Hermite Function

25 Pages·2012·0.35 MB·English

Chapter 13
Hermite and Laguerre
Polynomials
In this chapter we study two sets of orthogonal polynomials, Hermite and
Laguerre polynomials. These sets are less common in mathematical physics
than the Legendre and Bessel functions of Chapters 11 and 12, but Hermite
polynomialsoccurinsolutionsofthesimpleharmonicoscillatorofquantum
mechanicsandLaguerrepolynomialsinwavefunctionsofthehydrogenatom.
Because the general mathematical techniques are similar to those of the
precedingtwochapters,thedevelopmentofthesefunctionsisonlyoutlined.
Some detailed proofs, along the lines of Chapters 11 and 12, are left to the
reader.WestartwithHermitepolynomials.
13.1 HermitePolynomials
QuantumMechanicalSimpleHarmonicOscillator
For the physicist, Hermite polynomials are synonymous with the one-
dimensional (i.e., simple) harmonic oscillator of quantum mechanics. For a
potentialenergy
1 1
V = Kz2 = mω2z2, force F =−∂V/∂z=−Kz,
z
2 2
theSchro¨dingerequationofthequantummechanicalsystemis
h¯2 d2 1
− (cid:5)(z)+ Kz2(cid:5)(z)= E(cid:5)(z). (13.1)
2mdz2 2
Our oscillating particle has mass mand total energy E. From quantum me-
chanics,werecallthatforboundstatestheboundaryconditions
lim (cid:5)(z)=0 (13.2)
z→±∞
638
13.1 HermitePolynomials 639
restricttheenergyeigenvalue E toadiscreteset E = λ h¯ω, whereω isthe
n n
angular frequency of the corresponding classical oscillator. It is introduced
by rescaling the coordinate z in favor of the dimensionless variable x and
transformingtheparametersasfollows:
mK m2ω2
x=αz with α4 ≡ ≡ ,
h¯2 h¯2
(cid:2) (cid:3) (13.3)
2E m 1/2 2E
2λ ≡ n = n.
n h¯ K h¯ω
Eq.(13.1) becomes [with (cid:5)(z) = (cid:5)(x/α) = ψ(x)] the ordinary differential
equation(ODE)
d2ψ (x)
n +(2λ −x2)ψ (x)=0. (13.4)
dx2 n n
Ifwesubstituteψ (x)=e−x2/2H (x)intoEq.(13.4),weobtaintheODE
n n
H(cid:9)(cid:9)−2xH(cid:9) +(2λ −1)H =0 (13.5)
n n n n
ofExercise8.5.6.ApowerseriessolutionofEq.(13.5)showsthat H (x)will
n
behaveasex2 forlargex,unlessλ =n+1/2, n=0,1,2,....Thus,ψ (x)and
n n
(cid:5) (z)willblowupatinfinity,anditwillbeimpossibleforthewavefunction
n
(cid:5)(z)tosatisfytheboundaryconditions[Eq.(13.2)]unless
(cid:2) (cid:3)
1
E =λ h¯ω= n+ h¯ω, n=0,1,2... (13.6)
n n
2
Thisisthekeypropertyoftheharmonicoscillatorspectrum.Weseethatthe
energyisquantizedandthatthereisaminimumorzeropointenergy
1
E = E = h¯ω.
min 0
2
This zero point energy is an aspect of the uncertainty principle, a genuine
quantumphenomenon.Also,with2λ −1=2n, Eq.(13.5)becomesHermite’s
n
ODEand H (x)aretheHermitepolynomials.Thesolutionsψ (Fig.13.1)of
n n
Eq.(13.4)areproportionaltotheHermitepolynomials1H (x).
n
Thisisthedifferentialequationsapproach,astandardquantummechani-
caltreatment.However,weshallprovethesestatementsnextemployingthe
methodofladderoperators.
RaisingandLoweringOperators
Thefollowingdevelopmentisanalogoustotheuseoftheraisingandlowering
operatorsforangularmomentumoperatorspresentedinSection4.3.Thekey
aspectofEq.(13.4)isthatitsHamiltonian
(cid:2) (cid:3)(cid:2) (cid:3) (cid:4) (cid:5)
d2 d d d
−2H≡ −x2 = −x +x + x, (13.7)
dx2 dx dx dx
1Notetheabsenceofasuperscript,whichdistinguishesHermitepolynomialsfromtheunrelated
HankelfunctionsinChapter12.
