechanics,
135.
108.
(New
Princeton
York,
1964)
University
o North-Holland Pzlblishing Co., Amsterdam
CROSSING
OF ENERGY
LEVELS
W. J. CASPERS Laboratorium VOOY vaste stof fysica, Universiteit van Groningen, Nederland Received 4 April 1968
Level crossing is discussed for a system with a hamiltonian 2 = A + HB, H being an external parameter. A crossing is a degeneracy of two or more energy eigenvalues for a discrete value of H. The number of crossings is related with the degree in H of the basis vectors of the space of operators commuting with &‘: every crossing reduces the degree of one of these basis vectors, if one starts with a basis consisting of the first n powers of 2, n being the dimension of the Hilbert space. It is possible to indicate the maximum number of crossings for a given n. This number being realized, the space of commuting operators is invariant.
1. Introduction. This paper is concerned with the necessary conditions for level crossing of a quantum mechanical system. We consider systems the motion of which is described by a hamiltonian that is a linear function of a parameter H, S@ = A + HB. The eigenvalues of &? are, in general, nonlinear functions of H, b,(H), a = 1, 2, 3,. . . n, PZbeing the dimension of the hamiltonian matrix. The dimension n is considered to be finite in this paper. A level crossing occurs if two eigenvalues are equal for a discrete value of H, Sa(H1) = aii(Hl), tz # ,!?,without being identical for all values of H. At the end of the paper the special case of degenerate levels LTLx(H)3 cd,(H) is shortly discussed. Level crossing was discussed for the first time in relation with the identification
of energy levels of a molecular system for different values of an see Hundi-3). This parameter is for instance the external parameter, distance between the atoms in a two-atomic molecule, or, in an abstract model, the threshold between two minima of the potential energy of a onedimensional system. Hunds) formulated the hypothesis that crossings only occur as a consequence of the symmetry of the system or of a separation of co-ordinates. Intimately related with our problem is the discussion of the change of state of a quantum-mechanical system for an adiabatic or infinitely slow change in an external parameter. Ehrenfest’s adiabatic principle4) states that an atomic system originally in a stationary state will occupy stationary states corresponding with the momentary values of such an external para125
126
W.
J. CASPERS
meter for adiabatic changes of this parameter. The principle originally formulated in terms of the old quantum theory was proved by Born5) for Schrodinger’s wave mechanics. Hund’s “non-crossing rule” implies that a system in its lowest energy state will stay in the ground state during adiabatic changes of an external parameter, except for the exceptional cases for which crossings occur. Von Neumann and Wigners) proved the important theorem that a coincidence of two eigenvalues of a general hermitean matrix only occurs as a consequence of the variation of three parameters. Level crossing is an exception and on the basis of this principle also properties of large systems were discussed. The magnetic response of a macroscopic system for a quasistatic change in the external magnetic field may be evaluated in two different ways. First one may calculate the susceptibility on the basis of Ehrenfest’s principle, which gives the isolated susceptibility. Secondly one has the adiabatic susceptibility as a result of a purely thermodynamical calculation. Perturbation methods applied by Broer7) and Wrights) for large spin systems gave a difference between the two susceptibilities, whereas several authors claimed that the two quantities should be equal (Yamamotoa), Rosenfeldia), Caspers ii)). The apparent controversy should have its origin in the perturbation calculus, which does not converge for these large systems with interaction. The “non-crossing rule” was used as a starting point by the author to prove that the two susceptibilities are equalrl). The contents of this paper are twofold. First a general method is indicated to determine the level crossings for a given hamiltonian. The maximum number of crossings for a given dimension n is shown to be &n(n - 1). The proof is given in section 2 in which also a complete set of diagonal operators is introduced, i.e. operators commuting with the hamiltonian 9. Every diagonal operator, or constant of motion, may be expressed as a linear combination of the operators of this set, which is simply the set of the first n powers of 2. In the third section we show that the given set of diagonal operators may be replaced by a reduced set if a level crossing occurs, i.e. one of the set may be replaced by a polynomial in H with a degree that is one lower. Every crossing reduces the sum of the degrees of the complete set of polynomials by one. In the extreme case of +n(n - 1) crossings all polynomials have degree zero, and the space of diagonal operators is invariant, i.e. independent of H. In section 4 it is shown that the conditions for level crossings as indicated by Hund may be interpreted as special cases of the ones derived in this paper. We finally discuss in the last section some simple cases as an illustration of the theory developed in this paper. A short analysis of the case of degenerate levels is also given.
