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NTES ON MOLECUMR COLLISIOIIS7’
Part I: Elastic Scattering
1. Classical Mechanics of o-bo Collisions.
The description of an elastIc collision of two particlessubject to a central force may be reduced to an equivalent one-bodyproblem (see, for example, HCB, pp. 45-51; Goldstein, Chap. 3).This reduction is carried oUt by introducing coordinates which areconjugate to P and L, the total linear and angular momenta of thesystem. As P and L are constants of the motion, in these coordInatesthe equations of motion become separable, and the description of atwo-body collision can be given in four parts:
(i) Three degrees of freedom which specify the translationalmotion of the center of mass of the two particles relative toa laboratory-fixed coordinate system.
(2) One degree of freedom which gives the azimuthal orientation of the angular momentum vector L about , the initialrelative velocity vector of the particles.
(3) One degree of freedom which describes the rotationalmotion of the interparticle axis, r, about the center of mass,in a plane perpendicular to L and containing y.
(4) One degree of freedom which describes the relative motionof the two particles along the radial direction of r.
Step (i), the separation of the center-of-mass motion, can becarried through for a quite general potential function; the onlyrestriction is that the potential should not depend on the locationof the particles with respect to any external body.
Fig. 1-1 shows the coordInate vectors and r2 which locatethe two particles in an arbitrary laboratory reference system andalso the relative position vector,
-.
(i-i)
and the center-of-mass vector,
= (miri + m2r2)/m (1-2)
where m m1 + m2 and the particles are numbered such that m1 in2.
—2--
1
£2
Fig. i-i
The transformation relations,
(1-3a)
- —rm-
hold also for the corresponding velocity vectors (which are denotedhere by a superior dot to indicate time differentiation).
The linear momentum, angular momentum, and kinetic energy aregiven by
P + m22 mR (1-4)
L= m1(r1xi)+m2(r2x2) = m(Rx)÷i.(rx) (1-5)
1 • • 1 • • 1 1T iLi’ Li + 2L2’ £2 £‘ (1-6)
where i = m1m2/m is defined as the “reduced mass” of the twoparticles. Since the potential energy does not depend on R, thereis no force acting on the center of mass, therefore
PmR=O,
and
P = mR = a costant vector. (i-i)
—3-
This provides the conservation law for the total linear momentum ofthe system. If we assume that the motion of the center of mass isknown, we can define a new coordinate system wiose origin remainsattached to the center of mass. In this traveling “CM system3”R and R vanish, and
LtL(rx)
T=Lr.r
TIe mechanics of the relative motion of the particles thus isequivalent to that for a single particle with. mass p. and positionvector r moving about a fixed center of force.
The further separation of degrees of freedom outlined insteps (2), (3), and (4) cannot be carried out unless a central forcepotential Is assumed, where
V = v(r) function of magnitude of r only.
In this case, the force, is always along r. Hence
and
L p.(rx) a constant vector. (i-a)This conservation law for the angular momentum vector is seen to bea consequenc’e of the spherical symmetry of the potential function.Since and must both be always perpendicular to the fixed direction of L in space, the motion must be confined to a plane perpendicular to L. This argument breaks down if L = 0, but then themotion must be along a straight line through the center .of force,as L 0 requires r to be parallel to . Thus, central force motionis always motion in a plane.
Step (2) may now be attained by choosing the z-axis of our CMcoordinate system to lie along the direction of L, This eliminatesthe z-coordinate from the equations of motion, as z = 0 always. The
orientation of L with respect to laboratory directions is determinedby the initial condItions of the motions so these must be known inorder to project trajectories evaluated In the CM system onto thelaboratory system (see Fig. 1-2). As L is perpendicular to theinitial relative velocity vector v, it Is only necessary to specifyin addition the azimuthal orientation of L about v.
L
z-axis
CM System
— N
•7•/
Fig. 1-2
The conservation of the angular momentum vector furnishes threeindependent constants of motion (corresponding to the three cartesiancomponents). In effect two of these3 expxessing the constantdirection of L, have now been used to reduce the problem from three
-5-.
to two degrees of freedom. The third, corresponding to the conservation of the magnitude of L, remains at our disposal. Also, wehave not yet called on the conservation of energy, which provides
E T + V = constant. (1-9)These constraints will enable us to complete the solution withsteps (3) and (4).
On introducing polar coordinates in the plane of the motion,
x = r coO, cos9 - r sine
y r sinO, Sr sinQ + re cosO
we obtain
L t(xr-yk) = p.r2 (i-io)
E (2+2)+V(r) =(2+r2b)+v(r) (i-u)
Finally, by eliminating e, we have an equation for the radial motionalone,
2E 2
+ p + V(r). (1-12)2ji.r
The complete solution can be obtained by integrating the energyand angular momentum equations,
dr = - [—(E_v_ 2)J dt (l-13a)2p.r
dO = Ldt,
(l-13b)
from an initIal point (t = 0, r = r0, 0 = e) to the point (t,r,U).From the energy equation
—1/2t =
- J I —(s-v- dr2iir J
—6—
and this may be inverted, at least formally, to give
r = r(t, r0, E, L). (l—la)
Integration of the angular momentum equation then gives
o = + f(L/r2) dt
e = e(t, r0, e0, E, L). (l-14b)
In summary, we note that the o’iginalto—body problem, withsix degrees of freedom, requires for its solution that twelve parameters be specified (equivalent to initial values of six coordinatesand six velocities). Conservation of the total linear momentumaccounts for six of these constants (the components of and R0),which describe the straight line trajectory of the center of mass,R .= + R0t. Conservation of the direction and magnitude of thetotal angular momentum provides another three constants, and con—ervation of energy gives one additional constant. The ‘wo remainingparameters appear as the integration constants, r0 and
