Quantum Mechanics as a Classical Theory X: Quantization in Generalized Coordinates

10 Pages·0.13 MB·English

by  L. S. F. Olavo

Quantum Mechanics as a Classical Theory X:
Quantization in Generalized Coordinates
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9 L.S.F.Olavo
1
Departamento de Fisica, Universidade de Brasilia,

n
70910-900, Brasilia-D.F., Brazil
a
J
February 1, 2008
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1
v Abstract
2
0 In this tenth paper of the series we aim at showing that our formal-
0 ism, using the Wigner-Moyal Infinitesimal Transformation together with
1 classical mechanics, endows us with the ways to quantize a system in
0 any coordinate representation we wish. This result is necessary if one
6
even think about making general relativistic extensions of the quantum
9
formalism. Besides, physics shall not be dependent on the specific rep-
/
h resentation we use and this result is necessary to make quantum theory
p consistent and complete.
-
t
n
a 1 Overview
u
q
The present paper is the tenth part of a series of papers on the mathematical
:
v and epistemological foundations of quantum theory. The papers of this series
Xi may fit into one (or more) of four rather different categories of interest.
They may be considered as reconstruction papers, where the existing for-
r
a malism is derivedwithin the (classical) approachdefined by the use of Wigner-
Moyal’s Infinitesimal Transformation. Into such division we might put papers
I,II,III[1, 2, 3] and also papers VI,VII and VIII[6, 7, 8].
There are also papers where we resize the underlying epistemology to fit
the (purely classical) mathematical developments of the theory. In these cases,
allegedpurely quantumeffects arereinterpretedonclassicalgrounds—fromthe
epistemological perspective. Into this category we might put papers III,VI,VII
and IX[3, 6, 7, 9].
Pertaining to another class of papers we have those trying to expand the
applicabilityofquantumtheorytootherfields ofinvestigation. Itisremarkable
that such a task was taken with paper II[2], where a quantum theory (of one
particle) that takes into account the effects of gravity was developed, and with
1
papers IV and V[4, 5], where this generalized relativistic quantum theory was
applied to show that it predicts particles with negative masses.
The last category is the one where we try to investigate the boundaries of
quantumtheory—therewhereitseemstogiveunsatisfactoryanswers,bothfrom
themathematicalandepistemologicalpointsofview. WemightcitepaperIXas
one example where the importantquestionofoperatorformationwas discussed
atlength. Papersbelongingtothiscategorytrytoremovefromthetheorysome
of its formalproblems, as was the case with paper IX, or misinterpretations,as
wasthe casewith paper III where the problemofnon-localitywas investigated.
The present paper pertains to this last class of interest and is particularly
relatedwithformalproblems. Wehavebeentaught,sinceourveryintroduction
to the study ofquantummechanics,that,forquantizinga system,weshallfirst
write its classical hamiltonian in cartesian coordinates. This quantization may
be mathematically represented by
p→−i¯h∇ , x→x, (1)
where p is the momentum and x the coordinate, and by the transformation
H(p,x)→H(−i¯h∇,x). (2)
It is only after this quantization has been performed that we may change to
another system of coordinates, distinct form the cartesian one[10].
Thisseemstobeaterribleproblem,althoughnotseemedassuchbymanyof
