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Lower bound for the ground state energy of the no-pair Hamiltonian
24 July I997
PHYSICS LETTERS B Physics Letters B 405
(I 997) 293-296
Lower bound for the ground state energy of the no-pair Hamiltonian C. Tix’ Matematisk institutt, Vniversitetet i Oslo, Postboks 1053, N-0316 Oslo, Norway
Received 21 January 1997; revised manuscript
received 13 May 1997
Editor: R. Gatto
Abstract
A lower bound for the ground state energy of a one particle relativistic Hamiltonian - sometimes called no-pair operator -
is provided. @ 1997 Elsevier Science B.V.
1991 MSC: 35P15; 35Q40 MC!? 03.65.Pm; 03.65.Ge Keywords: Relativistic Hamiltonian; Wave equation; Bound; Ground state
1. Intmduction
The multi-particle analogon of the operator *=A+
(Do-$A+
(1)
is often used to describe relativistic effects in atoms and molecules, see, e.g., Refs. [ 8,9]. Here DO is the free Dirac operator Do = C~(Y. V + /?mc2 and
h+(p) =;
1+
ca.p+#3mc2 E(P)
>
(2)
with E(p) = ( c2p2 +m2c4) ‘i2 the projection operator on the electron subspace of the free Dirac operator in momentum space. The underlying Hilbert space of B is the space of square integrable four-spinors + with A-& = +. ’ Present address: Mathematik, Universitit Regensburg, 93040 Regensburg, Gem~any. E-mail: [email protected] 0370-2693/97/.$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO370-2693(97)00622-9
The operator B was introduced for the two particle case by Brown and Ravenhall [2] (see also Refs. [ 1,141) to cure the continuum dissolution of the two-particle Dirac operator with Coulomb interaction: since the Dirac operator is unbounded form below the two-particle Dirac operator has the whole real line as spectrum, the eigenvalues of the one particle operator “dissolve”. B is called no-pair operator or in the description of meson models (usually with different potentials) “reduced Salpeter operator” [ 111. Recently, its multi-particle analogon was used in connection with the “stability of relativistic matter” question
[lOI. Similar to the Dirac operator with Coulomb potential, the no-pair operator has a critical nuclear charge 2,. Hardekopf and Sucher [6,7] observed that 2, = 2/[ (7r/2 + 2/72) cy] with the fine structure constant LY:= e2/(tic) and investigated the ground state energy of B numerically. They claimed that, as for the Dirac equation, the ground state energy vanishes for
294
C. Tix/ Physics Letters B 405 (2997) 293-296
2 = 2, [ 7, p, 20251. Evans et al. [ 31 proved that the energy ($, Bq) is bounded from below by CUZ( 1/rr~/4)mc*, if the nuclear charge does not exceed the critical charge Z,, otherwise it is unbounded. Tix [ 151 improved this bound to mc2( 1 - CXZ) 2 0.09. This last result shows the difficulties to obtain accurate numerically results near the critical coupling parameter and the need for sharp bounds. In this short note we explore the results from [ 151 by using some rather trivial numerics to obtain a reliable lower bound for the ground state energy for all Z < Z,.
functions hi(p) ($, W)
> 0 in the potential energy part of
JJu(p’>kj(P’,p)u(p)dPdp’ /g/g JJ 004P')$(P')& 0
0
coca
=
X
&
u(P>sj(P>dpdp’
DC) ,.