2. Collision Trajectory and Deflection Angle.
In the CM system, a collision maybe pictured as the interactionof a mass point t with a fixed scattering center. Fig. 2-1 shows a
, typical trajectory (for a case where both long range attraction and‘pshort range repulsion are evident), The initial and final
15 7asymptotic states of motion differ only in that the velocity vectorr’Iias bèen’btated th’rbugh aná le
jT’” observable result of the collision. The asymptotic speed v andimpact parameter of the projectile are related to its energy andangular momentum by
E = tv2 and L = Lvb. (2-1)
The transition to a quantum description is facilitated if theclassical treatment is given in terms of E and L. However, often
t—-+\\
Fig. 2-1
I,
-1•;:-’-
. ,cf?
= zn1rn2/(m1+rn2)., reduced mass
v initial relative speed of colliding particles
b impact parameter, hypothetical distance ofclosest approach in the absence of potentialenergy
X = angle of deflection between the initial and finalrelative velocity vectors.
r = polar coordinates of point of closest approach,or “turning point”
: ‘,? :‘.‘
. ‘.. ‘‘‘‘f” I:” ;i j: .s ‘3 j.1.:i” f,
the formulas are tidier if v and b are used, and this is done in
most of the literature. Therefore we shall usually state the main
results in both ways.
The trajectory is symmetric about the point of closest approach
(r=r, e=9). We may take t=0 at this point and also define
= G- so that e’= there. Then,’ since t—-t and e’—-e’
leave the equations of motion (1-10) and (i-il) invariant and do
-8--
not alter the initial conditions (t=o, r=r0, ‘=O), the orbit is
invariant under reflection about the vector r.
Another general property that holds regardless of the form of
the potential function (as long as it is a central potential) is
Kepler’s law: the radius vector sweeps out equal areas in equal
times. This is a consequence of angular momentum conservation. As
indicated in Fig. 2-2, the differential area swept out in time dt
is
dA = - r(rd6)
so that
t
A2-A1= f 2
(r2) dt
ti
or, from (1-11),
M = -Ltt. (2-2)
r
e
.,‘‘J::;?)•/• - ,• I ,?j: ‘.‘ ‘‘t,’i ‘•..
.. ‘—.• 2
Fig. 2-2
The collision trajectory can be determined without evaluating
the complete time-dependent solution. From (1-13) we have
e rf rQ(r’
—
dO -- I •/trJ’
— -b I — dr
_____
o - [(E-v-2r)]
(2-3)
(The square root must be taken wIth a negative sign, since dr/d9<O)Furthermore, sInce we usually need to calculate only the deflectionangle X. we need to locate only the turning point (r0,e) of thetrajectory. As seen In Fig. 2-1,
and thus the angle of deflection is given by
X(E,L) = 2f
__
= -2b
‘r r2 bv)l/2
(2-4)
The radial distance of closest approach Is determIned from thecondition that r be a mInimum, dr/dO = 0, or the equivalent conditionthat the radial velocity vanish, = 0. From (1-12) we have
2E L
2 + V(r0) (2-5a)
or, on introducing (2-1),
b2 r [i- V(rc)]
Thus, once the potential function is specified, we may readilyevaluate r(E,L), and compute the deflection angle from (2-4). Notethat rc<b if V(rc)<0 (potential attractive at the turniig point)and r0b if V(r0)>O (repulsive at the turning point).
To compute the deflection angle nothing needs to be known aboutthe potential energy V(r) for r <r0(E,L). That is, the angle of
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V(r)= [()12 ()6]
Here E is the depth of the attractive w11 and a the radius ati’ 6which V=O; the bottom of the well, V=-E, occurs a) r = 2 “ o. This
potential is plotted in Fig. 3-1, together with the cen(;rifugalpotential for several values of the angular mcmcntua (dashei curves).Dimensionless units are used, with
* * * 2 * i * 2u =v + (r,/r ) =v +E (b/r )
L’- ti.-
bO
cv
cvC)
C)
v = r(r)/e, E/€
=
r = r/a h = b/a
1
0.5
0
-0.5
—1
.. ,f-—’.
c/
,
‘ /.(.L
-1
0 1 2*
Reduced Distance, r
‘ Fig. 3-1
3
-12-
In Fig. 3-2t effective potential curvc are shown for severalvalues of L.
*
r-f‘- aS
- ‘V— 4.)
0
.._/,_;<.
. C)
Ci -
C)
—1
- .:-
Reduced Distance) r
.
