Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

0.15 MB·English

by  M. Lorente

Continuous vs. discrete models for the
quantum harmonic oscillator and the
hydrogen atom
Miguel Lorente1
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0 Departamento de F´ısica, Universidad de Oviedo, 33007 Oviedo, Spain
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Abstract
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The Kravchuk and Meixner polynomials of discrete variable are introduced for the
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discrete models of the harmonic oscillator and hydrogen atom. Starting from Ro-
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7 drigues formula we construct raising and lowering operators, commutation and an-
8 ticommutation relations. The physical properties of discrete models are figured out
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through the equivalence with the continuous models obtained by limit process.
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4 PACS: 02.20.+b, 03.65.Bz, 03.65.Fd
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p Key words: orthogonal polynomials; difference equation; raising and lowering
- operators; quantum oscillator; hydrogen atom.
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1 Introduction
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The methodof finite difference isbecoming morepowerful in physics for differ-
ent reasons: difference equations are more suitable to computational physics
and numerical calculations; lattice gauge theories explore the physical prop-
erties of the discrete models before the limit is taken; some modern theories
have proposed physical models with discrete space and time [1].
In recent papers [2], [3] we have presented the mathematical properties of hy-
pergeometric functions of continuous and discrete variable. We have worked
out general formulas for the differential/difference equation, recurrence re-
lations, raising and lowering operators, commutation and anticommutation
relations. The starting point is the general properties of classical orthogonal
1 E-mail: [email protected]
Preprint submitted to Elsevier Science 1 February 2008
polynomials of continuous and discrete variable [4] of hypergeometric type, in
particular the Rodrigues formula from which the raising and lowering opera-
tors are derived. Similar results were obtained with more sofisticated method
using the factorization of the hamiltonian [6] [7]. In those papers the term
“oscillator” is used for the hamiltonian that has equally spaced eigenvalues.
Inthis paper we study thephysical properties oftwo simple examples, thehar-
monic oscillator and the hydrogen atom on the lattice. We make the Ansatz
that the discrete model has the same elements (hamiltonian, eigenvalues,
eigenvectors, expectation values, dispersion relations) as the continuous one,
provided the difference equation, the raising and lowering operator the com-
mutation and anticommutation relations become in the limit the equivalent
elements in the continuous case. To this scheme we can add the evolution
of the fields with discrete time, using some difference equation that replace
Heisenberg equation. [8]
Some applications of discrete models have been presented elsewhere. Bijker
et al. [9] have apply the SO(3) algebra of the Wigner functions to theone-
dimensional anharmonic (Morse) oscillator. Bank and Ismail have apply La-
guerre functions to the attractiv Coulomb potential [10]
2 The quantum harmonic oscillator of discrete variable
We start from the orthogonal polynomials of a discrete variable, the Kravchuk
polynomials K(p)(x) and the corresponding normalized Kravchuk functions [1]
n
K(p)(x) = d−1 ρ(x) k(p)(x), (1)
n n n
q
where d2 = N! (pq)n is a normalization constant, ρ(x) = N!pxqN−x(pq)n is
n n!(N−n)! x!(N−x)!
the weight function, with p > 0, q > 0, p+q = 1, x = 0,1,...N +1.
The Kravchuk functions satisfy the orthonormality condition
N
Kn(p)(x)Kn(p′)(x) = δnn′, (2)
x=0
X
and the following difference and recurrence equations [2]:
pq(N x)(x+1)K(p)(x+1)
− n
q
+ pq(N x+1)xK(p)(x 1)+[x(p q) Np+n]K(p)(x) = 0, (3)
− n − − − n
q
2
pq(N n)(x+1)K(p) (x)
− n+1
q + pq(N n+1)nK(p) (x)+[n(q p)+Np x]K(p)(x) = 0, (4)
