Generalized Version of the Creation and Annihilation Operators for the Schr\"odinger Equation

0.12 MB·

by  L. C. N. Santos

Generalized Version of the Creation and Annihilation
Operators for the Schro¨dinger Equation
Santos, L. C. Na,∗, C.C. Barros Jra,∗
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1 aDepartamento de F´ısica - CFM - Universidade Federal de Santa Catarina, Florian´opolis -
0 SC - CP. 476 - CEP 88.040 - 900 - Brazil.
2

n
a
J
0 Abstract
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A generalized version of the creation and annihilation operators is constructed

]
h and the factorization of the Schr¨odinger equation is investigated. It is shown
p
- that the generalized version of factorization operators yield a factorization for
h
t the twelve different separable coordinates for the Schr¨odinger equation.
a
m
Keywords: , factorization method, Schr¨odinger equation, operators
[

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v 1. Introduction
1
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5 The factorization method, introduced by Schr¨odinger [1] and Dirac [2] and
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laterdevelopedbyInfeldandHull[3],isoneofthemethodsforsolvingquantum
0
.
1 mechanical problems. The idea is to consider a pair of first-order differential
0
7 5 equations which are equivalent to a given second-order differential equation.
1
Thecompletesetofnormalizedeigenfunctionscanbeobtainedbythesuccessive
:
v
i application of the ladder operators on the eigenfunctions, which are the exact
X
solutions of the first order differential equation.
r
a
Fromthe1970stotheearly1980,itwasacommonopinionthatthe method
10 was completely explored. However, Mielnik made an additional contribution
to the traditional factorization method in 1984 [4]. In that work, he did not
consider the particular, but the general solution to the Riccati type equation
∗Correspondingauthor
Email addresses: [email protected] (Santos, L.C.N),[email protected]
(C.C.BarrosJr)
Preprint submitted toArxiv January 31, 2017
connected with the Infeld-Hull approach. Mielnik factorization is a powerful
tool in the derivation of new Hamiltonians whose corresponding eigenproblem
15 is analytically solvable. On the other hand, the connection between the Infeld
methodandsupersymmetricquantummechanics(SUSYQM)hasbeenexplored
by many authors [5, 6, 7, 8]. For example, Witten noticed the possibility of
arranging the second-order differential equations into isospectral pairs, the so-
called supersymmetric partners.
20 In many works, the factorization method has been used as a tool to formu-
late algebraic approaches to many non-relativistic quantum problems, the idea
is build sets of one variable radial operators which are realizations for su(1,1)
Lie algebra[9,10]. The separable coordinatesystems for the Schr¨odingerequa-
tion are confocal quadric surfaces [11], and the potential is a function of the
25 coordinates [12].
In this paper, the possibility of factorization of the separated equations is
investigated,anditisshownthatageneralizedversionofthecreationandanni-
hilationoperatorscanbeconstructed. Indeed,theseoperatorsyieldafactoriza-
tionfor the twelve differentseparablecoordinatesfor the Schr¨odingerequation.
Inthe Infeldmethod, the originalformofthe second-orderdifferentialequation
d dψ
p +q(θ)ψ+λρ(θ)ψ =0, (1)
dθ dθ
(cid:18) (cid:19)
is transformed in the form
d2y
+r(x,m)y+λy =0, (2)
dx2
wherem=1,2,3...andp,ρarepositivefunctions. Thetransformationconnect-
ing these equations is
y =(pρ)1/2ψ, dx=(ρ/p)1/2dθ. (3)
In this work, we propose to apply a factorization method for the original form
of the separated Schr¨odinger equation. In this way, our approach yield fac-
torization operators for the twelve different separable coordinate systems. An
2
interesting feature of this work is that the original Hilbert space of theory is
30 sustained.
So, this paper will show the following contents: In section II, we enumerate
thecoordinatessystemswhichwillallowseparationoftheSchr¨odingerequation.
InsectionIII, ageneralizedversionofthecreationandannihilationoperatorsis
proposed. InsectionIVandV, we apply the method to the radialsecond-order