CROSSING
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LEVELS
127
2. Determination of level crossings. There are level crossings for those values of H for which the Vandermonde determinant of the eigenvalues 8, equals zero. The Vandermonde differences
E, -
determinant
is given by the product
of all
&,(cx > ,5)
The square of the Vandermonde determinant can be expressed in terms of symmetric functions of the eigenvalues. One has to realize that the determinant Ilbklll,bkl = tf$-‘, k, 1 = 1, . . . n has the same value as Ijatjlj and taking the product of the two determinants one finds
In this expression ?%
s=_
Tr Sm Tr 1
=
is a normalized
trace
LTrZm. n
This normalized trace is a polynomial in H of dregee m or lower. For even m the degree will be m, otherwise the hermitean operator B will be identical to zero, which case is of no physical interest. For the case that the eigenvalues of B are non-degenerate, it is easy to see that for asymptotic values of H we have Vi w constant x Hncnpl) and Vi will be a polynomial in H of degree ~z(n - 1). Because of the fact that Vi is non-negative all real roots of V:(H) = 0 have even multiplicity. From the definition of I”‘n it follows that a twofold root HI corresponds with a single crossing &, - &o = = c(H - HI) + O[(H - HI)~]. For a fourfold root HI one may have two coincident single crossings go1 - 8~ = c(H - HI)+ . . . . 6, - &‘a= c’(H -
--HI) + . . . or two levels that touch for H = HI, 8, - 8~ = c”(H - H1)2 + + O[(H - HI)~]. A sixfold root may indicate three degenerate eigenvalues. One may interpret a point of contact as two or more coincident crossings and in this way one may state that the maximum number of crossings equals $z(n - 1). The degree of Vi is always even: 2s < n(n - l), and the number of crossings is smaller than or equal to s. As a simple illustration we consider the case for which A and B commute. All levels are linear functions of H and for a non-degenerate spectrum of B all pairs of levels will have one and only one intersection point. The number of level pairs equals Qn(n - l), which corresponds with the maximum number of crossings. For the case 2s < n(n - 1) the number of level pairs that are asymptotically parallel equals &z(n - 1) - s. The Vandermonde determinant may be given a geometrical interpretation in the vector space of all operators commuting with 8, as will be done in the following section. If there are no degenerate levels the set I, Z(H), #z(H), . . . SF”-l(H), for a fixed value of H, is a
W. J. CASPERS
128
basis of this vector space. That this series of powers of S is really a basis can be seen in the following way: If we take a representation diagonalizing A? the matrix of the hamiltonian is given by lb,B,,.rl. This representation is unique and a commuting operator C should have the representation jC,&#/. The operator equation C = Cz:t uk#k, giving the components of C for the given basis, corresponds with the set of linear equations C, = xi:; ak &$ fx = 1, . . . PZ,which has a unique solution, because the determinant of this set, which equals the Vandermonde determinant, is unequal to zero. The n dimensional space of diagonal operators undergoes a transformation if the value of H changes. This transformation corresponds with a “rotation” in a space of higher dimension with a basis formed by a set of symmetrized products of A and B. The kth power of A? can be written in the following way Sk
= 2 Hl{Ak--1Bl}, z=o
the braces indicating a symmetrized B. We have, for instance,
product of k - 1 factors A and I factors
{A2B3)=A2B3+ABAB2+APAB+AB3A+BA2B2+ + BABAB
+ BAB2A
+ B2A2B + B2ABA
The number
of terms in a symmetrized
symmetrized space equals
products
n + (n -
are linearly
1) + (n -
product
independent
2) + . . . +
+ B3A2. {Ak-LBl) equals the dimension
1 = *n(~ +
k I . If all
0 of the larger
1).