<
2. The lower hound A short summary of the method to obtain a lower bound for (+, B$) given in [3,15] is provided. Any normalized four-spinor in the electron subspace of Do can in be written in momentum space as &P> = --L N(P)
(
CEO+ E(P)) U(P) CP . @U(P) )
(3)
with the normalized Pauli spinor u, the Pauli-matrices UjandEu=E(O), N(p) = [2E(p)(E(p)+&)]“*. In [ 31 it was shown that the minimizer of (+, B$) is radially symmetric, i.e. the minimizing u is of the form u(p) = a(lpI)(fi]p()-‘(l,O)* where a is normalized and positive. This yields for the energy ($9 BG)
JJ coo0
--
CYZ
2li-
a(p’>k(p’,p)a(p>dpdp’
(4)
0 0
:=Sj(p’)
u(P>*sj(P>*
is obtained. The use of the Schwarz inequality in this way goes back to Hardy and Littlewood. It remains to choose the trial functions hj. To do this two things are important to notice: (i) No error is introduced by using the Schwarz inequality when hj/sj is proportional to the ground state wave function of (4) ; (ii) The operator B is similar to the Herbst operator H = @$%&? - e2Z/]x1. The ground state energy of H is obtained by minimizing the quadratic form (4) with k = ko, without the square roots and with 22 instead of Z. Raynal et al. [ 121 obtained upper and lower bounds for the ground state of H by introducing a positive trial function - using a different method as done here. They choose the trial function to be the Fourier transform of I,la-1,-cll4~
with e(p) = E(p), p = JpI, k := ko + kl and kj(“,p)
1
Qj (&
+ f)
Sj(P).
(5)
The Legendre functions of the second kind Ql are positive (see Ref. [ 131 for the notation) and appear here for the same reason as in the treatment of the non-relativistic hydrogen atom in momentum space [4, problem 771 and Sj(p) = \/l + (-l)j/e(p). Because of scaling we have assumed fi = m = c = 1 in (4). Using the Schwarz inequality to introduce trial
(7)
Since they obtained excellent bounds this trial functions seems to be a good approximation for the ground state wave function of H. These two points suggest to take trial functions similar to(7)fortheoperatorB.Thiswascarriedoutin[15]. The lower bound
C&B+) 2 sup infE[.TP,pl(p). -Io P>O was obtained in [ 151 where
(8)
C. Tix/Physics
Table 1 Lower bound for the energy E = inf, E[Z, &PI (p)
Letters B 405 (1997) 293-296
295
ami the
corresponding parameters
0.62 1 0.808
az @Tg
= e(p) -
Ef.Z,/%~l(p)
7j-
sin[Parctm(p/pu>l
’ sin[(P+ -!+L-) x
(”-1 1
1) a=taNp/pcL)l
c
+
1 e(p)
0.2
+ F2) -1’2)
pi;/;*
+ #a
(9) -l/2)
with the associated Legendre functions P,,-3’2 (see [5, p. 10601) . To cancel the large momentum terms in (9) p 2 0 is chosen to fulfill 2 -= CUZ
tan(Pr/2) P
0.4az0.6
0.8
1.0
Fig. 1. Lower bound for the energy (+, Be,) of the no-pair Hamiltonian
f$_$(dP2 (A@
0 0.1 0.3 0.5 0.7
0.9843 0.9526 0.8652 0.7395 0.5552
0.106 0.215 0.423
0.9060 0.9003 0.8527 0.7500 0.5709
+ ( 1 - pq
P tan(7$/2).
(10)
The resulting function & [ Z, p, ~1 is numerically evaluated. The values for the parameters p and p for some values of aZ can be found in Table 1 and a plot of the energy over the coupling constant is shown in Fig. 1. I have been informed by the referee (A. Martin) that for the “Sucher” two-body equation J.C. Raynal finds similar conclusions, namely that, contrary to what is indicated in Fig. 2 of the Hardekopf-Sucher paper [ 71, the energy at the critical a = LY,= 87r/(7r2 + 4), is strictly positive: E( cu,) /E( 0) > 0.258.
Acknowledgements The author thanks the referee (A. Martin) for his interesting information about the two-particle “Sucher” operator. This work was partially supported by the European Union under grant ERB4OOlGT950214 and under the TMR-network grant FMRX-CT 96-0001.