35 Schr¨odinger equation.
2. Separable coordinate systems for the wave equation
Inthissection,westudytheseparablecoordinatesystemsfortheSchr¨odinger
equation, for more details and information see [13]. We consider the standard
differential equation for the scalar field
2ψ+k2ψ =0, (4)
∇ 1
where ▽2 is the Laplaceoperator. When k is a function of the coordinates,we
1
obtain the Schr¨odinger equation. The rectangular coordinates x, y, z and the
curvilinear coordinates ξ , ξ , ξ are related by the scale factors
1 2 3
2 2 2
∂x ∂y ∂z
h = + + , (5)
n
s(cid:18)∂ξn(cid:19) (cid:18)∂ξn(cid:19) (cid:18)∂ξn(cid:19)
where n=1,2,3. The Laplacian can be expressed in its generalized form
1 ∂ h h h ∂ψ
2ψ = 1 2 3 , (6)
∇ h h h ∂ξ h2 ∂ξ
n 1 2 3 n (cid:20) n n(cid:21)
X
thus we can rewrite the equation (4) into
1 ∂ h h h ∂ψ
1 2 3 +k2ψ =0. (7)
h h h ∂ξ h2 ∂ξ 1
n 1 2 3 n (cid:20) n n(cid:21)
X
In order to obtain the separated equations, we introduce the St¨ackel deter-
3
minant
Φ Φ Φ
11 12 13
S =|Φmn|=(cid:12)(cid:12)(cid:12) Φ21 Φ22 Φ23 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) Φ31 Φ32 Φ33 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
=Φ Φ Φ(cid:12) +Φ Φ Φ +Φ(cid:12) Φ Φ
11 22 (cid:12)33 12 23 31 (cid:12) 13 21 32
Φ Φ Φ Φ Φ Φ Φ Φ Φ , (8)
13 22 31 11 23 32 12 21 33
− − −
where Φ are functions of ξ alone.
mn n
If the separated equations for the three-dimensional case are
1 d dX
f (ξ ) m + Φ (ξ )k2X =0, (9)
f (ξ )dξ m m dξ mn m n m
m m m (cid:20) m (cid:21) n
X
we can relate the equations (7) and (9) by the Robertson condition [13]
h h h
1 2 3
=f (ξ )f (ξ )f (ξ ),
1 1 2 1 3 1
S
which limits the kinds of coordinates systems which will allow separation. The
equation(9) is supposedto be a separatedequation,thereforethe functions f ,
n
40 Φn1, Φn2, and Φn3 must all be functions of ξn alone.
3. Creation and annihilation operators
In this section, we propose a generalized versionof the creation and annihi-
lation operators. For the Schr¨odinger equation the constant k must have the
1
3
formk2 =ε v (ξ )[13],wherev (ξ )= 2mV (ξ ),substituting itintothe
1 − n n n n ~2 n n
n
equation (9) gPives
1 d dX
m
f (ξ ) +
m m
f (ξ )dξ dξ
m m m (cid:20) m (cid:21)
3
Φ (ξ ) ε v (ξ ) X +
m1 m n n m
" − #
n
X
Φ (ξ )k2X +Φ (ξ )k2X =0. (10)
m2 m 2 m m3 m 3 m
4
Introducing k′ =ε, k =k′, k =k′ into (10) we obtain
1 2 2 2 2
1 d dX
m
f (ξ )
m m
f (ξ )dξ dξ
m m m (cid:20) m (cid:21)
3
+ k′2Φ (ξ )X v X
m nm m n− n n
m6=n
X
=k′2Φ X . (11)
n nn n
These are the separated equations for the three-dimensional space. Now we
proposetoapplyafactorizationmethodfortheseparatedSchr¨odingerequation
(11). The idea is to define two ladder operators
d 1 df
m
A = + R(ξ ), (12)
m
dξ 2f dξ −
m m m
d 1 df
A+ = + m +R(ξ ), (13)
m
dξ 2f dξ
m m m
and to show that these operators yield a factorization of the equation (11).
Indeed, if we multiply Aby A+, we find
2
1 d dX 1 df
AA+ = f m m
f dξ m dξ − 4f2 dξ
m m (cid:18) m (cid:19) m (cid:18) m(cid:19)
1 d2f
+ m +R′ R2, (14)
2f dξ2 −
m m
and comparing with equation (11) we find a Riccati type equation
R′ R2 =ǫ+Γ(ξ ), (15)
n
−
where ǫ is a constant and
1 df 2 1 d2f 3
Γ(ξ )= m m + k′2Φ v . (16)
n 4f2 dξ − 2f dξ2 m nm− n
m (cid:18) m(cid:19) m m m6=n
X
The occurrence of the Ricatti equation in the factorization of second-order di-
fferentialequationsisatypicalphenomenon. Specifically,the factorizationope-
rators convert the equation (11) into product of A and A+ with a extra con-
45 dition, a Riccati type equation. The explicit solution of this type of equation,
in ge-neral, is not Known [14]. In the following, we will obtain two particular
solutions of the Riccati equation (15) in a spherical coordinate system for the
Coulomb and isotropic oscillator potentials.
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4. Application to hydrogen atom
WewanttoshowhowtosolvetheradialSchr¨odingerequationwithCoulomb
potential. Weapplythemethodtotheradialsecond-orderdifferentialequation,
inthis case,the sphericalcoordinatesaredenotedbyξ =r, ξ =θ, ξ =φand
1 2 3
the f functions are f =r2, f =1 cos2θ, f = 1 cos2φ . Therefore the
n 1 2 3
− −
s matrix is given by p
1 1 0
r2
S = 0 1 1 . (17)
cos2θ−1 (cos2θ−1)2
 0 0 cos21φ−1 
 