Every crossing and consequently every zero of the Vandermonde determinant corresponds with a linear relation between the first n powers of X, for a given H = HI. This is because there are only n - 1 or less different eigenvalues of S? for this value of the parameter H. For a single crossing, or a single zero of the Vandermonde determinant the coefficient of ~59-1 in this linear relation is unequal to zero. For degeneracies of higher order the coefficients of xn-1 x78-2 ) . . . etc. equal zero. These relations may be multiplied by SP,.YP ..: to find other independent linear relations; for a twofold crossing we find two relations etc. The linear relations between powers of X, for a given H = HI, correspond with linear relations between the symmetrized products {Ak-ZBZ}. The maximum number of these linear relations is &z(n - l), corresponding with the lowest possible dimension of the space of symmetrized products; this lowest possible dimension is n. It could not be smaller because we made the supposition that there are n different eigenvalues for general values of H. If the number of crossings is smaller the corresponding reduction of the dimension of the space of symmetrized products is also smaller, i.e. we find
CROSSING
OF ENERGY
LEVELS
129
a restricted number of linear relations between these operators. There may exist, however, more linear relations, but these cannot be derived from linear relations between powers of &?, for a given H = HI. Formulated in an other way: level crossing is not a necessary condition for the existence of one or more linear relations between the symmetrized products. There may be a reduction of the degree of one of the basis vectors without crossing. In the next section we show that the basis of the space of diagonal operators given by the first n powers of 2’ may be replaced by an other basis, the vectors of which are of lower degree in H, if there are level crossings. Even crossing corresponds with a reduction by 1 of the sum of the degrees of the basis vectors. 3. Red&ion of the degrees of the basis vectors. The space of diagonal operators for a given general value of H, is an n dimensional space for which we may introduce a metric. The inner product of two hermitean operators C and D is defined by Tr CD TrDC CD = ~ =r=
Tr 1
Dc >
and the square of the length of a vector in the operator space is
As was already stated in the previous section every diagonal operator may be expressed as a linear combination of I, Z(H),
. . . X’“-l(H),
which operators give a basis for the space of diagonal operators, except for those values of H for which degeneracies occur. The Vandermonde determinant, already introduced in the previous section may be interpreted as the volume of the hyperparallelepiped defined by the basis vectors et = &+1(H),
i = 1, . . . S.
The square of the Vandermonde determinant is given by
J’i(H) = Ilet
@4II.
That Vn(H) may be interpreted as the volume of a hyperparallelepiped follows from its properties: a. It is a linear function of all the basis vectors. b. It equals zero if there exists a linear relation between the basis vectors. First wezonly consider single crossings. Higher order degeneracies may be considered as limiting’cases for which two or more crossings coincide. For a single crossing there exists a linear relation between the et; the coefficient
W. J, CASPERS
130
of e,(H) may be chosen to be equal to 1. Suppose we have a degeneracy H = HI
for
n-1 e,(Hl)
+
X
afh-#l)
=
0.
k=l
So we can find a linear combination
of the et(H) that has a single zero for
H = HI n-1 eA(H)
=
e,(H)
+
C k=l
ai”e,_k(H)
= (H -
HI) e:‘(H).
It cannot be a zero of higher order because V:(H) has a double zero for H = = HI, as we only consider single crossings. From the determinant properties it immediately follows that in terms of a new basis we may write V:(H)
= (H -
H1)2 /leil)(H) eil)(H)ll, ell)(H) = es(H), e:‘(H)
= &H-
i =
1, . . . n-
1,
G(H). 1
The new basis vector e:‘(H) has a degree n - 1 in H. Now we consider a second value of H for which there is a crossing. Also for this value H = H2 there exists a linear relation between the vectors of the original basis n-1
e,(Hz)
+ C Z~2’e,_~(Hz) = 0. k=l
This linear relation may be translated into a relation between the vectors of the set eb”; the coefficient of ec’ may be chosen to be equal to 1. Again we have a single zero n-1
ei”(H)
+ x ai2’ec?l,(H)
= (H -
Hz) e:‘)(H),
k=l
and we may write Vi(H)
= (H -
H1)2 (H -
ei2’(H) = e;‘)(H),
Hz)2 llej2)(H) ei2’(H)ll, i=
1, . ..n-
1.
