ELEMENTS of the QUANTUM THEORY* V-A. THE RIGID ROTATOR SAUL DUSHMAN Research Laboratory, General Electric Co., Schenectady, New York TAE SCHROEDINGER EQUATION IN THREE
DIMENSIONS
I
N ORDER to discuss such problems as that of the rotational energy states of a diatomic molecule, or the energy states of a hydrogen-like atom, it is necessary to formulate the S. equation in three d i e n sions. As in the case of the equation in one dimension, we employ as a starting point the equation for the propagation of a wave motion, which in the case of rectangular axes has the form
_
where $ is the amplitude, V2is designated the Laplacian differential operator, and u is the velocity of propagation of the wave. This is a partial differentialequation of the second order, and the complete integral must define $ as a function of the three coordinate variables and of the time, t. As in the solution of equation (29) in chapter 11we assume $(x,y,z,t) = +(x.y,z).r-t-~v~
where vA = u, and consequently - = -~*~~.+(x,Y.z).~-P~~Y~uI atP = -4*ve*.
Hence, 1 .-a v = --4rv us atz X1
In problems involving rotation about an axis of symmetry, or motion of a particle in a central orbit (e. g., the motion of an electron about a positively charged nucleus), it is much more &nvenient to express the Laplacian operator in polar or spherical codrdinates. These are indicated in Figure 22a, where r = O P designates the radius vector, 9 corresponds to the "latitude" and 7 , to the "longitude." The connection between these and the rectanrmlar coijrdinates is piven bv the following relations: z = O N = OMcosn
-
- .
= *+(X.,Y,8)..-2dYt
A'
#
= OPsinBcosn= rsinecosq y=MN=rsintlsinq e = P M = r cos 8,
Since and
(146) (147)
where P M is the perpendicular from the end of the and similar relations apply for b2$/bys and aV/bz2, radius vector to the planeVXY (the equatorial plane). i t is evident that equation (143) may be written in the In terms of spherical coordinates the Laplacian operator is obtained as follows: form Let gi, q2, and qa denote such coordinates (known as generalized coordinates) that it is possible to express an element of length, ds, by a relation of the form: (ds)' =
-
* This is the fifth of a series of articles oresentine a more de-
tailed and extended treatment of the subj&t matter cowred in Dr. Dushman's contribution to the symposium on Modernizing the Course in General Chemistry conducted by the Division of Chemical Education at the eiehtv-eiehth meet in^ of the American Chemical Society. levela and; 6hiG ~eptembeg12. 1934. The author reserves the right to publication in book farm.
adda)'
+ ad&# + add%)'
where al, a,and as are co&cients involving one or Then it. mav be more of the variables QI, 09, - . and 03. shown that
-~
vs+
=
+
-
. - c / a l w a . 31 % 73% a
(148)
It will be observed that since al, @, and aa may each be functions of q,. qz, and m, the order of operations is important. That is, with
daz .-.boy2 a,
where .$ = +(r,6',~). Thus for periodic motions of a particle in three dimensions, equation (145b) with the value of the Laplacian operator given in equation (150) takes the place, in quantum mechanics, of the energy equation used in ordinary mechanics, which is of the form
The operators are non-commuta-
A-
tive, and in deriving the S. equation this is of extreme significance. Now in the case of spherical coordinates i t is seen from an inspection of Figure 22b and Figure 22c that the velocity of a particle in its path, is given by the relation*
(g)' (g)' + rs (&) + =
dB'
I* sin' 8
(2)'
(149a)
This relation involves merely the proposition that the velocity along the element is the resultant of three velocities along three mutually perpendicular directions. Equation (149a) is obviously equivalent to the relation = (dr)%f r2(dB),
+ r2 sinx 8 (an)'
Hence,
*
where the first term expresses the kinetic energy and the second term expresses the potential energy as a function of the coordinate variables. Under certain conditions equation (150) assumes simpler forms. Thus, for conrtant nalue of r (rotation of a sphere about an axis), b/br = 0, and equation (150) becomes
P+
1 = xe.$(sine.%)+m.G
..
:.
*
(152)
'
For the case of motion in a plane, the coordinates may be chosen that a/a6' = 0, sin = '3 and
(149b)
a, = 1; an = r2; a8 = ra sinP 8
G3
and = re sin 6'. Therefore the Laplacian becomes? Figure 22b shows a projection of the meridian plane containing the right-angled triangle OMP, while Figure 22c shows a similar projection of the equatorial plane containing the rightangled triangle ONM. I t is evident that rAe.Ar is the area of the element of PQP'Q'. From this i t follows that rA.9.Ar.r sin 8. An is the element of volume. I n rectangular caerdinates, the element of volume is Ax. A y Az. Hence, in the limit, dxdyds = ra sin 8 dr d8 dn, and it is seen that da,oler = r1 sin 8 is the coefficient by which drdRdq must be multiplied t o convert it into an element of "01ume. t This equation may also be derived directly from (145a) by making use of equations (146) and (147). This method of deriving the equation, which however is quite lengthy, is given in some treatises, e. g., B. WILLIAMSON'S "An elementary treatise on the differentialcalculus."