640 Chapter13 HermiteandLaguerrePolynomials
Figure13.1
Quantum
Mechanical
0.5
OscillatorWave y(x)
2
Functions.The
HeavyBaronthe
x
x-AxisIndicatesthe 5
AllowedRangeof
theClassical
Oscillatorwiththe
SameTotalEnergy
0.5
y (x)
1
5
almostfactorizes.Usingnaivelya2−b2 =(a−b)(a+b),thebasiccommutator
[p ,x]=h¯/iofquantummechanics[withmomentum p =(h¯/i)d/dx]enters
x x
asacorrectioninEq.(13.7).[Because p isHermitian,d/dxisanti-Hermitian,
x
(d/dx)† = −d/dx.] This commutator can be evaluated as follows. Imagine
thedifferentialoperatord/dxactsonawavefunctionψ(x)totheright,asin
Eq.(13.4),sothat
d d
(xψ)= x ψ +ψ (13.8)
dx dx
bytheproductrule.DroppingthewavefunctionψfromEq.(13.8),werewrite
Eq.(13.8)as
(cid:4) (cid:5)
d d d
x−x ≡ ,x =1, (13.9)
dx dx dx
aconstant,andthenverifyEq.(13.7)directlybyexpandingtheproduct.The
productformofEq.(13.7),uptotheconstantcommutator,suggestsintroduc-
ingthenon-Hermitianoperators
(cid:2) (cid:3) (cid:2) (cid:3)
1 d 1 d
aˆ† ≡ √ x− , aˆ ≡ √ x+ , (13.10)
2 dx 2 dx
with(aˆ)† =aˆ†.Theyobeythecommutationrelations
(cid:4) (cid:5)
d
[aˆ,aˆ†]= ,x =1, [aˆ,aˆ]=0=[aˆ†,aˆ†], (13.11)
dx
13.1 HermitePolynomials 641
which are characteristic of these operators and straightforward to derive
fromEq.(13.9)and
[d/dx,d/dx]=0=[x,x] and [x,d/dx]=−[d/dx,x].
ReturningtoEq.(13.7)andusingEq.(13.10),werewritetheHamiltonianas
1 1 1
H=aˆ†aˆ + =aˆ†aˆ + [aˆ,aˆ†]= (aˆ†aˆ +aˆaˆ†) (13.12)
2 2 2
andintroducetheHermitiannumberoperatorN =aˆ†aˆ sothatH= N+1/2.
Wealsousethesimplernotationψ =|n(cid:12)sothatEq.(13.4)becomes
n
H|n(cid:12)=λ |n(cid:12).
n
Nowweprovethekeypropertythat N hasnonnegativeintegereigenvalues
(cid:2) (cid:3)
1
N|n(cid:12)= λ − |n(cid:12)=n|n(cid:12), n=0,1,2,..., (13.13)
n
2
thatis,λ =n+1/2.From
n
(cid:2) (cid:3)
1
|aˆ|n(cid:12)|2 =(cid:13)n|aˆ†aˆ|n(cid:12)= λ − ≥0, (13.14)
n
2
weseethat N hasnonnegativeeigenvalues.
Wenowshowthatthecommutationrelations
[N,aˆ†]=aˆ†, [N,aˆ]=−aˆ, (13.15)
which follow from Eq. (13.11), characterize N as the number operator that
counts the oscillator quanta in the eigenstate |n(cid:12). To this end, we determine
theeigenvalueof N forthestatesaˆ†|n(cid:12)andaˆ|n(cid:12).Usingaˆaˆ† = N +1, wesee
that
(cid:2) (cid:3)
1
N(aˆ†|n(cid:12))=aˆ†(N+1)|n(cid:12)= λ + aˆ†|n(cid:12)=(n+1)aˆ†|n(cid:12), (13.16)
n
2
(cid:2) (cid:3)
1
N(aˆ|n(cid:12))=(aˆaˆ†−1)aˆ|n(cid:12)=aˆ(N−1)|n(cid:12)= λ − aˆ|n(cid:12)
n
2
=(n−1)aˆ|n(cid:12).