The vector ei2) is defined above. Now we have basis vectors ek2)(H) of degree i - 1 in H for i = 1, . . . n - 1, and the vector e:)(H) has the degree 1z - 2. For the first zero the vector e,(H) was replaced by e:)(H) and the degree of the nth basis vector was reduced by one; for the second zero e:)(H) was replaced by e:)(H) and again the degree of the nth vector was reduced by one. The ejl) as well as the ej2) form a basis for the space of all diagonal operators for those values of H for which no degeneracies occur. This procedure can be continued until V:(H) is factorized in the following way Vi(H)
=lcr
(H -
Hz)~
Ile$“)(H)ej”)(H)Il,
CROSSING
OF ENERGY
LEVELS
m being the total number of zero’s of Vn, corresponding of crossings. This factorization
is performed
131
with the total number
in m steps, each of these steps
corresponding with a linear transformation of the basis vectors. In this linear transformation one of the basis vectors is replaced by an other, the degree of which in H is one lower. After the kth step the expression for V:(H)
reads
v;(H)= ,bl (H - Hd2 Ileik’(H)ebk)(H) 11,
k=
1,2,...m.
The k + Ith step (k + 1 < m) is made in the following way. First we order the ekk)(H) according to climbing powers of H and perform an orthogonalization procedure for the @)(Hk+l). i.e. from every ei”)(Hk+l) we subtract components parallel with ej”)(Hk+l), i < i. We define the set @+l)(H) in the following way
t$k+l)(H) = ejk)(H) _ C i
ei”‘(Hk+l)e,!k)(Hk+l) e:“)(H), ey)(Hk+l)
ey)(Hk+l)’
1, 2, 3, . . .)
-(kfl)(H) until we arrive at a vector eP A new basis is defined by
with a vanishing length for H = Hk+l.
eik+l)(H)
= ei”‘(H)(i < p), e:+“(H)
eikfl)(H)
= eik’(H) (i > f~),
and the corresponding jleik’(H) er)(H)lj
determinants
= (H -
=
1 H _ Hk+l
@+“(H),
are related in the following ..Hk+1)2 Ileik+l)(H) t~i’+~‘(H)Il.
way
The set eikfl) could not be degenerate for H = H,++I because V:(H) only has a double zero for this value of the external parameter. Continuing the procedure for all zero’s of V:(H) we finally arrive at the factorized form given above and for a total number of crossings m the sum of the degrees of the basis vectors equals n-1
d = x I I=0
m = &Z(TZ-
1) -
m.
The number d equals 0 for the case of maximum number of crossings m = = $n(KZ- 1). The reduction procedure is not unique but in our scheme we have, for every zero of V:(H), a reduction of a basis vector of lowest possible degree. Still there is some ambiguity because there may be more than one basis vector of this degree in the linear combination that vanishes for the corresponding value of H.
1%‘. J. CASPERS
132
The basis vectors
e:“)(H)
form a complete
set of diagonal
operators
lowest possible degrees. As was stated before the linear relations powers of Z(H) for a fixed value of H, correspond with linear
with
between relations
between the symmetrized products of A and B. Every crossing reduces the dimension of the space of these products. The vectors elk’(H) undergo a transformation in this space if the value of H is changed. The vector space defined by this basis is also transformed if H changes, except for the case that m = &(PZ - 1) for which the space of diagonal operators is invariant, all basis vectors ep’ being independent of H. It may be possible that the degrees of the basis vectors may be further reduced, without this reduction being derived from zero’s of V:(H). For this case V:(H) will contain a factor that is the square of a polynomial, which polynomial has no real zero’s. This will be shortly discussed in relation with the example of section 5.1, i.e. the spin hamiltonian for S = 1. Multiple crossings for which more than two levels coincide for a given value of H or for which two pairs of levels cross for the same value of the parameter etc., can be discussed as limiting cases of the foregoing analysis. These cases correspond with fourfold, sixfold etc. roots of V:(H). 4. Hund’s colzditions for level crossing. Hunds) has discussed several cases for which crossings may occur. First of all he stated that for a sufficiently general mechanical problem the coincidence of two levels cannot be realized by varying one external parameter. Von Neumann and Wigners) showed that for such a case only the correct choice of three (external) parameters results in a coincidence or level crossing. The first case mentioned by Hund for which a crossing appears for one external parameter is the case of a separable co-ordinate, e.g. the component of the total angular momentum in a direction of axial symmetry. One of the basis vectors e:“)(H) may now correspond with this component of the momentum, being the operator for infinitesimal rotations about the symmetry axis. This operator is independent of H and our reduction in this case may result in a eim) # I that is also independent of H. Another possibility is the invariance of the hamiltonian for the interchange of two identical particles; also in this case one of the e:@(H) # I may be independent of H. These two cases may be characterized by the word symmetry. Another important case, however, not mentioned explicitly by Hund, is the one of two (or more) systems without interaction. Both systems are supposed to be coupled with the external field H, and the hamiltonian takes the form
9
= Xr
+ X2,
%r
= A(1) + B(1) H, 2’2 = A(2) + B(2) H,
in which the operators A(i) and B(i) only contain variables of the ith system. Here we also have separation of co-ordinates but, in general, there is not one single co-ordinate that commutes with 8 and the system does not show
CROSSING
OF ENERGY
LEVELS
133
a symmetry. A discussion of this last case of separation of co-ordinates is given in section 5 as an example of the theory developed in the preceding sections. We also discuss in this last section another case of level crossing not based on symmetry. First of all the relatively simple case of the spin hamiltonian for S = 1 is analysed. 5. Discussion
of
special cases.