The separation constants are then required to be of the form
k2 =ε, k2 = l(l+1), k3 =m.
1 2 − 3
So the radial Schr¨odinger equation with potential v = K/r is
n
−
d2X 2dX l(l+1) K
1 1
+ + = εX , (18)
dr2 r dr − r2 r − 1
and the Ricatti equation is
l(l+1) K
R′ R2 =ǫ + , (19)
− − r2 r
the particular solution for this equation is given by
l K
R= ,
r − 2l
where ǫ = K2. The creation and annihilation operators in (12) and (13) can
−4l2
be written in the form
d 1 l K
A+ = + + ,
l dr r r − 2l
d 1 l K
A = + + .
l
dr r − r 2l
The commutator of A and A+ is r dependent
l l
2l
A+,A =A+A A A+ = . (20)
l l l l− l l r2
(cid:2) (cid:3)
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Indeed, the A and A+ are creation and annihilation operators. We can prove
l l
this directly, the first step is to consider the product between the operators
A+A and the radial wave function X , i.e,
l l n,l−1
K2
A+A X = H X
l l n,l−1 l−1− 4l2 n,l−1
(cid:18) (cid:19)
K2
= ε X , (21)
n,l−1− 4l2 n,l−1
(cid:18) (cid:19)
in a similar way
K2
A A+X = H X
l l n,l l− 4l2 n,l
(cid:18) (cid:19)
K2
= ε X . (22)
n,l− 4l2 n,l
(cid:18) (cid:19)
A direct calculation shows that these operators satisfy
2l
[H ,A ]= A (23)
l l r2 l
and
2l
H ,A+ = A+, (24)
l−1 l −r2 l
so the action of the A and H(cid:2) on the s(cid:3)tates X is
l l n.l
2l
H AX =A +H X
l l n.l l r2 l n.l
(cid:18) (cid:19)
=ε A X , (25)
n,l−1 l n.l
this result imply
X A X
n.l−1 l n.l
∝
or
X =cA X , (26)
n.l−1 l n.l
where c is a constant. Equivalently
X =cA+X . (27)
n.l l n.l−1
These results imply that the action of the operators A and A+ on the states
l l
X and X is to change the quantum number l. In order to determinate
n.l n.l−1
c, we apply A to the left-hand side of eq. (27)
l
A X =cA A+X (28)
l n.l l l n.l−1
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Therefore, using equation (26) we find
1
c= . (29)
ε K2
n,l− 4l2
q
(cid:0) (cid:1)
5. Radial harmonic oscillator
50
Solutions of equation (11) are limited to a small set of potentials and the
radialharmonicoscillatorisoneofthefewquantumsystemswhereanexactand
analyticalsolutionisknown. Thetime-independentSchr¨odingerradialequation
for the isotropic oscillator v =kr2 reads
n
d2 2 d l(l+1)
+ kr2 X = εX , (30)
dr2 rdr − r2 − 1 − 1
(cid:20) (cid:21)
for this case, the Riccati equation is
l(l+1)
R′ R2 =ǫ kr2 (31)
− − r2 −
the particular solution of equation (31) is given by
l
R= √kr, ǫ=√k(2l 1). (32)
r − −
This solution R leads to factorizing operators
d 1 l
A = + +√kr, (33)
l
dr r − r
and
d 1 l
A+ = + + √kr. (34)
l dr r r −
Hence, we get the following commutation rules
2l
A+,A = +2√k
l l r
(cid:2) (cid:3)
and
[H ,A ]=A A+,A
l l l l l
=A (cid:2) 2l +2(cid:3)√k , (35)
l
r
(cid:18) (cid:19)
8
H ,A+ =A+ A,A+
l−1 l l l l
(cid:2) (cid:3)= A+(cid:2) 2l +(cid:3)2√k , (36)
− l r
(cid:18) (cid:19)
thus, the action of the A and H on the states X is
l l n.l
H A X =ε A X . (37)
l l n,l n,l−1 l n,l
Equivalently, the action of the A+ and H on the states X leads to
l l−1 n.l−1
H A+X =ε A+X , (38)
l−1 l n.l−1 n,l l n,l−1
thus, A+ andA are the raisingandloweringoperatorsfor the isotropicoscilla-
l l
tor. It follows that
X =cA X , X =cA+X , (39)
nl−1 l nl nl l nl−1
where
1
c= .
ε +√k(2l 1)
nl
−
q
6. Summary and conclusions
In this paper, we have determined the separable coordinate systems for the
Schr¨odingerequation. Fromtheseresults,factorizationoperators forthetwelve
different separable coordinates have been provided. We have determined the
55 Schr¨odinger equation in the presence of a Coulomb potential and a radial har-
monic oscillator potential. We have shown that, a new set of generalized cre-
ation and annihilation operators has been introduced. By using the apparatus
developed in this work, we believe that other potentials in different coordinate
systems can be solved. We conclude by mentioning that links between super-
60 symmetricquantummechanicsandnon-linearordinarydifferentialequationsas
the Riccati equation (15) can be established [15].
9
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