In other words, N acting on aˆ†|n(cid:12) shows that aˆ† has raised the eigenvalue n
of |n(cid:12) by one unit; hence its name raising or creation operator. Applying
aˆ† repeatedly,wecanreachallhigherexcitations.Thereisnoupperlimitto
thesequenceofeigenvalues.Similarly,aˆ lowerstheeigenvaluenbyoneunit;
hence,itisaloweringorannihilationoperator.Therefore,
aˆ†|n(cid:12)∼|n+1(cid:12), aˆ|n(cid:12)∼|n−1(cid:12). (13.17)
Applyingaˆ repeatedly,wecanreachthelowestorgroundstate|0(cid:12)witheigen-
value λ . We cannot step lower because λ ≥ 1/2. Therefore, aˆ|0(cid:12) ≡ 0,
0 0
642 Chapter13 HermiteandLaguerrePolynomials
suggestingweconstructψ =|0(cid:12)fromthe(factored)first-orderODE
0
(cid:2) (cid:3)
√
d
2aˆψ = +x ψ =0. (13.18)
0 0
dx
Integrating
ψ(cid:9)
0 =−x, (13.19)
ψ
0
weobtain
1
lnψ =− x2+lnc , (13.20)
0 0
2
wherec isanintegrationconstant.Thesolution
0
ψ (x)=c e−x2/2 (13.21)
0 0
can be normalized, with c = π−1/4 using the error integral. Substituting ψ
0 0
intoEq.(13.4)wefind
(cid:2) (cid:3)
1 1
H|0(cid:12)= aˆ†aˆ + |0(cid:12)= |0(cid:12) (13.22)
2 2
sothatitsenergyeigenvalueisλ = 1/2anditsnumbereigenvalueisn= 0,
0
confirmingthenotation|0(cid:12).Applyingaˆ†repeatedlytoψ =|0(cid:12),allothereigen-
0
valuesareconfirmedtobeλ = n+1/2, provingEq.(13.13).Thenormaliza-
n
tionsinEq.(13.17)followfromEqs.(13.14),(13.16),and
|aˆ†|n(cid:12)|2 =(cid:13)n|aˆaˆ†|n(cid:12)=(cid:13)n|aˆ†aˆ +1|n(cid:12)=n+1, (13.23)
showing
√ √
n+1|n+1(cid:12)=aˆ†|n(cid:12), n|n−1(cid:12)=aˆ|n(cid:12). (13.24)
Thus,theexcited-statewavefunctions,ψ ,ψ ,andsoon,aregeneratedbythe
1 2
raisingoperator
(cid:2) (cid:3) √
1 d x 2
|1(cid:12)=aˆ†|0(cid:12)= √ x− ψ (x)= e−x2/2, (13.25)
2 dx 0 π1/4
yielding[andleadingtoEq.(13.5)]
ψ (x)= N H (x)e−x2/2, N =π−1/4(2nn!)−1/2, (13.26)
n n n n
where H aretheHermitepolynomials(Fig.13.2).
n
BiographicalData
Hermite,Charles. Hermite,aFrenchmathematician,wasborninDieuze
in1822anddiedinParisin1902.Hismostfamousresultisthefirstproofthat
eisatranscendentalnumber;thatis,eisnottherootofanypolynomialwith
integercoefficients.Healsocontributedtoellipticandmodularfunctions.
Havingbeenrecognizedslowly,hebecameaprofessorattheSorbonnein
1870.
13.1 HermitePolynomials 643
Figure13.2
HermitePolynomials
10
H(x)
8 2
6
4 H1(x)
2
H(x)
0
0 x
1 2
–2
RecurrenceRelationsandGeneratingFunction
Nowwecanestablishtherecurrencerelations
2xHn(x)−Hn(cid:9)(x)= Hn+1(x), Hn(cid:9)(x)=2nHn−1(x), (13.27)
fromwhich
Hn+1(x)=2xHn(x)−2nHn−1(x) (13.28)
followsbyaddingthem.ToproveEq.(13.27),weapply
d d
x− =−ex2/2 e−x2/2 (13.29)
dx dx
toψ (x)ofEq.(13.26)andrecallEq.(13.24)tofind
n
N
Nn+1Hn+1e−x2/2 =−√2(nn+1)e−x2/2(−2xHn+Hn(cid:9)), (13.30)
thatis,thefirstpartofEq.(13.27).Usingx+d/dxinstead,wegetthesecond
halfofEq.(13.27).
EXAMPLE13.1.1 TheFirstFewHermitePolynomials WeexpectthefirstHermitepolyno-
mialtobeaconstant,H (x)=1,beingnormalized.Thenn=0intherecursion
0
relation[Eq.(13.28)]yieldsH =2xH =2x;n=1impliesH =2xH −2H =
1 0 2 1 0
4x2−2, andn=2implies
H (x)=2xH (x)−4H (x)=2x(4x2−2)−8x=8x3−12x.
