Volume 214, number 4 PHYSICS LETTERS B 1 December 1988

At the recent 1AMP Congress in Swansea, Thirring [ 11 ] ~ pointed out that there is a link between the lack of stability of non-relativistic systems and the collapse of corresponding relativistic systems. We want to present here a very simple argument to prove this statement and at the same time to give a new way to derive conditions of the type ( 1 ).

Consider a system of N non-relativistic particles of mass rn interacting by r -2 forces (gravitational or Coulomb). For scaling reasons, the binding energy is

ENR (m, N ) = -mF(N, coupling constants) (3)

Then, because of the operator inequality,

m p2 +too 2 + (4) ~< 2m 2

for arbitrary rn > 0, the ground state energy of the corresponding system with x/p2+m2o replacing p2/2mo in the hamiltonian will satisfy

Eac~(mo, N ) < N + - m E ( N , ...), (5)

for a system of particles of mass too. Now assume

EN a ( m, N) ~ - m N "f (coupling constants)

for N large. (6)

Then, if c~ > 1,

m m 2 ,~ ERel (mo,N)<N(_~+_~m_m N - i f ) ( 7 )

and it is obvious that for

N'~-'f>½ , (8)

the system will collapse since m can be taken arbi- trarily large.

In the case ot = 1, we may also get collapse if the coupling constants are such that

f > ½ . (9)

All this can be generalized in an obvious way to the

"~ I am grateful to W. Thirring for indicating me his original pub- lication after receiving the first version of this work [ 12 ]. The proof uses a different operator inequality on p and the theorem that if collapse occurs for zero mass particles, it also occurs for non-zero mass.

case with two species of particles with different masses.

We may expect that inequality (8) gives rather good results if the non-relativistic upper bounds are good. Inequality (4) was in fact used in a different context to get an upper bound on the energy of a re- lativistic three-quark system [ 13 ] and led to the cor- rect estimate of the energy for large angular momentum.

First we shall illustrate on known cases how (8) works.

Take a system of N particles interacting by purely gravitational forces. In the non-relativistic case, the Thomas-Fermi approximation gives [ 14 ] ~2

ENR(m, N) ~- --N7/3ml¢2/10.51 , (10)

where x = Gm 2, G being Newton's constant in units when h = 1.

I f we make particles relativistic, we get collapse for

Nx3/2> 3.471 . ( 11 )

This is to be compared to the rigorous lower limit ob- tained by Lieb and Yau [ 7 ], for q= 2:

N/¢3/2 > 3.09. (12)

The number on the right-hand side of (12) coincides with the estimate of Landau [ 15 ].

Another illustration is the relativistic hydrogen-like atom (with an infinitely heavy nucleus). We know that the critical strength of the coupling constant, in the relativistic case is Zot = 2/ft. I f we use inequality (5), we get Zot = 1.

Therefore, we have indications that our procedure not only gives a criterion for collapse, but that the numbers obtained may be of the same order of mag- nitude as the best possible answers.

We shall now consider a neutral atom of charge Z first with a fixed centre and later with a moving nu- cleus. A non-relativistic neutral atom is such that

E N R > - C Z 7 / 3 0 1 2 m . ( 1 3 )

In the appendix we prove, using the results of Thirring [ 16 ] and Spruch and collaborators [ 17 ], that

C> 0.447, (14)

~2 It is very difficult to trace back the original calculation.

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Volume 214, number 4 PHYSICS LETTERS B 1 December 1988

but in fact it is very likely that

C > 0 . 5 , (15)

which is saturated for the hydrogen atom, while C tends to the Thomas-Fermi value for large Z [ 18 ]:

C~0.77. (16)

In this way, we get that a neutral atom (or an assem- bly of neutral atoms) will collapse if

1 og2Z4/3 > - (17)

2)<0.447 '

or, using Lieb and Yau notation:

f lea>2.92 . (18)

This is to be compared to (1) and (2) with q = 2 which gives 57 560.

It must be said, however, that our derivation is re- stricted to integer Z, neutral atoms and q= 2. How- ever, all these restrictions can probably be eliminated. For q=2, there must exist a bound with the correct behaviour since we know [ 18 ] that it is the quantity

of.2Z4/3 q 2/3 (19)

which enters in the Thomas-Fermi limit. However, experience with q= 2 shows that the Thomas-Fermi limit is reached algebraically from above, and, there- fore, explicit calculations are needed.

It is also possible to incorporate the relativistic motion of the nucleus. In the non-relativistic case, we have, according to Thirring (ref. [ 16 ], p. 275 )

E ( M = o o ) (20) E ( M ) < 1 + Z m / M '

for a neutral atom. For Z = 1 this is optimal, but cer- tainly not for larger Z.

So, neglecting the irrelevant rest masses:

M Z m 0 . 4 4 7 ZV/3oLZm ERe~<'~ + 2 -- I + Z m / M ' (21)

for arbitrary M and m. It is easy to see that the best choice is Z m / M = 1.

Then collapse occurs for

fl2a > 23.36. (22)

This is again a very reasonable number. In conclusion, we would like to say that inequality

(4) allows to transform a "relativistic" problem into a non-relativistic problem. All what has to he done is to gather or obtain upper bounds on energies of non- relativistic systems. Naturally, we shall not discuss the very difficult question of principle whether the dom- inant relativistic effects are taken care of by the sub- stitution m+p2/2m-- - , x /p2+m 2. This is a zeroth- order approximation for which we have no idea of what is the next order.