On the basis of de Broglie's theory, X in equation (145b) represents a "wave-length" defined by the relation
Hence the S. equation is of the form
where U,the potential energy, is a function of r, 8, and q . As in the case of the S. equation for one coordinate variable, we seek solutions of equation (154) that will be physically rational. Thus $7 must not become infinite a t any point in space, and it must tend to vanish as those regions are approached in which the probability of occurrence tends to become zero. The exact form of these "boundary conditions'' must depend upon the nature of the particular problem. Thus, in the case of
a hydrogen-like atom, the probability of occurrence of the electron must decrease continuously to zero as r tends toward infinitely large values. We shall find that actually this probability becomes infinitesimally small for values of r exceeding only a few atomic radii. In the case of the angle variables, the limits are 0 I8 5 r and fl 5 1 15 27r, and the distribution function as we shall designate @& must exhibit some form of periodicity with respect to these variables. . Furthermore, because experimental observations show that any atomic or molecular system can exist only in a discrete series of states defined by the energy values E,, &, . . . and so forth, we must expect, if the solutions of the S. equation correspond to the observations, that "sensible" solutions of equation (154) will exist only for a discrete series of values of the energy E, which will constitute the eigennalues. The corresponding eigenfunctions,@, will represent, in the most general case, amplitudes of stationary de Broglie waves in three dimensions, and cannot therefore be visualized physically. In the case of constant values of r, c$ represents the amplitude of vibration of a spherical surface, and hence the functions are known as surface sfiherical harmonics. They are represented by expressions which are functions of the latitude (0) and longitude ( q ) and which exhibit nudes and loofis along both meridian circles and zonal circles (parallel to the equatorial plane). Consequently, the mathematical expressions are quite complicated and, in fact, appear formidable a t kit glanceTo some. indeed. it rnirht aooear that the mathe" matician has endowed nature with a complexity far beyond its needs. Yet the only reply to such an accusation must be that the "simple" solutions, those which are relatively easy to understand (because they involve no "higher mathematics"), do not correspond to the facts. Nature is complex in its fundamental elements, and the only feature that is astounding is this: that human intelligence has been able to devise a method of reasoning with symbols by which' a one-to-one correspondence is attained between the deductions from this reasoning and the experimental facts. This, to the mind of the writer, has always appeared the most marvelous aspect of all mathematical technic in dealing with the interpretation of nature. And because i t is stimulating to understand this "picture"; because i t must add a certain measure of pleasure to perceive, even though it be dimly, a t k s t , the results attained by combining transcendental imagination with the most exacting type of logicbecause of these rewards which the effort holds fortl-let the reader not he discouraged too readily. Patience and persistence alone will accomplish wonders, even in the comprehension of a symbolic mathematical technic.
form, so that the distance between the centers of the atoms ( r ~ rz) is fixed, and therefore we neglect the possibility that the atoms will vibrate along this axis in virtue of their mutual attractive and repulsive forces. (Of course, such vibrations, of frequency nvo actually occur and give rise to vibrational energy states a problem which was considered in the case of the harmonic oscillator.) Under these conditions we may regard the molecule as possessing, in general, two degrees of freedom or mobility. The molecule will have a rotational motion about an axis of symmetry passing through its center of gravity. This will be represented by an angular velocity ? = drl/dt in the plane YOX (see Figure 22a). Also there will be a precessional motion of the fixed axis of the molecule about the axis of symmetry, which is represented by the angular velocity, 8. Since there is no potential energy term, the total energy is all kinetic and is given by
+
Since the molecule is rotating about its center of gravity, Ihll = !M*.