Fig. 3—2..• -
*For sufficiently high angular momentum (L > 1.569), the effectivepotential becomes monotonc and everywhere repulsive. For L*= 1.569,a point of Inflection appears at U = 0.8, and for lower angularmomentum (r, < 1.569), the effective potential has zones of repulsionat large and small distances, separated by a zone of attraction.The effective potential curves for 0 < < 1. 569 thus show a ‘centrifugal barrier, ‘ as well as a potenial well, at values of r wherethe attractIve force provided by the intermolecular potential isbalanced by the centrifugal repulsion.
1 2 3
—13-
The po;itlons of these ext;ren;a for a given value of L nay be
found. fro;ri
(u/r) = (,/Er) - = O
where the two roots rrn = rrsx (at barrcr top) and rm = rrjn (at
the botCc of the well) satisfy2 2 P 2 it
(U/orm = V/rjm + 3L/IJr 0 at x’
> 0 at
The value of L=L1, above which the effective potential cu.c’ve becores
monotoni is found from
N2.’ 2(o U/ar ) = 0, or 1’max
To express these relations in terms of the reduDed variables, we
take the potential to have the form
- V(r) €f(r/a)
*and find. with x = r ,in
f(x)+ (*/)2
f’(x) - 2L*2/x3
= f11(x) + 6L*2/x4.
L = x f’(x) (3-5)
and*2 lit
L1,. - -x f”(x) (3-6)
For the Lennard-Jones potential,
= _24(213 -
= 24(2Sx4 - 7x8)
By equating (3—5) aid (5—L;) :c find tho infieeton p.Jrtt
= 1.3077
an3 this corropcncL; to
r—
4/2)24i -
- L / \ I7 \5 5= (xs1/3) 1.569
* * *U (L = 1569, r = 1.308) = 0.8
*(U/r) = (U/r) = 0
* *Fig. 3-3 sho:s the curves Of U versus
r=r and r=rmax rn:t.ri
*L
which correspond to
1.6 2.0
* 11 ] fTC\ ,4/3U = ± = 0..8
V / 3j\±/)J 5
This inflection rcin we shall refer to as the Htripie point ; there
1.
0.
0.4
0 0.4 0.8 1.2
I’ig. 3-3
4. Topo].o. of SeaL
The collision dyr .ics varies marL:edly for diffcrent ranges ofE and L Consider first the: Lurning point of the trajectory.
* *Fig. 4—1 sho;s the function E —U plott0J versus r , for E = 1.Since the turning point. Is defined by -
r= r),
accorc1ig to Eq. (2-5), the ercs of this function give the va].uesof r The dashed curves refer to the case V(r) = 0, the solId
*4
4-,0
cti
C)
cii
*L=0
v)
—
— i..569
- -
-
__
*
/
r
/
3
/
/
-1
—2
—3
//
/
*E1
//
/
/
Fig. 4-1
—16-
curves to the Len:arc1—Jonos co:entil. ie seo tth. effect ofthe potential :Ls to shift the zeros rr to’cr r for iOi values*of L anci. to srl cx’ r for ln:’ge va]es of L . I’ox’ very largevalues of L c:heru the infJuenDc of the fieki becomes ne.ligib2.eone e:-cpccts to fi.i that r0 becomes practically equal to the initialimpact parameter. In Fig. 4-2 r is plotted as a function of b forvarious values of Again the dashed line car as ‘,cncls to theV(r) = 0 case. At small irip’.2t parameters, rcpu].sioa veccmir:aaesat the turning point and r h; at lcxge impaet pare::ctCrS)attraction rredominates and r < b, ‘1’:lth r—’-b for sufficientlylarge impact parameters.
b2
C)
4-)
0p4
Impact Parar•eter, b/a
1
0 2
Fig. 4-2
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The boundary condition that the radial factor R1 remain finite atr=0 determines the asymptotic form of the soJution (except for thenormalization constant).
Xn the absence of a potential, v(r) = 0, the wave equationreduces to a form of Bessel’s equcttion. The corresponding planewave solution.
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Pig. 7-2 indicates the physical interpretation of this. The.incident plane wave is equivalent to a superposition of an infinitenumber of spherical waves, and each term in the series of (7-5)corresponds to an orbital angular momentum of magnitude
L [s(1+l)]u/2t1
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x/v (14)/ic (14)r.Xn Pig. 7-2 we visualize th& incident beam as divided up intocylindrical zones such that the Ith zone contains particles withimpact parameters between 1?C and (t÷i)*. Although in quantummechanics only integral values of £ are admitted, we cannot speakof a well-defined impact parameter in a beam whose particles havea well-defined velocity. However, it is approximately correct toregard the particles with angular momentum (14)n as moving in the£th zono.
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9. The JWK}3 or Scm1c.ass:Lca1proximation
For molecular scattering, a semiclassical description isusually applicable in the realm of “hard” co].].sions. In general,the semiclassical approximation is complementary to the Bornapproximation. Thus a complete3 if approximate, solution of mostmolecular collision problems can be obtaIned by a combination ofthese two limiting forms of quantum mechanics.
