PHYSICAL RE VIE% A UOLUME 23, NUMBER 4 APRIL 1981

Quantum mechanics of a constrained particle

R. C. T. da Costa
Paulo, Srasil
(Received 26 August 1980)
I

The motion of a particle rigidly bounded to a surface is discussed, cansidering the Schrodinger equation of a free
particle canstrained to move, by the action of an external potential, in an infinitely thin sheet of the ordinary three-
dimensional space. Contrary to what seems to be the general belief expressed in the literature, this limiting process
gives a perfectly well-defined result, provided that we take some simple precautions in the definition of the potentials
and wave functions. It can then be shown that the wave function splits into two parts: the normal part, which .

contains the infinite energies required by the uncertainty principle, and a tangent part which contains "surface
potentials" depending both on the Gaussian and mean curvatures. An immediate consequence of these results is the
existence of different quantum mechanical properties for two isometric surfaces, as can be seen from the bound state
which appears along the edge of a folded (but nat stretched) plane. The fact that this surface potential is not a
bending invariant (cannot be expressed as a function of the components of the metric tensor and their derivatives) is
also interesting from the more general point of view of the quantum mechanics in curved spaces, since it can never
be obtained from the classical Lagrangian of an a priori constrained particle without substantial modifications in the
usual quantization procedures. Similar calculations are also presented for the case af a particle bounded to a curve.
The properties of the constraining spatial potential, necessary to a meaningful limiting process, are discussed in some
detail, and, as expected, the resulting Schrodinger equation contains a "linear potential" which is a function of the
curvature.

I. INTRODUCTION transversal spreading of the wave packet, which

will be present even in the case of a particle mov-
The motion of a particle in a one- or two-dimen- ing along a perfectly flat surface. On the 'other
sional domain of our Cartesian three-dimensional hand, if we choose the second approach we can
space is a well-known problem in classical me- forget for the moment all properties related to
chanics. It is usually treated in two different the external space, but will still have to work out
ways. In the first, or Newtonian approach, the an adequate quantization procedure for the a priori
particle is first thought of as moving freely (that curved motion. ' The aim of this paper is to see
is, unconstrained) in the three-dimensional space, how far we can go following the first of these two
but subjected to spatial forces which maintain, at approaches. As mentioned above (and discussed
all instants of time, its velocity oriented along a in more detail below) we believe this idea to be
preselected range of directions (the tangent plane unjustly neglected. The main point involved here
of a surface or tangent line of a curve). In the is the choice of the spatial forces which simulate
second, or I agrangian approach, the constraint the mechanical constraint in a certain suitable
is introduced from the beginning through the well- limit. To better develop our reasoning let us con-
known generalized coordinates, and the calcula- sider a surface constraint: As is well known, in
tions proceed without any necessary mention to classical mechanics the constraint forces can only
the space in which our surface (or curve) is sup- be uniquely determined if we assume them to be
posed to be embedded. For the purely spatial nondissipative (or frictionless); that is, they have
constraints considered here these two treatments the direction of the normal in all points of the
yield the same equations of motion, the choice surface. Well, since in quantum mechanics we
between one of them being, in general, a matter can no more predict the position of the particle
of convenience. with pointlike accuracy it is perfectly natural to
In quantum mechanics, however, the situation consider only constraint forces which are orthog-
is much less clear. If we select the first approach onal to our surface in all points of the space where
to begin with we have the advantage of a ready- the particle can possibly be found (a similar pro-
made Schrodinger equation but will have to deal cedure will be later developed for curves). This
with an unfamiliar situation in which the con- idea can be readily put in practice considering a
straint can only be thought of as a kind of limiting potential which is constant over the surface but
process. In fact, due to the uncertainty relations increases sharply for every small displacement
we must have (besides the quantum analog of the in the normal direction, in such a way as to pro-
classical forces to bend the momentum of the vide a normal "reaction" in a thin neighborhood
particle) infinite squeezing forces, " to contain the of the surface in question. (Weaker requirements