us, sincephysics is supposedto treatany(mathematical)systemofcoordinates
on the same grounds. If not for this reason, one may wonder about the future
ofgeneralrelativisticextensionsofatheorythatneedsthe flatcartesiansystem
of coordinates to exist. Within a relativistic theory this need seems to be a
scandal that denies from the very beginning any such extension.
One mayfind in the literature [12,13, 14] sometrials to overcomethese dif-
ficulties, but eventhese approachesarepermeatedwith additionalsuppositions
as in ref. [12, 13] where the author has to postulate that the total quantum-
mechanicalmomentumoperatorp correspondingtothegeneralizedcoordinate
qi
q is given by
i
∂
p =−i¯h (3)
qi ∂q
i
and has also to write the classical hamiltonian (the kinetic energy term) as
1
H = p∗ gikp (4)
2m qi qk
i,k
X
whatever be the complex conjugate of the classical momentum function.
Theseapproachesseemtoberatherunsatisfactoryforwewouldliketoderive
ourresultsusing onlyfirstprinciples,without havingto addmorepostulates to
the theory.
2
This problem appears because quantum mechanics, as developed in text-
books, is not a theory with a clearly discernible set of axioms[11]. Indeed, the
rules (1) and (2) above are part of the fuzzy set of axioms one could append to
it.
We have developed a completed axiomatic (classical) version of quantum
mechanics which, we expect, does not depend on the specific set of coordinates
used.
The aimofthis paper is to showthat ourexpectations areconfirmedby the
mathematical formalism.
We will show in the second section how to quantize a hamiltonian with a
central potential in spherical coordinates using only the three axioms we have
already postulated, now written in this coordinate system. This will serve as
an illustrative example of how quantization in generalized coordinates shall be
done.
Thethirdsectionwillaimatgeneralizingthepreviousparticularapproachto
any set of orthogonalgeneralized coordinates; that is, to show how to quantize
in such coordinates.
2 Spherical Coordinates: an example
Webeginbyrewritingourthreeaxioms[1]intheappropriatecoordinatesystem:
Axiom 1: Newtonian particle mechanics is valid for all particles which consti-
tute the systems composing the ensemble;
Axiom 2: For an ensemble of isolated systems the joint probability density
function is a constant of motion:
dF(r,θ,φ,p ,p ,p ;t)
r θ φ
=0; (5)
dt
Axiom 3: The Wigner-MoyalInfinitesimal Transformation,defined as
δr δr i
ρ(r− ,r+ ;t)= F(r,θ,φ,p ,p ,p ;t)exp( p·δr)d3p (6)
r θ φ
2 2 ¯h
Z
may be applied to representthe dynamics of the system in terms of func-
tions ρ(r,δr;t).
Using expression (5) (Liouville’s equation), we may write
∂F
+{H,F}=0, (7)
∂t
where H is the hamiltonian and {,} is the classical Poissonbracket. By means
of the hamiltonian, written in spherical coordinates
1 p2 p2
H = p2+ θ + φ +V(r), (8)
2m r r2 r2sin2(θ)
" #
3
we find the Liouville equation
∂F p ∂F p ∂F p ∂F
r θ φ
+ + + −
∂t m ∂r mr2 ∂θ mr2sin2(θ) ∂φ
∂V p2 p2 ∂F p2 ∂F
− − θ − φ + φ cot(θ) =0. (9)
∂r mr3 mr3sin2(θ) ∂p mr2sin2(θ) ∂p
" # r θ
As a means of writing the Wigner-Moyal Transformation in spherical coor-
dinates, we note that
δr=δrr+rδθθ+rsin(θ)δφφ (10)
where (r,θ,φ) are the unit normbals, andb b
p p
b b b p=prr+ θθ+ φ φ, (11)
r rsin(θ)
giving b b b
p·δr=δr·p +δθ·p +δφ·p . (12)
r θ φ
Usingtherelationbetweentheunitnormalsincartesian(i,j,k)andspherical
(r,θ,φ) coordinates
b b b r=isin(θ)cos(φ)+jsin(θ)sin(φ)+kcos(θ)
θ =icos(θ)cos(φ)+jcos(θ)sin(φ)−ksin(θ) , (13)