The semiclassical description of scattering outlined here isbased on the treatment of K. W. Ford and J. A. Wheeler, Ann. Phys.7, 259, 287 (1959). As emphasized by Ford and Wheeler, the crosssection in the semiclassical approximatioii may be practically equalto, or very-different from, the classical cross section, dependingon the shape of the potential, the collision energy, and the angleof observation. For a r-atr restricted class of potentials, forwhich the classical deflection angle is a monotonic function of theimpact, parameter, the semiclassical cross section is identical tothe •classicaJ. result. However, for most potentials of interest, thedeflection angle is not a monotonic function, and even when verymany large phase shifts contribute to the scattering, the con-vergence of the quantum cross section to the classical result isnonuniform: for any wavelei-igth, no matter how small, there will bea substantIal angu.lar region where the quantum and classical crosssections differ by a large amount,
The semiclassical approximation may be obtained by introducingthe following three mathematical approximations into the quantumpartial wave formulation: -
(1) The phase shift is approximated by the Jl’IKB resultgiven in Table 5-1,
-
ill F1b2 v(r)11/2dr
;
EJ r- (9-i).‘ e
(2) The Legendre polynomial is approximated by an asymptoticexpansion valid for large ,
- -
• r, 1
P1(cosX) [( 1)7r for sinx (9-2a)
or
P9(cosx) (cosx)J0[(+-)X), for sinX ‘ i/i. (9-2b)
(3) The summation of scattering ariplitudes in Eq. (7-7) isreplaced by an integral, 2 —>f d.
These approximations convert the quantum ex’pression (7-7) into theseniicj.assical formula for the scattering amplitude:
CO
f(x) = ‘i (4)1”2Ee - e (9-3a)(27rsinX) / 0
and
i 2ir1r(x) = -3 J (M-) e J0[U÷-)sinx)c (9-3b)0
Eq. (9-3a) applies when x is not too near 0 or -r (that is, wheniX over the region which gives significant contributions tothe integral). The phases P and P are defined by
= 2i ±(9-4)
Eq. (9-3b) applies when X is near 0 or ir (when ix i/. for significant contributions). The factor of (cosx) which appears in(9-2b) has been omitted in (9-3b), since ‘then X is near 0 or sir, thisis essentially unity or (-i), respectively, and the relationp(cosX) = (-l)p(cos[7r-X)) allows us to account for the lattercase by replacing y with sinx in the argument of the Besselfunction, In both the expressions (9-3) we have omitted the forwarddelta function which arises froni tle -1 term in the (e2 -factor of (7-7),
Z (2+1)p(cos) 2ö(1-cbsX) \ (9-5)\
rYi.‘
-54-
since this term does not contribute to the differential crosssection (except for the singular point, x 0); it must he includedin calculating the total cross section, however.
The conditions for the validity of the semiclassical approxiinations are:
(i) The JWKB approximation for the phase shift, Eq0 (9—i),requires that the pot•;ribial vry slowly over distancescomparable to the de Brogue wavelength, so that
V(r+?c) v(r) + c + v(r)
or(9.-6)
V(r)>.
(2) The use of the asymptotic approximations of (9-2) in thepartial wave series requires that many £-values contributeto the scattering at a given angle or that the major contributions come from large £-values. However, theasymptotic formulas (9-2) themselves are good approximations even at rather small £. The range of the twoformulas overlaps sufficiently to cover the whole rangeOfX.
(3) The replacement of summation by integration in Eqs. (9-3)requires for its validity that many partial waves shouldcontribute and that the phase shift should vary slowly andsmoothly with £. This approximation rests on approximations (1) and (2), since it depends on defining thephase shift and the Legendre polynomial P as smooth,continuous functions of £.
As we shall see, these conditions are usually satisfied in molecularscattering.
The integral (9-3a) for the scattering amplitude may often beevaluated by the “method of stationa:’y phase. ‘ If, for a givenangle of observation X,there is a value of £ for which
c1cP±0, (9-7)
then in the neighborhood of £ the imaginary exponential factorn the Integrand of (9-3a) will not be oscillating, and we expectthat noncancellng contributions to the integral for f() will comemaInly from this region. According to (9-4), the condition (9-7)for stationary phase is equivalent to
or
-x S £ (9-8a)
= + if ap/d 0 at £ Lx (9—Sb)
From Table 5-1, the der.ivative 2(d/d) is seen to be eal to theclassical deflection function, x(.), so these conditions merelystate that constructive interference only occurs when the angle ofobservation (which is necessarily positive) satisfies the classicalrelation -= JX(L ), that is,
x= -x() for stationary (9-9a)or
X-: +(2) for stationary. (9-9h)The simplest case to treat is a classical deflection function
x(L) which varies monotonicaJJy between 0 and ±ir, ‘and this isconsidered in Table 9-1 and Fig.. 9-1. Since both phases cannot bestationary simultaneously, we assue that the contribution fromthe term containIng the nonstationary phase is negligible. Also,we approximate the contributing phase by a quadratic function ofthe angular momentum,
x’ -xx , (9—10)where, from (9-4) we have
2(d2iiL/d2)LL , (9-fl)x
since the derivative is taken for a fixed angle of observation,X X0.5. The relation between the semcJ.assica1 phase shift andthe classical deflecti’on function converts (9-n) to
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The quantities entering these equations are all to be evaluated for£ £ On rnaldng the correspondence with the classical impactparameter, b (.a+-), we recognize Eq. (9-16) as identical to theclassical result found in Sec. 6.
x()
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Fig. 9—i
£
The conditions under which the derivation leading to (9-16)applies are illustrated in Fig. 9-1. This shows a simple monotonicclassical deflection function and the correspondIng phase shift.(A repulsive potential Is assumed) àr an attractive potential,x.(Y would be negative and i positive instead..) The contributionsto the cross section at X Xobs colite mainly from the indIcatedrange of £ values about The important £ values are those for
&e
which the curve and the line tangent to the curve at Lx cUfferby less than a fe;i radians; ou.tside this region, the oscillatorycharacter of the integrand in (9—3a) produces destructive interference and thc net contribution is negligible. The result of(9-16), that the semiclassIcal and classical cross sections areequals requires that Lx arid L be large compared to unity and thatthe curve be we].l represented by a parabola over the interval AL.The phase factor in the scattering amplitude of (9-14) 4s simplyrelated to the intercept, of the tangent line in Fig. 9-1is negative in the example shown).