1982 1981 The American Physical Society

 QU'ANTUM MECHANICS OF A CONSTRAINED I'ARTICLE 198$

can possibly be found, but the one presented here call g, (&r/&q, )(&r/sq&), s, =1, 2, the covariant
& j
is perfectly adequate for the ends we have in components of the metric tensor of our surface S,
mind. ) The constraint will then be considered as g=det(g, ,) and k, , =h, the coefficients of the
the limit of an infinitely strong attractive potential second fundamental form. Since the derivatives
which maintains the particle permanently attached of the normal N(qi, qs) lie in the tangent plane we
to a pre-established surface. In order to have have
the limit independent of the type of attractive
potential we must have some kind of separation of
the Schrodinger equation in which the surface part (2)
of the wave function obeys a special equation
which does not contain the transverse variable with
appearing in the constraining potential. This is
1 1
in fact what happens, as we shall now proceed to +11, (g12I221 g22@11)s 12 (Is 11g21 I22 igii)
show.
1 1
+21 (I22?g12 @1?g22)t +22 (@21g12 @2?gii)
II. PARTICLE BOUNDED TO A SURFACE
I et us consider a particle of mass m perma- (3)
nently attached to the surface S of parametric
equations r = r(qqs), where r is the vector posi-
(Weingarten equations). From (1) and (2) we ob-
tain
tion of an arbitrary surface point P. The portion
of the space in an immediate neighborhood of S 2
8R ar
can be parametrized as (Fig. 1) (~+~, ,q, )
q)
A(qqs, qs)= r(qq, ) +qsN(qqs), (1)
BR
where N(qqs) is the value taken at P by a con- =N(qi qs) .
tinuous unit normal to S. The absolute value of
the coordinate qs gives, for points where (1) is In our three-dimensional neighborhood of S the
nonsingular, the distance between the surface S covariant components of the metric tensor are
and the point Q of coordinates (qi, qs, qs). Accord- given by
ing to the ideas presented in the Introduction we I

shall now consider the spatial potential V= Vs(qs), G~ G~ 8R 9R

j z~ j=1&~s 3 ~

where X is a "squeezing parameter" which mea- eq aqua

sures the strength of the potential: Using (4) and denoting the transposed matrix by
the superscript T, we have
limV(qs) =
oo
q g0 is=giy+ ~~g+( g) ~siqs+(~g )&?qs

G3 G3 =0, i=1, 2; G33 1

.
(If a specific example is required to guide our
intuition, we can imagine the harmonic binding We can now turn our attention to the Schrodinger
'
Vs(qs) =-, mX qs, with X eventually going to infinity, equation. Writing the Laplacian in the curvilinear
which gives (qs) ~ )2/mX). coordinates (qqs, qs) we obtain
Before going to the Schrodinger equation it is
worthwhile to briefly review the mathematical
properties of the coordinate system (1). Let us

where G=det(G, J). Due to the structure of the

+ V.(qs)t=sl

G, &'s given in (5) we can break up the Laplacian
into two parts: the surface part, denoted by
Z(q 1, qs, q 2), given by the terms i, = 1, 2, and the j
normal part, defined by i= We can then j=3.
write

2m
a)(qqqs)g 2m 2
+ (In~G
eq3 Bq3 eq3
FIG. 1. Curvilinear coordinate system based on the
surface S of parametric equation r =r(q~, q2).
+ V(qs)y=sa . (7)
 1984 R. C. T. DA COSTA

Since we are hoping for the existence of a surface where dS= Wgdq, dqz (the area element of the sur-
wave function, depending only on the variables q& face) and
and q2, we are naturally led to the introduction of
a new wave function y from which, in the event of f(q(, qz, qz) =1+ Tr(o.')q, +det(&)q3 .
a separation X(qqz, qz) = X,(q(, qz)X(q, ) we will be
able to define the surface density probability Expression (8) now gives the desired result:
Ix (q~qz) I'J Ix.(qz) I'dqz T"e ade(luate transf-
mation (t -x
can be readily inferred from the vol-
X(q ~(qzu qz) lf(q(&qzs q3)] (t((q(r qzo qz) (io)
ume dV expressed in terms of the curvilinear
coordinates qqz, q, . Really, using (4) we have
Introducing this substitution into (7) we are left
dV=f(q((qz(qz)dSdqz ~ with

k' (X I' &'X+

2+ 4fz
1" sf &f
2f & z
.
+V(qz)X zh
+ sX
SI~~ 2 s Lo
&
Old
X

We are now ready to take into account the effect Using (3) this term can be written in a more
of the potential V(,(q&). Since in the limit when useful form
X-~ the wave function "sees" two steep potential
barriers on both sides of the surface, its value v, (qq, ) =- 2m
(M'-tf}=-
will be significantly different from zero only for
a very small range of values of q3 around q3 0.
In this case we can safely take q3-0 in all coeffi- where k& and k2 are the principal curvatures of
cients of E(I. (11) [except of course in the term the surface S, and
containing V(qz}]. The result from (5) and (9) is
M = ,
'(k( + kz)
52
=
2g
(g((kzz+gzzk(( 2g(2k(z) (mean curvature),

k2
'Tr
, n'-det &X
E = k(kz = 1 det(k &)
((Gaussian curvature) . (17)