− n−1 − − n
q
Fromthe properties of the Kravchuk polynomials we canconstruct raising and
lowering operators for the Kravchuk functions [2]
L+(x,n)K(p)(x) = pq(x+n N)K(p)(x)
n − n
+ pq(N x+1)xK(p)(x 1) = pq(N n)(n+1)K(p) (x), (5)
− n − − n+1
q q
L−(x,n)K(p)(x) = pq(x+n N)K(p)(x)
n − n
+ pq(N x)(x+1)K(p)(x+1) = pq(N n+1)nK(p) (x). (6)
− n − n−1
q q
The raising operator satisfies
(pq)n(N n)! n−1
K(p)(x) = − L+(x,n 1 k)K(p)(x),
n s N!n! − − 0
k=0
Y
where K(p)(x) = N!pxqN−x is the solution of the difference equation
0 x!(N−x)!
r
L−(x,0)K(p)(x) = 0.
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It canbe proved that the raising and lowering operatorsL+(x,n) andL−(x,n)
are mutually adjoint with respect to the scalar product (2).
Ifwedefinethedifferenceequation(3)astheoperatorequationH(x,n)K(p)(x) =
n
0 , we can factorize this equation as follows
L+(x,n 1)L−(x,n) = pq(N n+1)n+pq(x+n 1 N)H(x,n), (7)
− − − −
L−(x,n+1)L+(x,n) = pq(N n)(n+1)+pq(x+n+1 N)H(x,n). (8)
− −
Now we make connection between the Kravchuk function and the Wigner
functions that appear in the generalized spherical functions [4]
( 1)m−m′dj (β) = K(p)(x), (9)
− mm′ n
where j = N/2 , m = j n, m′ = j x, p = sin2(β/2), q = cos2(β/2).
− −
Thenformulas(3)to(8)canbewrittendownintermsoftheWignerfunctions,
namely,
3
1
sinβ (j +m′)(j m′ +1)dj (β)
2 − m,m′−1
q
1
+ sinβ (j m′)(j +m′ +1)dj (β)+(m m′cosβ)dj (β) = 0,
2 − m,m′+1 − m,m′
q
(3a)
1
sinβ (j +m)(j m+1)dj (β)
2 − m−1,m′
q
1
+ sinβ (j m)(j +m+1)dj (β) (m′ mcosβ)dj (β) = 0,
2 − m+1,m′ − − m,m′
q
(4a)
β
L+(m′,m) dj (β)=sin2 (m+m′) dj (β)
m,m′ 2 m,m′
1
+ sinβ (j m′)(j +m′ +1) dj (β)
2 − m,m′+1
q
1
= sinβ (j +m)(j m+1) dj (β),
2 − m−1,m′
q
(5a)
β
L−(m′,m) dj (β)=sin2 (m+m′) dj (β)
m,m′ 2 m,m′
1
+ sinβ (j +m′)(j m′ +1) dj (β)
2 − m,m′−1
q
1
= sinβ (j m)(j +m+1) dj (β).
2 − m+1,m′
q
(6a)
Notice that (3a) and (4a) are equivalent if we interchange m m′ and take
↔
in account the general property of Wigner functions
dj (β) = ( 1)m−m′dj (β). (10)
m,m′ − m′,m
The same property is satisfied between (5a) and (6a).
Expresions (5a) and (6a) can be obtained directly from the properties of
Wigner functions. In fact, it is known [4, formula 5.1.19] that
d m′ mcosβ
dj (β)+ − dj (β) = (j m)(j +m+1)dj (β), (11)
dβ m,m′ sinβ m,m′ − m+1,m′
q
d m′ mcosβ
dj (β)+ − dj (β) = (j +m)(j m+1)dj (β),
−dβ m,m′ sinβ m,m′ − m−1,m′
q
(12)
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The last equation (11), after interchanging m m′ and using (9), can be
↔
transformed into
d m m′cosβ
dj (β) − dj (β) = (j +m′)(j m′ +1)dj (β).
dβ m,m′ − sinβ m,m′ − m,m′−1
q
(12a)
Combining (11) and (12a) we obtain (6a) and by similar method we obtain
(5a) In order to give a physical interpretation of the difference equation and
raising and lowering operators for the Kravchuk functions we take the limit
when N goes to infinity and the discrete variable x becomes continuous s .
First of all, we take the limit of Kravchuk functions. We write
1 1 n
n!(N n)!(Npq)n /2 N!pxqN−x /2 2 /2
K(p)(x) = − n!k(p)(x)
n ( N!(pq)n 2n(n!)2) (x!(N −x)!)  Npq! n 
1 1/2 1 1/2  
= e−s2 H (s) = ψ (s), (13)
−n−→−∞→ (cid:26)2nn!(cid:27) (√2πNpq ) n n
where the last braket becomes the weight for the Hermite functions and the
functions ψ (s) are the solution of the continuous harmonic oscillator (up to
n
1
the constant (2Npq)− /4 .
In order to get the continuous limit of (4) we multiply it by 2 √2Npq and
substitute x = Np+√2Npq s ; after simplification we get .
n
1 2(n+1) K(p) (x)
s − N n+1
(cid:18) (cid:19)
n 1 (2p 1)n
+ 1 − 2n K(p) (x) 2 s+ − K(p)(x) = 0.
s(cid:18) − N (cid:19) n−1 − √2Npq ! n
In the limit N this equation becomes the familiar recurrence relation
→ ∞
for the normalized Hermite functions
2(n+1) ψn+1(s)+√2n ψn−1(s) = 2s ψn(s) (14)
q
Before we take the limit of (5) we redifine the raising operator, substracting
from it one half equation (3), namely,
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L+(x,n)K(p)(x) = [(x Np)+n(p q)]K(p)(x)
n 2 − − n
(cid:26)
pq(N x)(x+1)K(p)(x 1)+ pq(N x+1)xK(p)(x 1)
− − n − − n −
q q (cid:27)
= pq(N n)(n+1)K(p) (x) (15)
− n+1
q
We divide this expression by h√2Npq√Npq where h√2Npq = 1. After sim-
plification we get
1 1 n(p q)
L+(x,n)K(p)(x) = s+ − K(p)(x)
√Npq n √2" √2Npq # n