The agreenent with the classical result depends upon twoconditions not inherent in the semiclassia1 approximation itself:
(a) The angle of scattering must not be too close to 0 or r(sinX several times greater than l/Lx and
(b) There must be one and only one point of stationary phase.The latter condition will he met for all x only if the classicaldeflection function, (L)., varies monotonically beteen 0 and ±7r.This occurs for very few potentials of interest for molecu:larscattering; in particular, the Coulomb potential is almost theonly simple attractive potential with this property. In latersectIons we shall consider various special features of the semic1asscal cross section wnich can arise witn deflecion functionsfor which there is not a one-to-one relation between £ and X orfor which X(L) passes through 0 or ±71.
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SEMICLASSICAL ANGULAR MOMENTUM
Asvmntotic Approximation for Legendre Polynomials (Landau Lifschitz, p. 166-168
r2 1+ cote + 9.(9.+1) IP(cose) = 0.
Lde J
Substitute
P(cosO) X(0)
Vs in e
Find
+ 4)2+ -csc26]X 0.
This has the form of a one-dimensional wave equation,
x” + = 0
where1
= [(+) +csc8] = (1)[l+2
has the role of a deBroglie wavelength. In order to apply the quasiclassicalapproximation, we require dA/dO to be small, or I(-+)sin0l large. For largeZ, this condition will hold except near e = 0 and 8 = ri, where sinGvanishes. Thus we must require 02 >> 1 and (ir-G)2. >> 1. When these conditions hold, solve
x” + + = 0
and obtain1
or
X(8) = A sin[( + + a]
sin[(+4)8+c]1 1P(cos8) = A
____
, for 8>> r
(ir-8) >>
V’S inS
To determine the constants A and a, proceed as usual in the NKB method byconrparing with result of an exact solution in the region where the quasi-classical approximation fails.
When 8 is small enough, the differential euqation for P9,(cosO) can be
solved in terms of the Bessel function of zero order, J0(x). Thus, for
SEMICLASSICAL ANGULAR MOMENTUM - 2/ \2
8 << 1, put cotO 1/8 and replace 2(9+l) by+
, to obtain
[+ ( + )2J= 0.
This is the Bessel equation, with the solution
P(cosO) = + )ei, 8 << 1.
This approximation holds at 0 = 0 and at larger angles as long as 0 << 1.
In particular it can be applied in the range l/ << 0 << 1, where it must
agree with our previous result. The asymptotic expansion of the Bessel
function is
J(x) /‘sin (+
for x >> 1.
Thus, with x= ( + )8we have
P(cos0)2 sin[ +
+
for 1R << 0 << 1.
So we find
/2 itA = and a = and hence
(+)
_______
If l\ it
fP(cos8) I I. for l/Z << 8
‘ine
+ .) sine] for 8 << 1 or (it-C) <<
Note that the normalized angular wave function for in = 0 is
P(co:0).
sin (Z+.e+.....for 8 >> , (r-e) >>
SEMICLASSICAL ANGULAR MOMENTUM - 3
For large 2., the sin2 factor in the numerator oscillates very rapidly and
can be replaced by its average value, 1/2. Then
I @I 7TsinO, for large 2. holds everywhere except very close
to 0 = 0 and Ti.
Physical Interpretation:
For large 2., the quasiclassical motion corresponds to particle rotating
about the angular momentum vector, . For m = 0, the vector is perpendi
cular to the z-axis, so the situation is:
If & is the azimuthal angle about theprobability of finding the particle in 0to 0+dO is uniform. But may have anyazimuthal orientation about z, i.e. planein which the particle moves may have any
4 with equal probability.
To obtain the full distribution, rotateabout z-axis: Note that the probabilitycontained in the interval 0 to 8 + dO for
the particular planepictured on the
• extreme left is spread
dOout over 2TisinOdO whenaveraged over •,
• hence the probabilityin the full distribution becomes proportional to
1sinO
Another way to visualize this result is to consider the density of
intersections of latitude and longitude lines on a sphere. Clearly, these
intersections crowd together near the poles and, from the solid angle
element, the density of intersections is seen to be inversely provortional
to sinG.