2m Bq3
By", +V,
(q, )x= k~.
.
Bt
B
(i2)
The dependence of Vz on q is especially remark-
able due to the presence of the mean curvature M,
Equation (12) can now be easily separated by since it cannot be obtained from the g, &'s and their
setting X=X((qqz, t) xX(qz, t), where the sub- derivatives alone (contrary to what happens with
. scripts t and n stand for "tangent" and "normal, " K). This result has an important consequence:
respectively. The usual procedure yields the Vs(q( qz} will not be the same for two isometric
following equations. surfaces (for which correspondent points can be
found with the same g(z's). This is in striking
O'B X
p" + V(, (qz)X=ia (13) contrast with the results of classical mechanics
2m Bg, Bt
where the Lagrangian of the free surface motion,
2 2 Z(q(o q2o q(o q2) = 2 m(dsldt) = z m+i, ((g(((q(qZ)q(q(t
depends only on the metric properties of the sur-
face. Strange as it may appear at first sight,
' this is not an unexpected result, since, indepen-
[-, Tr(n)]' etd( ()(I X, = ia
dent of how small the range of value assumed for
(i4) q3, the wave function always moves" in a three-
dimensional portion of the space, so that the par-
Expression (13) is just the one-dimensional ticle is "aware" of the external properties of the
Schrodinger- equation for a particle bounded by limit surface S. In order to illustrate the proper-
the transverse potential V(qz), and can be ignored ties of Vs(qq, ) let us consider, for example, a,
in all future calculations. Expression (14), how- bookcover shaped surface obtained by. bending a
ever, is much more interesting, due to the pres- plarie around the surface of a cylinder of radius a
ence of the surface potential V, (qq, ) (Fig. 2). Selecting as parameters the arc s of
= -(k '/2m )(-,' Tr(a)]' d et(((. )j. the cross section C and the Cartesian coordinate
 QUANTUM MECHANICS OF A CONSTRAINED PARTICLE 1985

from the beginning, since this kind of potential

prevents the' separation of the wave function X in
tangent and normal parts, as given by (13) and
(14). It goes also without saying that the fact that
the forces tend to be normal to S in the limit
E -0 does not imply the vanishing of the tangential
components since the forces themselves go to
infinity, precluding any direct comparison with
the classical situation. If we now take d(q f q2)
= &[1+ef (qi, q2}], then, again according to Cheng,
"everything depends on the higher order terms
"
O(a ). Here, however, we cannot forget that al-
FIG. 2. Cross section C of the "bookcover" surface: though the terms of order E may be a small per-
A plane bent around the surface of a cylinder of radius turbation for the total potential, they may still be
a. The middlepoint A was chosen for the origin of the important when compared with the energies in-
arc s. volved in the surface motion.

z perpendicular to the plane of the figure, we III; PARTICLE BOUNDED TO A CURVE

have from Eqs. (14)-(17) Let us consider a pointlike particle of mass m,
2 2 Q2 2 rigidly bounded to a curve C of arc q&, parametric
Xt + Xf
k( )2
' k Xt ' (19) equation r = r(q, ), and tangent normal and binor-
2m', es ez 8m Bt
mal denoted respectively by t(q, ), n(q, }, and b(qi}.
Following the same reasoning of Sec. II we shall
where k(s) is the curvature of C at the point of now introduce a, curvilinear coordinate system
arc s.
If we consider a solution X,(s, t), indepen-
based on the curve C (Fig. 3):
dent of z, we obtain a one-dimensional Schrodinger
equation in the presence of the square well poten- q3) = r(qi) + q2n2(q1) +qan3(qi),
R(qadi q2i (20)
tial, cos 8(q i) n(q i) s
n2 ine(q &) b(q f),
(21)
n3
sine(q, )n(q i) + c
os 8(q, ) b(q, ),
V(s) = (19) with
0, is i)ne,
d&
( = r(q|), (22)
which (since [(2m/k ) i Vo ) (ae) ]i i2 = 8/2 v/4} has dqi
only one bound state of energy Eo, Vo &Eo &0. In where r(q|) is the torsion of C. (For the sake of
the limit a-0,
8=constant, which corresponds simplicity we have introduced a Cartesian coordi-
to an i/initely sharp bend in our plane, the trans- nate system for each normal plane of C. )
mission coefficient of (19) goes to zero. The two From (20), (21), and (22) we get
sheets, s (0 and s &0, of Fig. 2 are then effec-
tively disconnected, in strong contrast with the dR
usual solutions where the term V(s) is absent. = [1-k(q )f(q, q, q, )]t(q,), (23)
One last remark must be made on previous re-
sults stating that the limit obtained here does not
actually exist. Our opinion is that those calcula- (ng)q, (24)
tions, although mathematically unimpeachable,
are badly conceived from the physical point of where
view since they involve potentials with nonzero
tangent forces in every neighborhood of the surface f (qq2, q3) = cos8(q&)q2+ sine(q, }q~, (25)
S. To use the procedure described by Cheng we and k(qi)= ddt/dqi i is the curvature of C at the
imagine our particle squeezed between two impen-
point of arc q&. Since our coordinate system is
etrable surfaces, our own surface S and another orthogonal, (BR/Sq, )(8R/Sqi) =k, b,.&, we can write
surface S', and let the distance d(qi, q2), between the classical force F due a, potential V(qq~, q, )
them, go steadily to zero. If we take d= sf(qiq2),
as
&-0, then, to quote Cheng's own words, "The
Schrodinger equation would acquire a term pro-
portional to [sf(qq,)] which varies wildly over
"
the q's. This, in fact, could already be expected
F = gradV= ~ i
eV 8R
hf Bqf aqf
(26)
 R. C. T. DA COSTA