5.1. Spin hamiltonian for S = 1. In the description of paramagnetic resonance spectra of salts of the iron group use is made of the concept of spin hamiltonian, which gives the spin splitting of the lowest orbital state. Effects of spin-orbit coupling are included in this effective hamiltonian, For S = 1 the most general which only depends on the spin variables 12-9. spin hamiltonian in the case of an isotropic g-tensor is given is) by X,s
= D[S2, -
$S(S + l)] + E(S;
-
S;) + g,!$,H.S,
in which /?ois the Bohr magneton. The X, y and z axes correspond with the principal directions of the spin splitting, which can be described by a tensor of the second rank if S = 1. So far the magnetic field H has an arbitrary direction with respect to the principal axes. If the polar angles of H are defined in the usual way (6 being the angle between H and the z axis and q~the angle between the projection of H on the xy-plane and the x axis) the secular equation of Xs reads g3 +
P=
PS + 4 = 0, -(&D2 + E2 + hz),
h = gBo IHI,
q = &Da - jDE2 - QDh2(3 cos 26 + 1) -$Eh2(1
- cos 26)cos 2~.
It is well-known from the theory of the algebraic equations of the third degree that a necessary and sufficient condition for a twofold root is
&P3 + q2= 0, which is equivalent with the criterion &#‘!‘! ‘+i-2]/ = 0 of section 2, because the value of this determinant is : - (&fi” + 42). This quantity is non-negative and it is only zero for those values of h, 6 and q~for which there is a degeneracy. The easiest way to locate these degeneracies is to differentiate with respect to the three variables h, cos 26 and cos 29. Differentiation with respect to cos 26 and cos 2g, gives
2q(+Dh2 - $Eh2 cos 297) = 0, 2q.$Eh2(1 - cos26)
= 0.
The case q = 0 is almost trivial levelcrossing. Thisconditiongives
because it implies p = 0 as a condition for D = E = 0, with a crossing for h = 0. (I)
W.
134
The other solutions
E = 0;
J. CASPERS
of the second equation
h = 0;
cos26
= 1.
For the first case we find as a nontrivial with a zero of Vandermonde’s crossing is
are
determinant,
solution
h = 0, corresponding
so the second possibility
of level
DfO,E=h=O.
(II)
For k = 0 there is another possibility This case is equivalent with (II) apart Finally the most interesting case is for redundant for this case because q doesnot -A$”
-
q2 zz &(+D2
+ E2 +
@)3
of level crossing given by E = fD. from an interchange of two axes. cos 26 = 1. The condition for v is depend on v. There is a crossing for -
(&D3
-
gDE2
-
tDj$)2
zz
0,
or
(E2 + h2)(02 - E2 - h2)2 = 0, which gives a level crossing for ks = 02 -
IDI > /El
if
cos28
=
E2,
1.