The situation for be considered in the same fashion. Suppose
projection of on the , then the quasiclassical metion corresponds
to: where cosa = M/L. Again, for any particu
lar orientation of the vector, the
probability of finding the particle is
uniform in the azimuthal angle about
and may have any azimuthal orientation
about z, but with a fixed.
z
orbit ofparticle
0 canz—axis is
z
SEMICLASSICAL ANGULAR MOMENTUM - 4
If we let t,Ls be the angle of rotation in the plane perpendicular to
then according to the argument in the center of page 3, we have
P0(8)sin0d8
where P0 refers to the probability of finding the particle between 0 and
0 + do and P, to that of finding it in iP to b + dti, and N is a normalization
constant. Since P,Op) = constant, we have
p (0)= N(d/d0)
0 sinO
In the m = 0 case = 0 and the previous result follows.
we must determine the relation between qi, 0, and a.In the m 0 case,
Thus we have
cosO sinacosi
sinOdO = sinasind
Note is dihedral angle between planescontaining (,, and ) and andHence we use the spherical trigonometryformula -
.2 21/2 .2(sin a — cos 0) (s.n 02 1/2
- cos a)
‘p
cosa = cosb cosc + sinb sinc cosA
71Herea=0, ba, c=-,
soN
P0(8)= sinasin’p =
Note that the region where sin20
N N
< cos2a, corresponding to either 0 < - - a
or 0 > + a, is classically forbidden.
SEMICLASSICAL ANGULAR MOMENTUM - 5
Pictorial summary:
m=O
0 \
V+l)>AC = (sjn2O -
IT
I/IClassical
Problem
____
Density sinO I
I
______________________________ __________________________________
00 0 1800 00 ‘‘‘ 1800c . f. c . f
c.f. c.f. classicall
forbidden
IpI2
QuantalI NotProblem
“SiDensity -______
_______
...widher
fl e
unctycip
As compared with the classical limit, the quantal result (for large 2.) has
rapid oscillations nd vuts some intensity in the classically forbiddenregions (although this decays very quickly).
QUASI-CLASSICAL CASE
WKB or JWKB Approximation
ii± - jp(x)dx(x) = F(x)e “ -‘
icd2F+QdF+1dPF\ =0ip2 dx pdx )dx
2 dF + 1 dp = dpdx
-n(F2p) 0
1/2F = const/p
This approximate form for ‘(x) can be obtained more formally by expanding anexponential form in powers of !. It turns out the first term leads to theclassical result, the second to the “old quantum theory” (Bohr-Sommerfeldcondition) result, and the higher terms to corrections which provide an asymutoticapproach to the exact solution.
with S = S + -S0 ii
22 2 2
____
_ + (E - V) = 0 or + = 0, p = v’2m(E-V)2md2 dx2 if2
±- pxIf V is constant, solutions are = constante . For the case where Xis small compared with region in which V changes appreciably, try an approximation of the form
where p(x) and F(x) are slowly varyingin Schrödinger Equation gives
Since we assume = Ti/p is small and Fterm and have
functions of position. Substitution
varies slowly with x, we omit the first
and hence
Thus, use
=
1
e
‘ 2i r s\2
+ (E - V) ‘
dx- ilc —.-j + (V - E) = 0
Terms without give:
QUASI-CLASSICAL - la
iS(x)See also Park, p 89ff, who uses (x) = F(x)e where F and S are real inthe classically allowed region,
rxit
j-I p (x)dxFor our case, P(x) = F(x)e 0
i-lower limit can be any constant
/ l \J 0
ip’ ip ip if= [CF” ± F ± _2- F) ± (FT ± _2 F)] e wj
2[F” ±
°F’ ± ._2. - 4— F] efPo
LHS of Schrdinger Equation:
4,”+ 4’ =
± °F’ ± F) e - fpdx
Neglect as amplitude slowlyvarying over wavelength
. 2if l \—
çpF1 +- P01F) 0
Factor out p01”2, which 0 if not at classical turning point.
1 -1/2p ‘F = 0o 20 o
(p 1’2F) = 0dx o
1/2p F = const
- 1/2F = const p0
I2 1p V
0 0
QUkSI—CLASSICAL - 2
,asito
(1)
Linear in f:
as aso 2
+L O_02 2
ax
a2S/ax2
- 2S /ax0
Potential Well Problems:
E = V(a) = V(b) locates classicalturning points x = a and x = b.
In the region of classically allowed motion,II, we have E > V, p real, and will besome linear combination of and . Wewrite it as
The phase angle specifies the relative proportion of and ‘ and will bedetermined below by comparison of the approximate ip with the exact solutionnear x = a.
= E -v
(2)as as
0
ax ax
Quadratic in :
2\ax I
C
a2s1=0
2 2ax
etc.
Thus we find
as0
ax
as1
= ±/m(E-V) and S = ±fpdx, p =0
ia= - —
— 2.ni —
2 ax ‘ax1Si = - - .np
Then, if only S and S terms are retained,
etc.
4- fpdx -
-1/2 4 fpdx±(x) = e e = p e
II= pli2 sin(
J:Pdx +
Suppose
V(x) -
I LLU-
a bx
Figures from Powell and Craseman “Quantum Mechanics”.
N
Fic. 5—22. The Airy function Ai(z) (1/jr) f° cos (3/3 + z) ds.
