A general comparison theorem

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by  Claude Semay

A general comparison theorem
Claude Semay∗
Service de Physique Nucl´eaire et Subnucl´eaire, Universit´e de Mons,
Acad´emie universitaire Wallonie-Bruxelles, Place du Parc 20, 7000 Mons, Belgium
(Dated: February 17, 2011)
UsingtheHellmann-Feynmantheorem,ageneralcomparison theoremisestablishedforaneigen-
value equation of the form (T +V)|ψi = E|ψi, where T is a kinetic part which depends only on
momentumsandV isapotentialwhichdependsonlyonpositions. WeassumethatH(1) =T+V(1)
and H(2) = T +V(2) (H(1) = T(1) +V and H(2) = T(2) +V) support both discrete eigenvalues
E(1) and E(2) , where {α} represents a set of quantum numbers. We prove that, if V(1) ≤ V(2)
{α} {α}
(T(1) ≤T(2))forallposition(momentum)variables,thenthecorrespondingeigenvaluesareordered
1
E(1) ≤E(2) . Some analytical applications are given.
1 {α} {α}
0
2
Thecomparisontheoremofquantummechanicsstates WeconsidertwoHamiltoniansH(1)andH(2)suchthat
n
that,forsomeeigenvalueequations,iftworealpotentials
a
J are ordered, V(1) V(2), then each corresponding pair φH(2) H(1) φ 0, φ . (4)
4 of eigenvalues is or≤dered E(1) E(2) ( α represents a h | − | i≥ ∀ | i
{α} ≤ {α} { } Let us assume that the Hamiltonian
2 set of quantum numbers). This can be shownfor Hamil-
tonianswhichareboundedfrombelow byusing the Ritz H(a)=(1 a)H(1)+aH(2) (5)
] −
h variational principle. But, such a procedure is not ap-
p plicable for the corresponding Dirac problem since the possesses a number (finite or infinite) of well defined
t- Dirac Hamiltonian is not bounded from below. Never- eigenvalues E{α}(a) characterized by a set of quantum
n theless,thecomparisontheoremhasalsobeenprovedfor numbers α , for 0 a 1 [6]. If a; α is the cor-
{ } ≤ ≤ | { }i
a a Dirac equation with a potential monotone in a param- responding eigenstate, the Hellmann-Feynman theorem
u
eter [1–3]. directly yields
q
UsingtheHellmann-Feynmantheorem[4],weshallsee
[ ∂E (a)
that the comparison theorem can be applied to a great {α} = a; α H(2) H(1) a; α . (6)
2 class of eigenvalue problems written in the form ∂a { } − { }
v D (cid:12) (cid:12) E
5 (T +V)ψ =E ψ , (1) It is a positive number due(cid:12)(cid:12) to the hyp(cid:12)(cid:12)othesis (4). So,
5 | i | i E{α}(a) is an increasing function of a and
3
where T is a kinetic part which depends only on mo-
.5 mentums and V is a potential which depends only on E{α}(0)=E{(1α)} ≤E{(2α)} =E{α}(1), (7)
2 positions. No assumption is made about the number of
1 particles, and it is not necessary that the Hamiltonian where E(1) and E(2) are respectively eigenvalues of
0 {α} {α}
1 is bounded from below. We have not seen this presenta- Hamiltonians H(1) andH(2). Condition (4) is not neces-
tionelsewhere,althoughitisrelatedtoideaspresentedin
: sarilyeasytoverifyforarbitraryHamiltonians. Itisthen
v [3]. As the Klein-Gordonequationis notofthe form(1),
interesting to look at two particular simpler situations.
i
X the comparison theorem presented here does not apply, LetusfirstconsidertwoHamiltoniansH(1) =T+V(1)
r but results about the ordering of the spectra can been andH(2) =T +V(2) such that V(1) V(2) for all values
a obtained [3, 5]. ≤
of the position variables appearing in these potentials.
The Hellmann-Feynman theorem states that if the
Relation(4) is satisfiedsince the meanvalue is takenfor
Hamiltonian of a system is H(a) where a is a parame- thepositivequantityV(2) V(1). So,thetheoremapplies
ter, and that the eigenvalue equation for a bound state inthis case. Strictly speak−ing,the conditionV(1) V(2)
is ≤
mustnot be satisfiedeverywherein the position spaceof
the potentials. Indeed, some results can be obtained for
H(a)a =E(a)a , (2)
| i | i two-bodynonrelativisticproblemsinwhichthegraphsof
where E(a) is the energy and a the normalized associ- thecomparisonpotentialscrosseachotherinacontrolled
ated eigenstate, then | i way [7].