3 2 1
Comparing with Eq. (13.27) for n = 0,1,2, we verify that our results are
consistent:2xH − H(cid:9) = H = 2x·1−0, etc.Forconvenientreference,the
0 0 1
firstseveralHermitepolynomialsarelistedinTable13.1. (cid:2)
644 Chapter13 HermiteandLaguerrePolynomials
Table13.1
H0(x)=1
HermitePolynomials H1(x)=2x
H2(x)=4x2−2
H3(x)=8x3−12x
H4(x)=16x4−48x2+12
H5(x)=32x5−160x3+120x
H6(x)=64x6−480x4+720x2−120
TheHermitepolynomials H (x)maybesummedtoyieldthegenerating
n
function
(cid:6)∞ tn
g(x,t)=e−t2+2tx = H (x) , (13.31)
n
n!
n=0
whichwederivenextfromtherecursionrelation[Eq.(13.28)]:
(cid:6)∞ tn ∂g
0≡ n!(Hn+1−2xHn+2nHn−1)= ∂t −2xg+2tg. (13.32)
n=1
IntegratingthisODEintbyseparatingthevariablestandg,weget
1∂g
=2(x−t), (13.33)
g ∂t
whichyields
lng =2xt−t2+lnc, g(x,t)=e−t2+2xtc(x), (13.34)
wherecisanintegrationconstantthatmaydependontheparameterx.Direct
expansionoftheexponentialinEq.(13.34)gives H (x) = 1and H (x) = 2x
0 1
alongwithc(x)≡1.
EXAMPLE13.1.2 Special Values SpecialvaluesoftheHermitepolynomialsfollowfromthe
generatingfunctionforx=0;thatis,
(cid:6)∞ tn (cid:6)∞ t2n
g(x=0,t)=e−t2 = H (0) = (−1)n .
n
n! n!
n=0 n=0
Acomparisonofcoefficientsofthesepowerseriesyields
(2n)!
H2n(0)=(−1)n , H2n+1(0)=0. (13.35)
n!
(cid:2)
EXAMPLE13.1.3 Parity Similarly,weobtainfromthegeneratingfunctionidentity
g(−x,t)=e−t2−2tx =g(x,−t) (13.36)
thepowerseriesidentity
(cid:6)∞ tn (cid:6)∞ (−t)n
H (−x) = H (x) ,
n n
n! n!
n=0 n=0
13.1 HermitePolynomials 645
whichimpliestheimportantparityrelation
H (x)=(−1)nH (−x). (13.37)
n n
(cid:2)
Inquantummechanicalproblems,particularlyinmolecularspectrosopy,a
numberofintegralsoftheform
(cid:7)
∞
xre−x2H (x)H (x)dx
n m
−∞
areneeded.Examplesforr =1andr =2(withn=m)areincludedintheex-
ercisesattheendofthissection.ManyotherexamplesarecontainedinWilson
etal.2 Theoscillatorpotentialhasalsobeenemployedextensivelyincalcula-
tionsofnuclearstructure(nuclearshellmodel)andquarkmodelsofhadrons.
ThereisasecondindependentsolutiontoEq.(13.4).ThisHermitefunction
isaninfiniteseries(Sections8.5and8.6)andofnophysicalinterestyet.
AlternateRepresentations
Differentiationofthegeneratingfunction3 ntimeswithrespecttot andthen
settingtequaltozeroyields
dn
H (x)=(−1)nex2 e−x2. (13.38)
n dxn
This gives us a Rodrigues representation of H (x). A second representation
n
maybeobtainedbyusingthecalculusofresidues(Chapter7).Ifwemultiply
Eq.(13.31)byt−m−1andintegratearoundtheorigin,onlythetermwithH (x)
m
willsurvive:
(cid:8)
m!
H (x)= t−m−1e−t2+2txdt. (13.39)
m 2πi
Also,fromEq.(13.31)wemaywriteourHermitepolynomial H (x)inseries
n
form:
2n! 4n!
H (x)=(2x)n− (2x)n−2+ (2x)n−41·3···
n (n−2)!2! (n−4)!4!
(cid:2) (cid:3)
[(cid:6)n/2]
n
= (−2)s(2x)n−2s 1·3·5···(2s−1)
2s
s=0
[(cid:6)n/2] n!
= (−1)s(2x)n−2s . (13.40)
(n−2s)!s!
s=0
ThisseriesterminatesforintegralnandyieldsourHermitepolynomial.
2Wilson, E. B., Jr., Decius, J. C., and Cross, P. C. (1955). Molecular Vibrations. McGraw-Hill,
NewYork.Reprinted,Dover,NewYork(1980).
3Rewritethegeneratingfunctionasg(x,t)=ex2e−(t−x)2.Notethat
∂ ∂
e−(t−x)2 =− e−(t−x)2.