+3
(:2)
Fro. 5—23. W K 13 nppro,imation to the harnionc-osciIIattr wave functionin the [ate o 1. To the acelt racy of the graph, t lt ‘\ k 3 wave function(heavy lint’) tinritlts tvil It the exact wave lunction (Itrokea [‘it) in the interiurof the well. Near (1w tbsiit’aI turning Innunt 12 = ?, the \VKI3 :npproxinn:t&iunhre:nks duty n. Thn.\ ry Inn ntt inn (liln I inn) cni nein hit n itln the exact tva yefn,nctinii at 12 anti ,unn,’n:ts ti, 3VK 3 tIninnnn’dnnati,tn in, Iii. ci ,,, ital anti non,—cI:ceninal regloics. Al unnail anti l:tcgc’ :, the Airy ftninntion tieviatis frucin thent wave function,.
z
p4(x)
— win Ipproximation— Airy [anchor,
Exact wave function,
QUASI-CLASSICAL - 3
In the region of classically forbidden motion, I and III, we have E <V, p = ilpi imaginary. Here the requirement that P remain finite as lx!°° determines the choice of ib or P. Thus, take
;ftpldx= const.p”2 e
-1/2 IadX-1/2 -
Jp= Alp! e = Alpi e , or ‘P for x < a
cx1i
-
- I iP dx-1/2
= Bipi e , or for b <
Need to determine A, B, C and to connect solutions. The WKB solutionsfail near the turning points a or b. However, close to a or b we can approxi—mate the potential as linear and use the exact solutions (Airy functions) tointerpolate between regions I and II and II and III.
Near x = a, V(x) E - c(x-a)=’ —J- +(x-a)’P 0
dx
Near x = b, V(x) E + (x-b)=--- - --- (x-b)’P = 0dx
Put
f2ira\1’13z = - —-
(x-a) orz =ç—--) (x-b)
2mc 1/3
= (—-) (a-x)
and then have in either case
z’P = 0.
The solution which vanishes asymptotically for large positive z (z > 0corresponds to either x < a or to x > b) is
Ai(z) = cos ( + sz)cis
For large Izi this has the asymptotic forms
QUASI-CLASSICAL - 4
23/2
z>o24z1’‘
Ai(z) “.. isin[ (-z)3”2 +
, z < 03 4
We consider values of x close enough to a or b that the linear approximation to V(x) is adequate but large enough that these asymptotic forms forAi(z) can be used. Such values of x always exist if the motion is quasi-classical throughout (i.e. change in X = ft/p small compared to itself).
Near x = a we have
p = 12m(E-V:f 12m[g - + ct(x-a)J ‘Tma(x-a)1/3 1/2p = (2nicd) (-z)
For x < a:
dx = - (a-x)3 =- 2 3/2
For x > a:
= dx = / (xa)3/2
Hence we see that near x = a,. x
(exact) Ne
Z= ma)6N
lpI2eaI2v’z” 2V
*11(exact) )1/4 (-z)312
+
=(2mO)’N •1/2i[lf
d +
Thus we find that (WKB) will match (exact.) if we take = and also
that p1 (WKB) will match i4 (WKB) and (exact) if we take A = C.
Hence a
2IpJ2e
II = 1/2sin (fPdx
+
approximation to left of a approximation to right of a
x<a a<x
QUASI-CLASSICAL - 5
Similar analysis near x = b shows
pII = c sinc.jb pdx +
approx to left of bx<b
CI IpIdx= 1e2)p
approx to right of bb <x
Since the two approximations to must be the same (except perhaps for theconstant factors Cand C’), we have
sin{ Pdx+) = sin{Pdx+)
denote by 0 denote by 0’
In order that C sinO = C’ sinO’ be an identity in x, the sum of the phases,0 + 0’, which is a constant, must be an integer multiple of 7T, or
8 + 0’
b
apdx + = (n+l)Tr, n = 0, 1, 2,
with C = (-l)C’. Hence
or
1ç- pdx= (n+-)’iT
I = (n +
fbwhere J = 21 pdx and 27T1 = h. Itds interesting to note
only the S0erm (rather than S0 and S1) in carrying out
dure, we would have found instead pdx = nh.
1C -
l/22)p
a < x <b
The integer n = number of nodes of , since the phase 0 =-
f:Pdx +
increases from at x = a to (n + at x = b and therefore the
sine must vanish n times in this range (whereas outside a < x <
decreases monotonically and has no zeros at a finite distance).
In general, we only expect the WKB approximation to be highly accuratewhen n is large. However, in some cases the exact have the same functional dependence on n for small and large n. In such cases (e.g. Coulomb
The relation J(E)of E which will yieldcal action integral.is
that if we had kept
the matching proce
(n + )h allows us to determine the discrete valuessuitable wavefunctions by merely computing the classiThe corresponding WKB approximation to the wavefunction
dx
x< a
J pdxa
- LP1dxiT” (-1)TCe4 1/221p1
b <x
QUASI-CLASSICAL - 6
field, harmonic oscillator), the WKB quantization rule, although reallyapplicable only for large n, gives the exact result for E.
In normalizing the WKB wavefunction, we can restrict the integrationrange to a < x < b, since at large n falls very rapidly outside this range.Then
i,X2 dx .21 11
C
a
SiflQa1X
+
i.