Proceeding as in the case of the surface con- always maintain the force gradV, in the normal
straint, we shal1 select, from (26), a binding planes of C. The Schrodinger equation is then
potential V(q2, q3) independent of q in order to written as

" ' (27)

2m 1-kf Bq~ 1-kI Bq~/ & Bq& Bq& Bq& Bt

The volume element is given by dV= (1 kf)

xdq, dq2dq3, which suggests the introduction of the 2m ~q, Sm
k(q, )')t, =fr ' . (32)

new wave function Z(qq2, q3) =(1 kJ)' Here equation (32) has the same property of
x(q&, q2, qs). Equation (27) is then transformed equation (14): although all curves are isometric
into each one has, depending on the curvature, its
own distinct quantum mechanics. It must also
I 1 B 1 B be noted that Eq. (32) does not depend on the
2m (1-kf)'" Bq& 1-kf Bq& (I-kf) ') detailed behavior of the potential V, (qq, ) (its
k equipotentials around the curve C can be circles,
Sm (1-kf)' elipses, rectangles, ect. ), provided that once it
is defined in one normal plane it is known in all
I' ~8'X
2~~8q2
2+
8'X'I
~. l+V.(q2
eq, &
.
qs)&=fk .
~X
(28) points of the space by giving the same potential
to all "parallel" curves with the same values of
Assuming for V the expected properties of a con- q, and q, (Combescure transforms to the mathe-
maticaliy minded). In a certain sense it can be
straining potential:
said that in V, (q2q, ) we have introduced a general-
2 2 ization of the ordinary two-dimensional potential
Op q2+q3=O
2+ (obtained when C is a straight line).
p

mV. (q2, q3)= 2 (29)

q2+q300, 0 One last remark must still be made about the
possibility of binding a particle to a curve in two
we can directly take f -0 in (28), obtaining successive steps: First using a surface con-
straint of the type employed in Sec. II and, after
I
2m
By,
aqua Sm
8, , 2 k
2m eq2
By2+
Balll
2
eq3j
that, assuming an extra surface potential. to reduce
the motion to a curve. It is not difficult to see
that the result obtained in this way will, in gen-
+ V, (qq3)g = ik
~x
.
BI;
(30) eral, depend on the intermediate surface selected
in the process. The reason is that the normals
Equation (30) is now readily separated by setting to this intermediate surface are not necessarily
Xg(qf f) xZ(qt, qq, t) The resul. t is contained in a normal plane of the curve. This
means that the potential responsibl. e for the sur-
2 2
2
X + Xll~V( X face constraint can give rise to forces with non-
) ~ (31)
vanishing tangential components in a neighborhood
of the curve, contrary to the definition of
V~(qq, ). It can also be shown that the same
result (29) can be obtained if the chosen surface
belongs to the following family:
R(qS) = r(q, ) + q, (s)A, (q, ) + q, (s)n, (q, ), (33)
where q, (s), q, (s) gives the intersection of the
surface with the normal planes of C. Notice that,
since q, and q, do not depend on q the surface is
completely determined from the knowledge of its
intersection with one of the normal planes.
(q,
ACKNOWLEDGMENTS
0
FIG. 3. Curvilinear coordinate system based on the The author wishes to thank H. Koberle and N.
curve C of parametric equation r =r(q~). Cartesian Tebphilo de Oliveira for helpful and stimulating
coordinates g and y were used for the normal plane I'. comments.
 QUANTUM MECHANICS OF A CONSTRAINED PARTICLE 1987

B. S. De Witt, Rev. Mod. Phys. 29, 377 (1957). Cambridge, Mass. , 1950).
~D. J. Struik, Differential Geomet~ (Addison-Wesley, K. 8. Cheng,J.
Math. Phys. 13, 1723 (1972).