Evenifthevarietyofkineticoperatorsismuchsmaller
∂E(a) ∂H(a) than for potentials, it is worth comparing the spectra of
= a a . (3)
∂a ∂a two Hamiltonians H(1) =T(1)+V and H(2) =T(2)+V
(cid:28) (cid:12)(cid:12) (cid:12)(cid:12) (cid:29) such that T(1) T(2) for all values of the momentum
(cid:12) (cid:12) ≤
(cid:12) (cid:12) variables appearing in these kinetic operators. For simi-
larreasonsastheonespresentedabove,thetheoremalso
∗ E-mail: [email protected] applies in this case. It is not really surprising that the
2
comparisontheoremworksforbothpotentialandkinetic between brackets has one minimum in x = Q(c), with
n,l
operator, because one can indifferently consider a posi- 2
a value equal to 2 Q(c) Q(ho) Q(c) > 0. So, we
tion or an momentum representationfor Hamiltonians. n,l n,l − n,l
Wenowgiveseveralanalyticalillustrationsofthisthe- haveE(2) E(1) >(cid:16)0,ina(cid:17)cc(cid:16)ordancewith(cid:17)the comparison
orem for different two-body systems, using in particular theoremn,.l − n,l
the following well known cases:
Provided the potential V(r) does not depend on the
massm, thetheorempredicts thatthe eigenvaluesofthe
p2 2λ
H(ho) = +λr2, E(ho) = Q(ho), Hamiltonianp2/m+V(r)decreaseforanincreasingmass
2µ n,l s µ n,l m,whiletheeigenvaluesofthespinlessSalpeterHamilto-
(ho) nian 2 p2+m2+V(r) increase with the mass m. This
Q =2n+l+3/2; (8)
n,l already known result [8] can be immediately checked on
H(c) = p2 κ, E(c) = µκ2 , (8) andp(9) for the nonrelativistic case.
2µ − r n,l −2 Q(c) 2 Let us now consider the two kinetic operators T(1) =
n,l 2 p2+m2 and T(2) = 2m+p2/m which is the non-
Q(c) =n+l+1. (cid:16) (cid:17) (9) relativistic limit of T(1). We have T(1) T(2) since
n,l Tp(2) 2 T(1) 2 = p4/m2 0. We can c≤onclude that
Ifthe potentialpartofaHamiltonianpossessingadis- − ≥
the replacement of the semirelativistic kinetic part in a
cretespectrais ofthe formgv(r) wherev(r) is apositive (cid:0) (cid:1) (cid:0) (cid:1)
Hamiltonianbyitsnonrelativisticcounterpartimpliesan
function and g a coupling constant(positive ornegative,
increaseofthespectra[9]. Letusconsideranexplicitex-
according to the structure of v(r)), the theorem states
ample. TheeigenvaluesE(2) ofH(2) =2m+p2/m κ/r
that the eigenvalues increases with g. This is a well n,l −
knownresultwhichcanbe immediatelycheckedwith(8) are given by (9), while only upper bounds E˜(1) of the
n,l
and (9). exact eigenvalues E(1) of H(1) = 2 p2+m2 κ/r can
With T =p2/(2µ), let us now consider n,l −
be obtained (the value of κ is assumed to be low enough
p
κ to allow the existence of bound states) [10]:
V(1)(r)= , (10)
−r
κ 3κ
V(2)(r)= 2r03r2− 2r0, (11) E˜n(1,l) =2mv1− 4 Qκ(2c) 2, (14)
u n,l
where r0 is an arbitrary positive distance. These poten- u
tials are tangent at r , and the quantity t mκ(cid:16)2 (cid:17)
0 E(2) =2m . (15)
2 n,l − 4 Q(c) 2
κ r r n,l
V(2)(r) V(1)(r)= 1 +2 (12)
− 2r r − r (cid:16) (cid:17)
(cid:18) 0 (cid:19) (cid:18) 0 (cid:19)
2 2 4
isnon-negativeforallphysicalvaluesofr. Using(8)and Since En(2,l) − E˜n(1,l) =m2κ4/ 16 Q(nc,)l >0,we
(9),thedifferencebetweentwocorrespondingeigenvalues have E(cid:16)(2) >(cid:17)E˜(1)(cid:16) E((cid:17)1). (cid:18) (cid:16) (cid:17) (cid:19)
is given by n,l n,l ≥ n,l
2 2
√κ x3 3 Q(c) x+2Q(ho) Q(c)
− n,l n,l n,l
En(2,l)−En(1,l) = (cid:20) (cid:16)2r0√µ(cid:17)r0 Q(nc,)l 2 (cid:16) (cid:17) (cid:21), ACKNOWLEDGMENTS
(cid:16) (cid:17) (13) The author thanks the F.R.S.-FNRS for financialsup-
where x=√κµr0 is an arbitrary positive quantity. It is port. He alsothanksF.Buisseretforacarefulreadingof
easytocheckthat,forpositivevaluesofx,thepolynomial the manuscript.
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