In the quasi-classical domain, the argument of the sine is rapidly varyingso we replace sine2( ) by its mean value of 1/2 and obtain
1c2 i.2 pa
In terms of the frequency Cii = 2ir/t of the classical periodic motion, tfbd
2m i — , we havep
C = (2(iim/Tr)1”2. Note cii varies with energy E
J(E ) -J(E)n+l n h 2ir
Since t = dJhjdE, have t =o o E -E E -E w
n+1 n n+1 n n+1,n
Can also obtain convenient estimate of spacing of levels from
E -E=h h h
n+1 n J(E + ) - J(E ) dJ(E) tdE
n+l n
Penetration Through a Potential Barrier
In treating the potential well I 1 : ri
problem, we discarded theincreasing real exponential I Eterms in the nonclassical regions,in order to keep ‘(x) finite at V(x)
±. For a potential barrier, I- x
the increasing exponentials must -be retained, since the nonclassical region is of finite width. Hence weneed to use the connection formulas derived via Bi(z) as well as those viaAi(z).
Suppose beam of particles is incident from the left. Thus in region Iwill have incident and reflected waves, in II will have decaying and increasing exponentials, and in III only a transmitted wave. We therefore “workbackwards” from to find via the connection formulas the proper linear
OUTLINE OF POTENTIAL T3ARRIER PROBLEM
III=
p1/2[cos ( I:÷
a
+ ( , x > a
I w Iptdx w IpIdxI 1
eb<x<a
A F1 -
fXJpJdx
= p1/2 LT e + T e
Pi -
ib
_
I I1/2
[2T1sin ( fbpdx+ +
T cos ( 1: +, x < h
QUASI-CLASSICAL - 7
/ çX
__
i(J pdxA \a— 1/2
e
p
I pdx+ )+ . SiI f pdx
+ .‘)], x > a
- I IpIdxl1 1X J,x<a
Now we rewrite in a form convenient for derivingtion formulas for the “barrier to right” case, whichix lb
or . Thus, useb x
ra 1a jX
I IiI’x I IPIdx - IpIdx.
and hence rewrite as
Il = 1/2e
A
p
WI pdx
+ -Te
rXiiW1 pdx
b < x < a.
Now obtain via the connection formulas
A= 1/2
[2T-1
p
pdx+
i /1+ 1 COSQc pdx
+
x < b.
combinations to represent and P1. Form of is
4 , x>a
where we have inserted the in the exponential to facilitate application of
the connection formulas. Since A is complex such a phase factor may beabsorbed in it. To apply the connection formulas, we first write
AIII — 1/2p I (i
Then, using the formulas for a “barrier at the left,” we find in the non-
classical region
A!pldx
x
by use of connecinvolve
Define -T=e
QUASI-CLASSICAL - 8
Finally, in order to identify the incident and reflected parts of it isconvenient to rewrite it in terms of imaginary exponentials,
= 1/2[2T1
(e18 e)+ T (e +e0= fPdx +
4 [(T - -T)e’0 - (T + -T)e’0] E+
Now see that the e0 term represents a wave moving to the left, hence
the reflected wave, and the i4i ft elO term represents a wave moving to the
right, the incident wave. Amplitudes of interest are:
- IAI -1 1 -Incoming wave :1/2
(T + - T) a1p1
Reflected wave (T - T) E a1p1
Transmitted wave p111:1/2 a111
pill
The transmission coefficient is defined as ratio of transmitted to incident flux, where flux is given by velocity times intensity, or vial2, andthus
2 2v1111a1111 p1111a1111
Transmission coefficient =.i = =
v11a11 p1a1
2- 12m(V-E)dx
= (1 + T2)2= e
as T must be small for WKB treatment to be valid
Similarly,+ 2 + 2
,. v1a1 a1Reflection coefficient =7 =
- 2 = a1
cJASI-CLASSICAL - 9
/ 1 2\21 -
- T2
+for T small
Airy Functions
Solutions of the differential equation
9 - ZJ) = 0
are called Airy functions. The solution which vanishes for large positivez is
Ai(z) = ‘cos(. + sz)ds
with the following asymptotic forms for large Izi,
2 3/21
Ai(z) -‘ , e , z > 02v z /
Ai(z)÷ 1 (-z)3”2
+
:7!-], <
A second solution, which diverges for large positive z, is
13 3Bi(z) =
—
+ (+ sz) ds
with the asymptotic forms
2 3/21
Bi(z) -‘ , e , z > 0vcz’
Bi(z)+ 1
cos [- (-z)3”2 + , z <4
Note the factor of ½ that appears in the exponential forms for Ai(z) and Bi(z).
StJARY OF CONNECTION FORfluJAS
“Barrier to left”
raii i
iPdxx
a
B
Idx 3/2wI I
ix
2z=—-p
(2
1
:- pdx =
PdX÷)1Ai(z)____i e1/22 JpJ
V(x) = E - a(x-a) +
2IpI2e ÷ Ai(z)
- 1/2 pdx+
IpI2 Bi(z) ÷11/2
p
‘lf
cos (\W JaiTpdx
+
“Barrier to right”
V(x) = B + (x-b) +
E
11/2
• fisin
b
x
b 1x
W1 pfdx
SUMMARY CONNECTION FORMULAS - 2
1
___
1 1bkx
1/21
fb
_______
)- e(
pdx+
1/2Ii’i
21
1bpdx =
3/2 -P 1 2 3/2(-z)X
(2)2/3 1bjpldx = z