 bφ=−isin(φ)+jcos(φ)
b
we find the followibng relation between the momenta:
p =p sin(θ)cos(φ)+(p /r)cos(θ)cos(φ)−(p /r)(sin(φ)/sin(θ))
x r θ φ
p =p sin(θ)sin(φ)+(p /r)cos(θ)sin(φ)+(p /r)(cos(φ)/sin(θ)) .
y r θ φ

p =p cos(θ)−(p /r)sin(θ)
 z r θ
(14)
TheJacobian relating the two infinitesimal volume elements
dp dp dp =kJk dp dp dp (15)
x y z p r θ φ
is given by
1
kJk = . (16)
p r2sin(θ)
It is now possible to rewrite expression (6) as
δr δr dp dp dp
ρ(r− ,r+ ;t)= F(r,p;t)eh¯i(δr·pr+δθ·pθ+δφ·pφ) r θ φ. (17)
2 2 r2sin(θ)
Z
4
With equation (9) and expression (17) at hands we may find the equation
satisfiedbythedensityfunctionρ(r,δr;t)inexactlythesamewayaswasprevi-
ously done for cartesian coordinates[1]—it is noteworthy that now we have the
jacobian (16) that will change slightly the appearance of this equation terms.
After some straightforwardcalculations we arrive at
¯h2 1 ∂ ∂ρ 1 ∂ ∂ρ 1 ∂2ρ
− r2 + sin(θ) + +
m r2∂r ∂(δr) r2sin(θ)∂θ ∂(δθ) r2sin2(θ)∂φ∂(δφ)
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
¯h2 δr ∂2ρ δr ∂2ρ δθcot(θ) ∂2ρ ∂V ∂ρ
+ + + +δr ρ=i¯h .
m r3 ∂(δθ)2 r3sin2(θ)∂(δφ)2 r2sin2(θ)∂(δφ)2 ∂r ∂t
(cid:20) (cid:21)
(18)
To go from this equation to the equation for the amplitudes we may write
δx δx δx δx
ρ(x− ,x+ ;t)=ψ†(x− ;t)ψ(x+ ;t) (19)
2 2 2 2
and also write
ψ(x;t)=R(x;t)exp(iS(x;t)/¯h) (20)
in cartesian coordinates, for example. We then expand expression (19) around
the infinitesimals quantities to get, until second order,
R 3 ∂2R 1 3 ∂R ∂R
ρ(x,δx;t)= R2+ δx δx − δx δx ·
i j i j
 4 ∂x ∂x 4 ∂x ∂x 
i j i j
 iX,j=1 iX,j=1 
 i ∂S ∂S ∂S 
·exp δx +δy +δz , (21)
¯h ∂x ∂y ∂z
(cid:20) (cid:18) (cid:19)(cid:21)
where x ,i=1,2,3 implies x,y,z, and use the relations
i
∂ =sin(θ)cos(φ)∂ +(1/r)cos(θ)cos(φ)∂ −(1/r)(sin(φ)/sin(θ))∂
x r θ φ
∂ =sin(θ)sin(φ)∂ +(1/r)cos(θ)sin(φ)∂ +(1/r)(cos(φ)/sin(θ))∂ ,
y r θ φ

∂ =cos(θ)∂ −(1/r)sin(θ)∂
 z r θ
(22)
where ∂ is an abbreviation of ∂/∂u, to write the density function in spherical
u
coordinates as
R ∂2R ∂2R ∂R
ρ(r,δr;t)= R2+ δr2 +δθ2 +r +
4 ∂r2 ∂θ2 ∂r
(cid:26) (cid:20) (cid:18) (cid:19)
∂2R ∂R ∂R
+δφ2 +rsin2(θ) +cos(θ)sin(θ) +
∂φ2 ∂r ∂θ
(cid:18) (cid:19)
∂2R 1∂R ∂2R 1∂R
2δrδθ − +2δrδφ − −
∂r∂θ r ∂θ ∂r∂φ r ∂φ
(cid:18) (cid:19) (cid:18) (cid:19)
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∂2R ∂R 1 3 ∂R ∂R
+2δθδφ −cot(θ) − δx δx ·
i j
∂θ∂φ ∂φ 4 ∂x ∂x 
(cid:18) (cid:19)(cid:21) iX,j=1 i j
i ∂S ∂S ∂S