(III)
For the trivial case (I) we may choose for the z axis the fixed direction of H. The energy eigenvalues are h, 0, -h, and for h = 0 there is a threefold degeneracy, or a coincidence of three single crossings. The sum of the degrees of the basis vectors of the space of commuting operators is 3 for the general case (3 = &Z(PZ- 1)). Case (I) shows a reduction of 3, corresponding with the threefold degeneracy for h = 0, giving a simple illustration of the general theory of sections 2 and 3. Level crossing here is a consequence of the high symmetry of the system, which is invariant for all rotations about the z axis. Commuting operators are E, S, and Sz. In the second case it is quite clear that the degeneracy for h = 0 is independent of the direction of the field; H need not be parallel with a symmetry element of the system. The energy eigenvalues for lz = 0 are QD and -$0, the first one is twofold. The hamiltonian for h = 0 obeys the equation &?s + +DX - $02 = 0, and a nontrivial commuting operator is given by
e3 =
X;(h)
+ QDXs(h) h
- $D2 ,
and a complete set is ei = I, e2 = A?(h), es, the sum of the degrees is 2. In case (III) it is again the symmetry of the hamiltonian that leads to level crossings. For cos 26 = 1 (6 = 0 or 7~) the eigenvalues of Xs are pi = -go,
b2 = QD + JEZ + h2,&3 = &D -
JEW + h?
For E # 0 &2 and 8s do not cross, but 4’1 and 83 have intersection
points
CROSSING
OF ENERGY
LEVELS
135
for 02 = E2 + h2, h = *dD2
-
E2
if
D > IEl.
In this case br and 8s have no crossing. For -D > IE) 61 and &s have intersections for two values of 12. The maximum number of crossings is two, corresponding with a nontrivial constant of motion es = Si, and a basis is ei = I, es, es = 2; the sum of the degrees is 1. For IDI < IE 1 we have the same basis and there are no crossings. But it was already stated that the existence of coincidencies is not a necessary condition for the reduction of the degrees of the basis vectors. For E = 0 the system shows maximum symmetry, i.e. axial symmetry and a complete set of commuting operators is ei = I,
e2 = S,
and
es = Si,
all of degree zero. There is a reduction of 3 corresponding with level crossings for h = 0, 12=-&D. In the trivial case we have the same set of commuting operators. 5.2. S$Gn hamiltonian for higher values of S. We do not have the intention to discuss in detail the general case for these higher spin values. We only want to draw the attention to two special cases the first of which is rather wellknown. For half-odd spin values there are always degeneracies for h = 0 because of Kramers’ theorem. All levels have even degeneracy if there is no external field, and this degeneracy is not related with geometrical properties of the system, but has its basis in the time reversal symmetry. For S = 2 one can find examples with degeneracies that are not a consequence of symmetry or of a general physical principle like Kramers’ theorem. The general spin hamiltonian for this case takes the form
*s =M=--2 ; B;M TsM(S)
+ ;
BzM TIM(S) + h-S>
M=-4
the z axis corresponding with the direction of the external field. The TIM and TJM are the wellknown tensor operators for the spinls), and BUM and BUM are complex constants obeying the symmetry relations BLM = (---I"B~-~,
L = 2, 4.
The BLM as well as the TLM transform like the spherical harmonics YLM, as far as the rotations are concerned. The TLM may be normalized in such a
136
way
W.
J. CASPERS
that the matrix elements are given by <~M~ITLMI
2M2)
=
<2MzLMI2Ml>,
the symbol in the right member being a Clebsch-Gordan coefficientle). For a suitable choice of the constants BLM the Hilbert space falls apart into two subspaces corresponding not connected by Zs.