References [II Hans A. Bethe and Edwin E. Salpeter, Quantum mechanics of one- and two-electron atoms, in: Handbuch der Physik, XXXV, ed. S. Fliigge, first edition, (Springer, Berlin, 1957) p. 88. [21 GE. Brown and D.G. Ravenhall, On the interaction of hvo electrons, Proc. Roy. Sot. London A. 208 (A 1095) 552, September 1951. [31 William Desmond Evans, Peter Perry and Heinz Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter, Commun. Math. Phys. 178 (1996) 733. [41 Siegfried Plilgge, Practical Quantum Mechanics I, Vol. 177 of Gnmdlehren der mathematischen Wissenschaften, first edition (Springer, Berlin, 1982). [51 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, fourth edition (Academic Press, New York, 1965). [61 G. Hardekopf and J. Sucher, Relativistic wave equations in momentum space, Phys. Rev. A 30 ( 1984) 703. [7] G. Hardekopf and J. Sucher, Critical coupling constants for relativistic wave equations and vacuum breakdown in quantum electrodynamics, Phys. Rev. A 31 (1985) 2020. [ 81 Y. Ishikawa and K. Koc, Relativistic many-body perturbation theory based on the no-pair Dirac-CouIombBruit Hamiltonian: Relativistic correlation energies for the noble-gas sequence through Rn (Z = 86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence, Phys. Rev. A 50 (1994) 4733. [9] Hans Jiirgen Aa. Jensen, Kenneth G. Dyall, Trond Saue and Knut Faegri, Jr, Relativistic four-component multiconfigurational self- consistent-field theory for molecules: Formalism, J. Chem. Physics 104 ( 1996) 4083.
296
C. Tix/Physic.s Letters 3 405 (19971293-296
[IO] Elliott H. Lieb, Heinz Siedentop and Jan Philip Solovej, Stability and instability of relativistic electrons in classical electromagnetic fields, J. Stat. Phys., to appear ( 1997). [ 1I] M. G. Olsson, S. Veseli and K. Williams, Validity of the reduced Salpeter equation, Phys. Rev. D 53 ( 1996) 504. [ 121 J.C. Raynal, SM. Roy, V. Singh, A. Martin and J. Stubbe, The “Herbst-Hamiltonian” and the mass of boson stars, Phys. l-&t. B 320 (1994) 105. [ 131 Irene A. Stegun, Legends functions, in: Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables , eds. Milton Abramowitz and Irene A. Stegun, Ch. 8 (Dover, New York, 1965) p, 331. ( 141 J. Sucher, Relativistic many-electron Hamiltonians, Phys. Scripta, 36 ( 1987) 271. [IS] C. Tix, Strict positivity of a relativistic hamiltonian due to Brown and Ravenhall, Bull. London Math. Sot., submitted (1997).
PHYSICS LETTERS B Physics Letters B 405
(I 997) 293-296
Lower bound for the ground state energy of the no-pair Hamiltonian C. Tix’ Matematisk institutt, Vniversitetet i Oslo, Postboks 1053, N-0316 Oslo, Norway
Received 21 January 1997; revised manuscript
received 13 May 1997
Editor: R. Gatto
Abstract
A lower bound for the ground state energy of a one particle relativistic Hamiltonian - sometimes called no-pair operator -
is provided. @ 1997 Elsevier Science B.V.
1991 MSC: 35P15; 35Q40 MC!? 03.65.Pm; 03.65.Ge Keywords: Relativistic Hamiltonian; Wave equation; Bound; Ground state
1. Intmduction
The multi-particle analogon of the operator *=A+
(Do-$A+
(1)
is often used to describe relativistic effects in atoms and molecules, see, e.g., Refs. [ 8,9]. Here DO is the free Dirac operator Do = C~(Y. V + /?mc2 and
h+(p) =;
1+
ca.p+#3mc2 E(P)
>
(2)
with E(p) = ( c2p2 +m2c4) ‘i2 the projection operator on the electron subspace of the free Dirac operator in momentum space. The underlying Hilbert space of B is the space of square integrable four-spinors + with A-& = +. ’ Present address: Mathematik, Universitit Regensburg, 93040 Regensburg, Gem~any. E-mail: [email protected] 0370-2693/97/.$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO370-2693(97)00622-9
The operator B was introduced for the two particle case by Brown and Ravenhall [2] (see also Refs. [ 1,141) to cure the continuum dissolution of the two-particle Dirac operator with Coulomb interaction: since the Dirac operator is unbounded form below the two-particle Dirac operator has the whole real line as spectrum, the eigenvalues of the one particle operator “dissolve”. B is called no-pair operator or in the description of meson models (usually with different potentials) “reduced Salpeter operator” [ 111. Recently, its multi-particle analogon was used in connection with the “stability of relativistic matter” question
[lOI. Similar to the Dirac operator with Coulomb potential, the no-pair operator has a critical nuclear charge 2,. Hardekopf and Sucher [6,7] observed that 2, = 2/[ (7r/2 + 2/72) cy] with the fine structure constant LY:= e2/(tic) and investigated the ground state energy of B numerically. They claimed that, as for the Dirac equation, the ground state energy vanishes for
294
C. Tix/ Physics Letters B 405 (2997) 293-296
2 = 2, [ 7, p, 20251. Evans et al. [ 31 proved that the energy ($, Bq) is bounded from below by CUZ( 1/rr~/4)mc*, if the nuclear charge does not exceed the critical charge Z,, otherwise it is unbounded. Tix [ 151 improved this bound to mc2( 1 - CXZ) 2 0.09. This last result shows the difficulties to obtain accurate numerically results near the critical coupling parameter and the need for sharp bounds. In this short note we explore the results from [ 151 by using some rather trivial numerics to obtain a reliable lower bound for the ground state energy for all Z < Z,.