·exp δr +δθ +δφ , (23)
¯h ∂r ∂θ ∂φ
(cid:20) (cid:18) (cid:19)(cid:21)
where now x ,i=1,2,3 means r,θ,φ.
i
The next step is to take expression (23) into equation (18) and to collect
the zeroth and first order terms in the infinitesimals1 to get, as usual[1], the
equations (written in spherical coordinates)
∂ ∂S 1 ¯h2
δr· + (∇S)2+V(r)− ∇2R =0 (24)
∂r ∂t 2m 2mR
(cid:20) (cid:21)
and
∂R2 R2
+∇· ∇S =0. (25)
∂t m
(cid:18) (cid:19)
Equation (25) is the continuity equation while equation (24) is the same as
writing
∂ ∂ψ 1
δr· Hψ−i¯h =0, (26)
∂r ∂t ψ
(cid:20)(cid:18) (cid:19) (cid:21)
where the ψ is as given in expressbion (20) and the hamiltonian operator H is
given by
−¯h2 1 ∂ ∂ 1 ∂ ∂ 1 ∂2
H = r2 + sin(θ) + +V(r).
2m r2∂r ∂r r2sin(θ)∂θ ∂θ r2sin2(θ)∂φ2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(27)
b
In this sense, we may say, since the infinitesimals are all independent, that
we have derived the Schr¨odinger equation written as
−¯h2 1 ∂ ∂ 1 ∂ ∂ 1 ∂2
r2 + sin(θ) + ψ+
2m r2∂r ∂r r2sin(θ)∂θ ∂θ r2sin2(θ)∂φ2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
∂ψ
+V(r)ψ =i¯h , (28)
∂t
wheretheconstantcomingfromexpression(26)mightbeappendedintheright
hand term above and reflects a mere definition of a new reference energy level.
We thus have quantized the system using only spherical coordinates from
the very beginning as was our interest to show.
In the next section we will generalize this result to any set of orthogonal
coordinate systems.
1thiscumbersomeexercisewasperformedusingalgebraiccomputation.
6
3 Orthogonal Coordinates
In this case we have the transformation rules
x =x (u ,u ,u ) ; α=1,2,3, (29)
α α 1 2 3
wherex arethe coordinateswritteninsomesystem(notnecessarilythe carte-
α
sian one), u are the coordinates in the new system and the differential line
i
element is given by[15]
dr=h du e +h du e +h du e , (30)
1 1 1 2 2 2 3 3 3
wherethe e arethe unitnormalsinthe new (u ,u ,u )-coordinatesystemand
i 1 2 3
∂r
h e = . (31)
i i ∂ui
The momenta (p ,p ,p ) canonically conjugate to the u-coordinate system
1 2 3
are given by
du du du
p=m h 1e +h 2e +h 3e =
1 1 2 2 3 3
dt dt dt
(cid:18) (cid:19)
p p p
= 1e + 2e + 3e , (32)
1 2 3
h h h
(cid:18) 1 2 3 (cid:19)
or
du
p =mh2 i, (33)
i i dt
such that
p·δu=p δu +p δu +p δu . (34)
1 1 2 2 3 3
The Hamiltonian may be written as[16]
1 p2 p2 p2
H = 1 + 2 + 3 +V(u) (35)
2m h2 h2 h2
(cid:20) 1 2 3(cid:21)
and the Liouville equation becomes
∂F 3 p2 ∂F 3 3 p2 ∂h ∂V ∂F
+ i + i i − =0. (36)
∂t mh2∂u mh3∂u ∂u ∂p
i=1 i i j=1"i=1(cid:18) i j(cid:19) j# j
X X X
Sincethecoordinatetransformation(29)isaspecialtypeofcanonicaltrans-
formation, we shall have the Jacobian of the momentum transformation given
by
1
kJk = , (37)
p h h h
1 2 3
for the Jacobian of the coordinate transformation is
kJk =h h h (38)
u 1 2 3
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and the infinitesimal phase space volume element is a canonical invariant[17].
With the density function given by
δu δu i dp dp dp
ρ u− ,u+ ;t = F(u,p;t)exp p·δu 1 2 3, (39)
2 2 ¯h h h h
(cid:18) (cid:19) Z (cid:18) (cid:19) 1 2 3
it is straightforwardto find the differential equation it satisfies as
∂ρ ¯h2 1 ∂ h h ∂ ∂ h h ∂
2 3 1 3
−i¯h − + +
∂t m h h h ∂u h ∂(δu ) ∂u h ∂(δu )
(cid:26) 1 2 3 (cid:20) 1 (cid:18) 1 1 (cid:19) 2 (cid:18) 2 2 (cid:19)
∂ h h ∂ 3 δu ∂h ∂2ρ 3 ∂V
1 2 i j
+ − + δu ρ=0. (40)
∂u h ∂(δu ) h3 ∂u ∂(δu )2 i∂u
3 (cid:18) 3 3 (cid:19)(cid:21) iX,j=1 j i i  Xi=1 i
The reader may easily verify that, with the coordinate transformationgiven by

x=rsinθcosφ h =1
1
y =rsinθsinφ ; h =r , (41)
2
 
z =rcosθ h =rsinθ
  3
we recover the result of equation (18). 
We now write the density function as
δu δu δu δu
ρ(u− ,u+ ;t)=ψ†(u− ;t)ψ(u+ ;t), (42)
2 2 2 2
with the, generally complex, amplitudes written as
ψ(x;t)=R(u;t)exp(iS(u;t)/¯h) (43)
and expand these amplitudes until second order in the infinitesimal parameter
δu, to find (until second order)
R 3 ∂2R 3 ∂R
ρ(u,δu;t)= R2+ δu δu − Γk −
 4  i j ∂ui∂uj ij∂uk!
 iX,j=1 Xk=1
 