Hence, if we put
. A
THE RIGID ROTATOR
Let us consider the problem of a diatomic molecule constituted of two atoms of masses, PI and m, located a t distances, rl and re, from the axis of rotation. We shall assume that the molecule is of the "dumb-bell"
and we can write (155a) in the form
where I = moment of inertia of the molecule about its center of gravity, ro = mean radius of gyration, and l/,., = 111~1 f 1/k. where ji is known as the "reduced" mass. Thus ro2(b2 sin28.& = v2, whereiv is the velocity of rotation, and the corresponding de Broglie wave-length is given by
+
Hence, the S. equation
-
becomes
v s + + 8raEI+ - h= *a, ~ ~ Since ro is a constant for the molecule, we may use the form of the Laplacian operator given in (152) with rz = ro2. Hence, multiplying both terms by ro2the equation to he solved is
There are two cases which may occur. In the first
of these, the molecule is free to revolve only about an axis a t right angles to the axis of the molecule. This is known as the case of the rigid rotator with fixed axis. In the second case; the molecule may exhibit a motion of precession, as well as that of rotation. The latter is the case for which equation (156) applies and will be discussed in a following section. In the k t case, however, b/b0 = 0, sin 6' = 1, and the equation reduces to the ordinary differential equation
where m2 = 8r2El/h2 is used to indicate that the coefficient is always positive. The solution of this equation has been discussed previously. It is
+
= A~~~
+BG-~~V
(15%)
which may he written in the form of a sine or cosine function. Thus we may use the form
+ = C sin (mq + 6)
(1585)
where 6 is a phase angle. Now this equation has physical significance only if m = 0, 1, 2, etc. Consequently, theeigenvalues of the discrete energy states are given by the relation
artet"). Physically this is interpreted, in the present case, as indicating that actually there are two energy states which have become merged (degenerated) into what appears to he one state, because the energy is the same irrespective of the direction of rotation of any molecule with respect to other molecules. However, if the molecules are placed in a magnetic field, the energy will vary (because the molecules possess magnetic moments) with the direction of rotation of the molecule. For one direction of rotation the energy will be slightly greater, and for the opposite direction slightly less than the value Emwhich exists in absence of a magnetic field. We shall now consider the case of the rigid rotator with free axis, for which the S. equation is that given in (156). The solution must represent 4 as +(B,q), that is, as a function of the two angle variables. As in all cases of partial differential equations we attempt to separate the equation into two ordinary differential equations. That is, we postulate a solution for 4 of the form d(0.d = X ( E ) . Z h )
(161)
where X is a function of 6' only, and Z of q only. We have an indication that this is possible from the fact that in the case of the rotator with fixed axis we have already found a limiting case of this problem. Evidently
where m is an arbitrary integer. The corresponding eigenfunctions are given by either &, = Arimn
and
since m may be either positive or negative. The reason for not using the solution given in (158a) is the requirement that
+
and its derivatives in equation Substituting for (156) and using the symbol a ' = 8rr~~r/h',
we obtain the relation
(162)
:,
Substituting for 4, from either of the epuations (160), it is seen that
6
That is, the normalizing factor is irrespective of and + ,, the value of m. Obviously, the functions n Z m, are orthogonal, since
+,
where Z = Z(?) and X = X(6'). Since sin2B/X,Znever becomes infinite, we can multiply through by this factor and thus obtain the equation
It will be observed that the left-hand side does not involve q, and the right-hand side does not involve 6'. Since this equality must be valid for all possible values of 0 and q, it follows that each side of the equation must be equal to a constant, which we shall designate by ma. We thus obtain the two ordinary differential equations
and as shown in Chapter 111, each of these integrals is identically equal t o zero. It will be observed that in this case there are two eigenfunctions, even by equations (160), corresponding to any given eigenvalue, as defined by (159). We have here an illustration of a condition that is met with frequently in the solution of problems in quantum mechanics. Such energy states, for which there are and available more than one eigenfunction for any given eigenvalue, are known as degenerate (German, "ent-
The first of these has been solved already. The solutions are given by
where m = 0, 1, 2, etc. Now let us consider equation (164), and as a first step in the process of solving it we change to a variable x, such that x = cos 0
Therefore and
(166)
1
- xa = sina0
d
d
and in consequence of (1686) each power of x must vanish. Hence,
Thus the ratio
If k can increase beyond limit, ah + z/ak = 1 for very large values of k, and consequently, if the series for X is to converge for x = * 1, the highest power of x must be given by k; therefore ak + = 0, whence
,
2 = -sine.- d r Before carrying through the transformation to the new variable, we may divide through by sin2& This gives
k(k
+ 1) = 'a
*I
EI =W -
(170)
Since hZ
i t follows that E can assume only the discrete series of values given by the relation
Introducing the variable x, this becomes
This equation is one of 'the most important in mathematical physics and is known as Legendre's equation of order m, in x, where - 1 S x S 1. That is, the equation has physical significance between the limits x = cos 0 = * 1. These limits constitute so-called singular points, since 1 - x2 = 0 a t these points. Since m can have any integral value, including 0, we shall consider first the solution of equation (167) for the case m = 0, that is, the Legendre equation of order zero,
where k = 0, 1, 2, etc. This relation is different from that deduced for the case of the rotator with rigid axis which was stated in equation (159), and is actually in much better agreement with the spectroscopic observations on the rotational energy levels of molecules, than the relation given in equation (159), which is identical with that derived by means of classical mechanics.