Another problem is that in cases when the "exact" answer is known, our method gives results of the fight order of magnitude, i.e., numbers, which, divided by two, give conditions for the absence of collapse. It is not very easy to understand why it is so except in sys- tems where the kinetic energies of the particles are concentrated around their mean value.

I am very grateful to the organizers of the 1AMP Congress in Swansea, in particular to R. Benguria, where this work was initiated. I thank J.D. Jackson for several suggestions, in particular concerning references.

Appendix

Proof of the inequality

E N R ( Z ) < -0 .447 zT/3oLem,

where ENR is the non-relativistic energy of a neutral atom with an infinitely heavy nucleus of charge Z.

(i) For Z = 1.

ENR = - ½ m ( Z a ) 2 = - ½ m Z 7/3 a 2 .

(ii) For 2 ~ Z ~ 9. The helium-like ion of charge Z has, according to Thirring (ref. [ 16], p. 222) an en- ergy which is less than

--Z2oL2m ( 1 - 5 / 1 6 Z ) 2 .

Now the energy or an atom E ( Z , Z ) is certainly less than the energy of a positively charged ion E ( Z , N ) , N ~ Z, because one can take a trial function which is the antisymmetrized product of the ion and of N - Z electrons localized far away for which the kinetic en- ergy can be made much smaller than the (negative) potential energy. The minimum of I E(Z ,2) I Z - 7/3 )< a - 2 m - 1 is reached for Z = 9 and is 0.447.

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Volume 214, number 4 PHYSICS LETTERS B 1 December 1988

( i i i ) For Z = 10, 20, 40, 60, 80; 86, 118, 168, 218, 268, 290, the values o f E ( Z , Z ) Z - 7 / 3 o t - 2 m -1 are,

respectively, less t han (ref. [ 16] p. 267)

- 0.594, - 0.620, - 0.646, - 0.648, - 0.656

a n d (ref. ( 1 7 ) )

+ 0.669, - 0.678, - 0.697, - 0.699 .

We use now, for Z > z

E( Z, Z ) < E ( Z , z) < ( Z 2 / z 2 ) E ( z, z ) .

So, for ins tance,

E ( 19, 19) < (Z/z) 7/3 (~9) l /3E( 10, 10)

< -0 .479ZT/3o t2m.

In this way, one covers all Z ' s f rom Z = 1 up to

ZM = 1108. ( i v ) Beyond that we use an inequa l i ty va l id for

closed shells which is an a d a p t a t i o n of eq. (4 .5 .14) of ref. [ 13 ] tak ing in to account concav i ty propert ies:

E < - Z2o~2m no

× [ 1 - ( 1 / 6 Z ) (n3o+4n2o+5no+2)]2,

where no is the n u m b e r of shells

Z = ~ (2no 3 + 3n~ + n o ) ,

and, start ing with no= 11, Z = 1012, we can prove that we can go f rom one shell to the next wi thou t ever going above - 0.447.

Natura l ly one m a y th ink that all this is r id icu lous

because the fo rmula p roposed by Schwinger [ 19 ]

E = - 0 . 7 6 8 745 Z 7 / 3 + ½ Z 2 - 0 . 2 6 6 Z 5/3 ,

or its r e f i nemen t in ref. [ 17 ] are cer ta in ly very good for these values of Z.

References

[ 1 ] F.J. Dyson and A. Lenard, J. Math. Phys. 8 (1967) 423; 9 (1967) 698.

[ 2 ] E.H. Lieb and W. Thirring, Phys. Rev. Lett. 35 ( 1975 ) 687, 1116 (E).

[3] L.D. Landau, Nature 141 (1938) 333; J.R. Oppenheimer and R. Serber, Phys. Rev. 54 (1938) 540.

[4] J.M. Lrvy-Leblond, J. Math. Phys. 10 (1969) 806. [5] J.G. Conlon, E.H. Lieb and H.T. Yau, Commun. Math.

Phys., to be published; and Princeton University preprint (1988); J.G. Conlon, plenary talk 1AMP Congress (Swansea, 1988 ).

[6] S. Chandrasekhar, Phil. Mag. 11 ( 1931 ) 592. [7] E.H. Lieb and W. Thirring, Ann. Phys. (NY) 155 (1984)

494. [8] E.H. Lieb and H.T. Yau, Princeton University preprint

(1987) [9] I. Herbst, Commun. Math. Phys. 53 (1977) 285; 55 (1977)

316 (E). [ 10] E.H. Lieb and H.T. Yau, Princeton University preprint

(1988); H.T. Yau, invited talk lAMP Congress (Swansea, 1988 ).

[ 11 ] W. Thirring, oral communication Session on Schri3dinger operators, IAMP Congress (Swansea 1988).

[12]W. Thirring, Naturwissenschaften 73 (1986) 605 [in German].

[ 13] A. Martin, Z. Phys. C 32 (1986) 359. [ 14 ] J.D. Jackson, private communication. [15] L.D. Landau, Phys. Z. Sovjetunion 1 (1932) 285, trans-

lated by K.R. Lang and O. Gingrich, A source book in as- tronomy and astrophysics (Harvard U.P., Cambridge, MA, 1979) pp. 1900-1975.

[ 16 ] W. Thirring, Quantum mechanics of atoms and molecules (Springer, Berlin, 1979 ).

[17] R. Shakeshaft, L. Spruch and J.B. Mann, J. Phys. B 14 (1981)121.

[ 18 ] See, e.g., E.H. Lieb, Rev. Mod. Phys. 53 ( 1981 ) 603. [ 19] J. Schwinger, Phys. Rev. A 22 (1980) 1927.

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