3 3
1 ∂R ∂R i ∂S
− δu δu ·exp δu , (44)
i j i
4 ∂ui∂uj ¯h ∂ui!
iX,j=1  Xi=1
 
where Γ is the Christoffel Symbol[18]. The reader may verify that expression

(44) gives the correct (23) result when expressed in spherical coordinates.
We then insertthis expressioninto equation(40) and collect the zeroth and
first order terms on the infinitesimals2 to get the equations (written in general
orthogonalcoordinates)
∂ ∂S 1 ¯h2
δu· + (∇S)2+V(u)− ∇2R =0 (45)
∂u ∂t 2m 2mR
(cid:20) (cid:21)
2Again,algebraiccomputation wasusedthroughout.
8
and
∂R2 R2
+∇· ∇S =0. (46)
∂t m
(cid:18) (cid:19)
Equation (46) is the continuity equation while equation (45) is the same as
writing
∂ ∂ψ 1
δu· Hψ−i¯h =0, (47)
∂u ∂t ψ
(cid:20)(cid:18) (cid:19) (cid:21)
where the ψ is as shown in expresbsion (43) and the hamiltonian operator H is
given by
b
−¯h2 1 ∂ h h ∂ ∂ h h ∂
2 3 2 3
H = +
2m h h h ∂u h ∂u ∂u h ∂u
1 2 3 (cid:20) 1 (cid:18) 1 1(cid:19) 2 (cid:18) 1 2(cid:19)
b
∂ h h ∂
+ 1 2 +V(u). (48)
∂u h ∂u
3 (cid:18) 3 3(cid:19)(cid:21)
Because the infinitesimals are all independent, the term inside brackets in
equation(47)mustvanishidentically. Thisallowsustosaythatwehavederived
the Schr¨odinger equation written as
−¯h2 1 ∂ h h ∂ ∂ h h ∂
2 3 2 3
+
2m h h h ∂u h ∂u ∂u h ∂u
1 2 3 (cid:20) 1 (cid:18) 1 1(cid:19) 2 (cid:18) 1 2(cid:19)
∂ h h ∂ ∂ψ
+ 1 2 ψ+V(u)ψ =i¯h , (49)
∂u h ∂u ∂t
3 (cid:18) 3 3(cid:19)(cid:21)
wheretheconstantcomingfromexpression(47)mightbeappendedintheright
hand term above and reflects a mere new definition of the reference energy
level—or a new definition of the function S by adding this constant factor.
Wethushavequantizedthesystemusingonlygeneralorthogonalcoordinates
from the very beginning as was our aim to show. The extension of this result
for non-orthogonal coordinate systems is straightforward and will not be done
here.
4 Conclusions
We have shown that the process of quantization is coordinate system indepen-
dent. This result does not bring anything new to the machinery of quantum
theory, since its formal apparatus for application on computational problems
remains untouched.
However,when showing this coordinate system independence, we also show
thatthe formalismisinconformitywithourexpectationsthatphysics shall not
depend on the way we choose to represent it.
9
Although this seems to be sterile from the point of view of calculations,
it gives the formalism coherence and wideness. Coherence for the reasons ex-
plained above and wideness for now it is possible to justify any trial to find a
general relativistic extension of this formalism.
Theseresultsmayalsobe seenasanotherconfirmationofourguessesabout
the classical nature of quantum theory.
References
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[3] Olavo,L.S.F, quant-ph/9503022
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[5] Olavo,L.S.F, quant-ph/9503025
[6] Olavo,L.S.F, quant-ph/9509012
[7] Olavo,L.S.F, quant-ph/9509013
[8] Olavo,L.S.F, quant-ph/9511028
[9] Olavo,L.S.F, quant-ph/9511039
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