1 2p  2q 1
1 s 1+ s+ K (x+1)
−2h vu −sNq ! sNp Np! n
u
t

2p 1 2q
1 s+ 1+ s K (x 1)
−vu −sNq Nq! sNp ! n − 
u 
t 

n 
= 1 (n+1) K(p) (x)
s − N n+1
(cid:18) (cid:19)
1 1
= s [ψ (s+h) ψ (s h)]
n n
−N−→−∞→ √2 − 2h − −
(cid:26) (cid:27)
1 d
= s ψ (s) = √n+1 ψ (s), (16)
n n
−h−→→0 √2 ( − ds)
Similarly from (6) we get
1 1 d
L−(x,n) K(p)(x) s+ ψ (s) = √n ψ (s). (17)
√Npq n −N−→−→∞ √2 ( ds) n n
h→0
Therefore the raising and lowering operators for the Kravchuk functions be-
come, in the limit, creation and annihilation operators for the normalized
Hermite functions.
We still have an other connection between the raising and lowering operators
of Wigner functions with the generators of the SO(3) algebra.
From (5a) and (6a) we define
1 (j +m)(j m+1)
A+dj (β) L+dj (β) = − dj (β), (18)
mm′ ≡ √Npq mm′ s 2j m−1,m′
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1 (j m)(j +m+1)
A−dj (β) L−dj (β) = − dj (β). (19)
mm′ ≡ √Npq mm′ s 2j m+1,m′
Multiplying both expressions by the spherical harmonics Yjm′ , adding for m′
and using the property of Wigner functions
Yjm = djm,m′ Yjm′, (20)
m′
X
we obtain
(j +m)(j m+1) 1
A+Yjm = s 2j− Yj,m−1 = √2jJ−Yjm (21)
(j m)(j +m+1) 1
−
A Y = − Y = J Y (22)
jm s 2j j,m+1 √2j + jm
where J+,J− are the generators of SO(3) algebra.
For the commutation relations of these operators we have
m 1 n
(AA+ A+A)Y = Y = 2J Y = 1 Y , (23)
jm jm z jm jm
− j 2j − j!
we substitute (20) in this expression and then take the limit:
n
AA+ dj (β) = 1 dj (β) a,a+ ψ (s).
mm′ − j! mm′ −j−→−∞→ n
h i h i
For the anticommutation relation we have from (7) and (8):
1 1
AA+ +A+A Y = j(j +1) m2 Y = J~2 J2 Y . (24)
jm j − jm j − z jm
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Again substituting (20) and taking the limit
AA+ +A+A dj (β)
mm′
(cid:16) (cid:17) n2
= (2n+1) dj (β) aa+ +a+a ψ (s) = (2n+1)ψ (s)
( − j ) mm′ −j−→−∞→ n n
(cid:16) (cid:17)
This correspondence suggests that the operator algebra for the quantum har-
monic oscillator on the lattice is expanded by the generators of the SO(3)
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groups. The commutation relations [J+,J−] = 2Jz play the rol of the Heisen-
berg algebra, and the anticommutation relation multiply by ~ω/2 play the rol
of the Hamiltonian. In order to complete the picture we define the position
and momemtum operators on the lattice as follows:
1 1
~ /2 ~ /2 m′ mcosβ
X : i A+A+ dj (β) = i − dj (β)
2Mω! mm′ 2Mω! √jsinβ mm′
(cid:16) (cid:17)
1 1
M~ω /2 M~ω /2
P : A A+ dj (β) =
2 ! − mm′ 2 ! ×
(cid:16) (cid:17)
(j +m′)(j m′ +1) (j m′)(j +m′ +1)
− dj (β) − dj (β)
× s 2j m,m′−1 − s 2j m,m′+1 
 