∂t ∂x
646 Chapter13 HermiteandLaguerrePolynomials
EXAMPLE13.1.4 The Lowest Hermite Polynomials For n = 0, we find from the series,
Eq.(13.40), H (x) = (−1)0(2x)0 0! = 1;forn= 1, H (x) = (−1)0(2x)1 1! =
0 0!0! 1 1!0!
2x;andforn=2,thes=0ands=1termsinEq.(13.40)give
2! 2!
H (x)=(−1)0(2x)2 −(2x)0 =4x2−2,
2
2!0! 0!1!
etc. (cid:2)
Orthogonality
The recurrence relations [Eqs. (13.27) and (13.28)] lead to the second-order
ODE
H(cid:9)(cid:9)(x)−2xH(cid:9)(x)+2nH (x)=0, (13.41)
n n n
whichisclearlynotself-adjoint.
ToputEq.(13.41)inself-adjointform,wemultiplybyexp(−x2)(Exercise
9.1.2).Thisleadstotheorthogonalityintegral
(cid:7)
∞
H (x)H (x)e−x2dx=0, m=(cid:18) n, (13.42)
m n
−∞
withtheweightingfunctionexp(−x2)aconsequenceofputtingthedifferen-
tial equation into self-adjoint form. The interval (−∞,∞) is dictated by the
boundaryconditionsoftheharmonicoscillator,whichareconsistentwiththe
Hermitianoperatorboundaryconditions(Section9.1).Itissometimesconve-
nienttoabsorbtheweightingfunctionintotheHermitepolynomials.Wemay
define
ϕ (x)=e−x2/2H (x), (13.43)
n n
withϕ (x)nolongerapolynomial.
n
Substituting H = ex2/2ϕ intoEq.(13.41)yieldsthequantummechanical
n n
harmonicoscillatorODE[Eq.(13.4)]forϕ (x):
n
ϕ(cid:9)(cid:9)(x)+(2n+1−x2)ϕ (x)=0, (13.44)
n n
whichisself-adjoint.Thus,itssolutionsϕ (x)areorthogonalfortheinterval
n
(−∞ < x < ∞)withaunitweightingfunction.Theproblemofnormalizing
thesefunctionsremains.ProceedingasinSection11.3,wemultiplyEq.(13.31)
byitselfandthenbye−x2.Thisyields
(cid:6)∞ smtn
e−x2e−s2+2sxe−t2+2tx = e−x2H (x)H (x) . (13.45)
m n
m!n!
m,n=0
13.1 HermitePolynomials 647
Whenweintegrateoverxfrom−∞to+∞thecrosstermsofthedoublesum
dropoutbecauseoftheorthogonalityproperty4
(cid:7) (cid:7)
(cid:6)∞ (st)n ∞ ∞
e−x2[H (x)]2dx= e−x2−s2+2sx−t2+2txdx
n
n!n!
n=0 −∞ −∞
(cid:7)
∞ (cid:6)∞ 2n(st)n
= e−(x−s−t)2e2st dx=π1/2e2st =π1/2 . (13.46)
n!
−∞ n=0
Byequatingcoefficientsoflikepowersofst,weobtain
(cid:7)
∞
e−x2[H (x)]2dx=2nπ1/2n! (13.47)
n
−∞
whichyieldsthenormalization N inEq.(13.26).
n
Hermitepolynomialsaresolutionsofthesimpleharmonicoscillatorofquan-
SUMMARY
tummechanics.TheirpropertiesdirectlyfollowfromwritingtheirODEasa
productofcreationandannihilationoperatorsandtheSturm–Liouvilletheory
oftheirODE.
EXERCISES
13.1.1 In developing the properties of the Hermite polynomials, start at a
numberofdifferentpoints,suchas
(1) Hermite’sODE,Eqs.(13.5)and(13.44);
(2) Rodrigues’sformula,Eq.(13.38);
(3) Integralrepresentation,Eq.(13.39);
(4) Generatingfunction,Eq.(13.31);and
(5) Gram–Schmidtconstructionofacompletesetoforthogonalpoly-
nomials over (−∞,∞) with a weighting factor of exp(−x2),
Section9.3.
Outline how you can go from any one of these starting points to all
theotherpoints.
13.1.2 Fromthegeneratingfunction,showthat
[(cid:6)n/2] n!
H (x)= (−1)s (2x)n−2s.
n (n−2s)!s!
s=0
13.1.3 Fromthegeneratingfunction,derivetherecurrencerelations
Hn+1(x)=2xHn(x)−2nHn−1(x),
Hn(cid:9)(x)=2nHn−1(x).
4Thecrossterms(m=(cid:18) n)maybeleftin,ifdesired.Then,whenthecoefficientsofsαtβareequated,
theorthogonalitywillbeapparent.