functions hi(p) ($, W)
> 0 in the potential energy part of
JJu(p’>kj(P’,p)u(p)dPdp’ /g/g JJ 004P')$(P')& 0
0
coca
=
X
&
u(P>sj(P>dpdp’
DC) ,.
<
2. The lower hound A short summary of the method to obtain a lower bound for (+, B$) given in [3,15] is provided. Any normalized four-spinor in the electron subspace of Do can in be written in momentum space as &P> = --L N(P)
(
CEO+ E(P)) U(P) CP . @U(P) )
(3)
with the normalized Pauli spinor u, the Pauli-matrices UjandEu=E(O), N(p) = [2E(p)(E(p)+&)]“*. In [ 31 it was shown that the minimizer of (+, B$) is radially symmetric, i.e. the minimizing u is of the form u(p) = a(lpI)(fi]p()-‘(l,O)* where a is normalized and positive. This yields for the energy ($9 BG)
JJ coo0
--
CYZ
2li-
a(p’>k(p’,p)a(p>dpdp’
(4)
0 0
:=Sj(p’)
u(P>*sj(P>*
is obtained. The use of the Schwarz inequality in this way goes back to Hardy and Littlewood. It remains to choose the trial functions hj. To do this two things are important to notice: (i) No error is introduced by using the Schwarz inequality when hj/sj is proportional to the ground state wave function of (4) ; (ii) The operator B is similar to the Herbst operator H = @$%&? - e2Z/]x1. The ground state energy of H is obtained by minimizing the quadratic form (4) with k = ko, without the square roots and with 22 instead of Z. Raynal et al. [ 121 obtained upper and lower bounds for the ground state of H by introducing a positive trial function - using a different method as done here. They choose the trial function to be the Fourier transform of I,la-1,-cll4~
with e(p) = E(p), p = JpI, k := ko + kl and kj(“,p)
1
Qj (&
+ f)
Sj(P).
(5)
The Legendre functions of the second kind Ql are positive (see Ref. [ 131 for the notation) and appear here for the same reason as in the treatment of the non-relativistic hydrogen atom in momentum space [4, problem 771 and Sj(p) = \/l + (-l)j/e(p). Because of scaling we have assumed fi = m = c = 1 in (4). Using the Schwarz inequality to introduce trial
(7)
Since they obtained excellent bounds this trial functions seems to be a good approximation for the ground state wave function of H. These two points suggest to take trial functions similar to(7)fortheoperatorB.Thiswascarriedoutin[15]. The lower bound
C&B+) 2 sup infE[.TP,pl(p). -I
(8)
C. Tix/Physics
Table 1 Lower bound for the energy E = inf, E[Z, &PI (p)
Letters B 405 (1997) 293-296
295
ami the
corresponding parameters
0.62 1 0.808
az @Tg
= e(p) -
Ef.Z,/%~l(p)
7j-
sin[Parctm(p/pu>l
’ sin[(P+ -!+L-) x
(”-1 1
1) a=taNp/pcL)l
c
+
1 e(p)
0.2
+ F2) -1’2)
pi;/;*
+ #a
(9) -l/2)
with the associated Legendre functions P,,-3’2 (see [5, p. 10601) . To cancel the large momentum terms in (9) p 2 0 is chosen to fulfill 2 -= CUZ
tan(Pr/2) P
0.4az0.6
0.8
1.0
Fig. 1. Lower bound for the energy (+, Be,) of the no-pair Hamiltonian
f$_$(dP2 (A@
0 0.1 0.3 0.5 0.7
0.9843 0.9526 0.8652 0.7395 0.5552
0.106 0.215 0.423
0.9060 0.9003 0.8527 0.7500 0.5709
+ ( 1 - pq
P tan(7$/2).