with M = 2, -2
and M = 1, 0, - 1, which are
Taking B4s = B44 = 0 we find that there are no matrix elements connecting 122) with 12 - 2) and 12 - l), and 12 - 2> with 12 2> and 12 1). The matrix elements (2 2 I&‘sl 2 1) and (2 2 IL%?s~ 2 0) are zero if B;,<2121
/22)+B;,<2141
122>=0,
B;,<2022~22)+Bf,(2042~22)=0, and the same equations guarantee that there are no matrix elements connecting the states 12, -2), 12, -1) and 12, -2>, 12, 0). The four ClebschGordan coefficients in the equations are non-zero and for an arbitrary choice of the BUM we can always find B~M(M = j= 1, *2), in such a way that the Hilbert space is separated in two subspaces and the eigenstates of X’s are vectors in these subspaces. The eigenvalues for the subspace (12, -2>, 12, 2)) are linear functions of h given by Bso(2 2 2 012 2) + B40<2 2 4 012 2) f
2k,
which have a crossing for h = 0. These levels will certainly have an intersection with all triplet states, because the asymptotic slopes of these states is 0, 1 or - 1. So we have a total of 7 intersections, which cannot be interpreted in terms of symmetry or of separation of coordinates. 5.3. Systems &thoztt interaction. If we have two systems without interaction we may expect, in general, a number of crossings, which disappear, at least partially, if an interaction between the systems is introduced. We suppose that the levels of the individual systems do not cross, to make-the discussion as simple as possible. The dimensions of the Hilbert spaces of the two, uncoupled, systems are supposed to be nl and ns. If the total system with interaction has no special symmetries, resulting in level crossings, the sum of the degrees of the basic commuting operators is +1n2@v2-
I),
corresponding with the first nias powers of the total hamiltonian. For the composite system without interaction we have the following basic set &$Z~,
k = 0, ...n1 - 1, I = 0, . . . ns -
1,
CROSSING
OF ENERGY
LEVELS
137
21 and 9s being the hamiltonians of the two individual systems. This basic set cannot be reduced and the sum of the degrees in this case equals m-1 c s=o
m-1 z (s + t=o
q
and the decoupling @r?zs(n@s
-
=
~W2(~1
+
n2 -
4,
of the systems results in a maximum 1) -
~12&+zr + ns -
number of crossings
2) = 913&+21122 -
921-
ns +
1).
This reduction will be illustrated with a simple case. We consider two spins 8 with different g values, in one and the same external magnetic field h superposed on constant local fields different for the two spins. X1 and 2s may be put in the general form #1=
g1ho1*&+
g1h.L
22
=
gzhoz*Sz
+
g2hSzz,
for a suitable choice for the unit of the magnetic field. The individual energy eigenvalues are G
= zttg&w&,,
+
%,,
+
(ho,,, +
W2)"*
8:
=
+
G2.y
+
@02,2 +
42)i.
f&2(%2,,
We take gl and gs > 0. There are two energy levels of the system of the two spins, 8: + 6; and 8; + 6’ i, that will have two intersection points for a certain range of values for the components of her and has. The trivial case her = ho2 = 0 will not be considered. The two levels intersect for
which has real solutions for (&or,,
-
&o,,J2
-
(g; -
gMh,2,
-
&%)
2 0.
The general conditions for which this inequality is obeyed are rather complicated but it is easy to show that for h 02(h 01) sufficiently small and gs > > gr(gr > gs) there are two crossings. This agrees with the predicted value #nrns(7zr~zs- 1~1- ns + 1) = +.2*2(4 - 2 - 2 + 1) = 2. 5.4 Degenerate levels. For the case of levels identical for all values of the external parameter H the situation is essentially different from the one in which the levels intersect for a finite number of values of the external field. The Vandermonde determinant is equal to zero for all values of H and the operator space corresponding with the basis IA? . . . Xn-1 is only n - 1 dimensional, except for those values of H for which additional intersections occur, resulting in a dimension still smaller. From a study of a diagonal representation of X one may conclude that the space of hermitean operators
CROSSING
138
OF
ENERGY
LEVELS
commuting with X is 1z + 2 dimensional, for all values of H. For the case of more degenerate levels this dimension is correspondingly increased. A systematic study of the algebraic properties of this space of larger dimension has not been performed so far. Con&.&on. An analysis has been made of the properties of the vector space of all hermitean operators commuting with a given hamiltonian Z@ = = A + BH, H being an external parameter. This analysis was made to find conditions for level crossing. The vector space is 1zdimensional, n being the dimension of the hamiltonian. A basis is formed by the first n powers of S : I, 2, 82 . . . Zfi-1. The basis vectors are polynomials in the parameter H of degree 0, 1, 2, . . . n - 1. For every crossing of levels one of the basis vectors may be replaced by an other polynomial, with a degree that is one lower. The maximum number of intersection points is $n(n - 1) corresponding with a basis independent of H, or with an invariant vector space. The study of the non-crossing rule is made especially as a basis for a future analysis of the properties of the energy spectrum of large systems of particles with interaction. For these systems the response in slowly varying external fields has our special interest ii). Acknowledgements. The author is greatly indebted to Drs. H. P. van de Braak, Ir. J. Hennephof and Drs. G. Vertogen for critical reading of the manuscript and for stimulating discussions.
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