with dispersion with respect to the state K(p)(x)
n
~ ~ n2
(∆X)2 = X2 = AA+ +A+A = 2n+1
n n 2Mω n 2Mω − j2!
D E D E
M~ω M~ω n2
(∆P)2 = P2 = AA+ +A+A = 2n+1
n n 2 n 2 − j2!
D E D E
from which the uncertainty relation follows:
~ n2
(∆X) (∆P) = 2n+1
n n 2 − j2!
The eigenvalues of the Hamilton operator on the lattice are connected with
the index m = j n of the eigenvectors dj (β). These eigenvalues are equally
− mm′
separated by ~ω but finite (m = j,...+j). The eigenvalues of the position
operator on the lattice are connec−ted with the index m′ = j x of dj (β).
− mm′
These eigenvalues are equally separated by ~ but finite (m′ = j,...+j).
Mω −
Therefore the Planck constant ~ plays a roleqwith respect to the discrete space
coordinate similar to the discrete energy eigenvalues.
3 Wave equation for the hidrogen atom with discrete variables
Our model is based on the properties of generalized Laguerre polynomials as
continuous limit of the Meixner polynomials of discrete variable.
We start from the generalized Laguerre functions
ψα(s) = d−1 ρ (s) Lα(s) (25)
n n 1 n
q
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with α > 1, d2 = Γ(n+α+1)/n!, ρ (s) = sα+1e−s, that satisfy the or-
− n 1
thonormality condition
∞
ψnα(s) ψnα′(s) s−1ds = ρnn′ (26)
Z0
from the differencial equation and Rodrigues formula for the Laguerre poly-
nomials [4] we deduce the following properties for the Laguerre functions (25)
i) differential equation
λ 1 α2 1 1
ψα′′(s)+ − ψα(s) = 0, λ = n+ (α+1). (27)
n "s − 4 − s2 # n 2
ii) Recurrence relations
(n+α+1)(n+1)ψα (s)
− n+1
q
(n+α)nψα (s)+(2n+α+1 s) ψα(s) = 0. (28)
− n−1 − n
q
iii) Raising operator
1 d
L+(s,n) ψα(s)= (2n+α+1 s) ψα(s) s ψα(s)
n −2 − n − ds n
= (n+1)(n+α+1) ψα (s) (29)
− n+1
q
iv) Lowering operator
1 d
L−(s,n) ψα(s)= (2n+α+1 s) ψα(s)+s ψα(s) =
n −2 − n ds n
= n(n+α) ψα (s) (30)
− n−1
q
from (29)and (30) we get the factorization of (27)
d2 λ 1 α2 1
L−(s,n+1)L+(s,n) = (n+1)(n+α+1) s2 + − (31)
− (ds2 s − 4 − s2 )
d2 λ 1 α2 1
L+(s,n 1) L−(s,n) = n(n+α) s2 + − (32)
− − (ds2 s − 4 − s2 )
Notice that (27) corresponds to some self adjoint operator of Sturm-Lioville
type. Also (29) and (30) are mutually adjoint operators with respect to the
scalar product (26).
Similarly, in the discrete case, we defined the normalized Meixner functions
M(γ,µ)(x) d−1 ρ (x) m(γ,µ)(x) (33)
n ≡ n 1 n
q
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where m(γ,µ)(x) are the Meixner polynomials,
n
n!Γ(n+γ) µxΓ(x+γ +1)
d2 = , ρ (x) = ,
n µn(1 µ)γΓ(γ) 1 Γ(x+1)Γ(γ)
−
and γ,µ are some constants 0 < µ < 1, γ > 0. The functions (33) satisfy
the orthonormality condition
∞
1
Mn(γ,µ)(x) Mn(γ′,µ)(x)µ(x+γ) = ρnn′ (34)
x=0
X
and the following properties:
i) Difference equation
µ(x+γ)(x+1)(x+γ)
M (x+1)
n
s x+γ +1
+ µ(x+γ)xM (x 1) [µ(x+γ)+x n(1 µ)]M (x) = 0(35)
n n
− − − −
q
ii) Recurrence relation
µ(n+γ)(n+1)Mn+1(x) µ(n+γ 1)nMn−1(x)
− − −
q +q(µx+µn+µγ +n x)M (x) = 0 (36)
n
−
iii) Raising operator
L+(x,n) M (x)= µ(x+γ +n) M (x)+ µ(x+γ)x M (x 1)
n n n
− −
q
= µ(n+γ)(n+1) M (x) (37)
n+1
q
iii) Lowering operator
−
L (x,n) M (x)= µ(x+γ +n) M (x)
n n
−
µ(x+γ)(x+1)(x+γ)
+ M (x+1)
n
s x+γ +1
= µ(n+γ 1)n Mn−1(x) (38)
− −
q
A redefinition of (37) and (38) can be obtained substracting one half the
difference equation (35) from both:
10