(10)
The resulting function & [ Z, p, ~1 is numerically evaluated. The values for the parameters p and p for some values of aZ can be found in Table 1 and a plot of the energy over the coupling constant is shown in Fig. 1. I have been informed by the referee (A. Martin) that for the “Sucher” two-body equation J.C. Raynal finds similar conclusions, namely that, contrary to what is indicated in Fig. 2 of the Hardekopf-Sucher paper [ 71, the energy at the critical a = LY,= 87r/(7r2 + 4), is strictly positive: E( cu,) /E( 0) > 0.258.
Acknowledgements The author thanks the referee (A. Martin) for his interesting information about the two-particle “Sucher” operator. This work was partially supported by the European Union under grant ERB4OOlGT950214 and under the TMR-network grant FMRX-CT 96-0001.
References [II Hans A. Bethe and Edwin E. Salpeter, Quantum mechanics of one- and two-electron atoms, in: Handbuch der Physik, XXXV, ed. S. Fliigge, first edition, (Springer, Berlin, 1957) p. 88. [21 GE. Brown and D.G. Ravenhall, On the interaction of hvo electrons, Proc. Roy. Sot. London A. 208 (A 1095) 552, September 1951. [31 William Desmond Evans, Peter Perry and Heinz Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter, Commun. Math. Phys. 178 (1996) 733. [41 Siegfried Plilgge, Practical Quantum Mechanics I, Vol. 177 of Gnmdlehren der mathematischen Wissenschaften, first edition (Springer, Berlin, 1982). [51 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, fourth edition (Academic Press, New York, 1965). [61 G. Hardekopf and J. Sucher, Relativistic wave equations in momentum space, Phys. Rev. A 30 ( 1984) 703. [7] G. Hardekopf and J. Sucher, Critical coupling constants for relativistic wave equations and vacuum breakdown in quantum electrodynamics, Phys. Rev. A 31 (1985) 2020. [ 81 Y. Ishikawa and K. Koc, Relativistic many-body perturbation theory based on the no-pair Dirac-CouIombBruit Hamiltonian: Relativistic correlation energies for the noble-gas sequence through Rn (Z = 86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence, Phys. Rev. A 50 (1994) 4733. [9] Hans Jiirgen Aa. Jensen, Kenneth G. Dyall, Trond Saue and Knut Faegri, Jr, Relativistic four-component multiconfigurational self- consistent-field theory for molecules: Formalism, J. Chem. Physics 104 ( 1996) 4083.
296
C. Tix/Physic.s Letters 3 405 (19971293-296
[IO] Elliott H. Lieb, Heinz Siedentop and Jan Philip Solovej, Stability and instability of relativistic electrons in classical electromagnetic fields, J. Stat. Phys., to appear ( 1997). [ 1I] M. G. Olsson, S. Veseli and K. Williams, Validity of the reduced Salpeter equation, Phys. Rev. D 53 ( 1996) 504. [ 121 J.C. Raynal, SM. Roy, V. Singh, A. Martin and J. Stubbe, The “Herbst-Hamiltonian” and the mass of boson stars, Phys. l-&t. B 320 (1994) 105. [ 131 Irene A. Stegun, Legends functions, in: Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables , eds. Milton Abramowitz and Irene A. Stegun, Ch. 8 (Dover, New York, 1965) p, 331. ( 141 J. Sucher, Relativistic many-electron Hamiltonians, Phys. Scripta, 36 ( 1987) 271. [IS] C. Tix, Strict positivity of a relativistic hamiltonian due to Brown and Ravenhall, Bull. London Math. Sot., submitted (1997).