We must now consider the form of the eigenfunction Xk corresponding to each of the eigenvalues, Ea, as defined by (171). From (169) it follows that
that is, dPX
dX
(I - x ~ dz' ) - - - ~ z dx . - + ~ ~ X = O
(1685)
Let us assume, as in the case of equ&on (114) for the linear oscillator, that X may be represented as a polynomial of degree k, so that Similarly, it is readily shown that where k = 0 , 1, 2 , .
Then
2 -2x
.. .(k - I ) , k.
= Zai.k.&-1
dX .ax = - Z2at.k.s'
and
Also, in terms of coefficients of 2.
Therefore, the coefficient of 2 in (168b) is given by ar+z(k 2)h 1) - ar(k(k - 1) Zk - a']
+
+
+
* W. E. BYERLY,"Fourier's series and spherical harmonics," Ginn and Co., Boston. T. M. M~cRossnT."Soherical harmonics."E. P. Dutton and Co., New York. D. HnaapHn~y,"Advanced mathematics" (see references in Chaoter 11). ~ h e s ;Gee e treatises will be found extremely useful for further consultation on the differential equations dealt with in this and the following chapter.
SEPTEMBER, 1935
441
If k is even, the power series beginning with 2 will end with ao, if k is odd, with alx. We thus obtain the series,
where the coefficients of the various powers of x are given by the relations (172), and an is arbitrary. If we assign the value* QX
=
(2k
- 1)(2k - 3) .. . 1 k!
the resulting function X is known as a Legendre coe&cient of order zero and degree k or a surface zonal harmonic and designated by the symbol Pk(x) = Pa(cos 8). Thus the complete expression for the Legendre coefficients has the form Pdx) =
(2k
- 1)(2k - 3) .. . 1 k! k(k
-
Px(-2) = (-1)k
PA(Z)
i t follows that for k even, Pa(-%) = Pa(x), that is, the function is symmetrical about x = 0 or 8 = r/2, resembling in this respect sin 8, while fork odd, Pk(-X) = -Pk(x), and the functions change sign in passing through x = 0 or 8 = r/2, which also is characteristic of cos 8. ,,o
k(k - 1) .xr-2 - 2(2k - 1)
- 2)(k - 3) , X1-4
+ 2.4.(2k - 1)(2k - 3)
sponding limits are 1and - 1. Figure 23 and Figure 24 (taken from the curves plotted in Byerly's book) show graphs of the functions PI(cos 8) to P7(cos 8), inclusive, as functions of 8. It will be observed that a t the limits all the functions have the identical absolute value 1. Furthermore, since
-
,,
,]
(173)
The !%st few members of this series are as follows:t Po@) = 1 P&) = x 1 P,(x) = (3x2 - 1) 2 1 Pa(,) = - (5ra - 32) 2 1 P,(z) = - (352' - 302' 8 1 Ps(x) = (63r6 - 70x5
-
or PO( c o s ~ ) . =1 P,(COS8) = cos a 1 or P,(cos 8) = (3 ens28 - 1) 2 or P~(COS 8) = -1 (5 C O S ~a - 3 cos 8) 2
-
+ 3) + 15%)
FIGURE
24.-PLoTS
i
i
F r c w g 23.-PLOTS
OF
RE
LEGENDREPOLYNOMIALS, PcP7. INCLUSIVE
As pointed out in connection with the Hermitian polynomials, the number of roots, corresponding to nodes, along the axis of 2 or 8, is equal .to that of the highest power of x. Thus PI(%) = cos'8 passes through 0 a t 8 = r/2, while Pz(x) exhibits two nodes which may be determined from the quadratic relation, Pdx) =
-0.5-
OF
POLYFIRSTFOURLEGENDRE
oma as AS F U N ~ ~ I O08 N Se
At 8 = 0, cos 8 = x = 1,while a t 8 = n, cos 8 = -1, and a t 8 = n/2, cos 8 = x = 0. The functions may be plotted as functions of either 8 or x. In the former case, the limits are 0 and n, in the latter case the corre-
-
*This value, as will appear from (174), makes the first Legendre function, P, (*.) = 1. t Tables of values of Pr t o PT(inclusive), as functions of 8, are .piven in L. S ~ ~ ~ ~ ~ s ~ ~ ~ ~ ' s " M a tables" t b e m and a t iin c aanl appen.dix in BYERLY'S "Fourier's series."
;(A+(61) = 0: ..
1)
HencePz(x) = Oforcos8 = *I/&= *0.$'75, that is, for 81 = 54'45' and Oz = 125"15". In a similar manner it is possible to calculate the k values of 8 a t which any of the functions Pk(cos 8) becomes equal to zero. As in the case of the other polynomials, it is readily shown that the Legendre polynomials -are related by a recurrency formula of the form (k
+ l)Pk+l(x) = (2k + l)xP&c) - kPk-l(x)
(175)
which makes it possible to calculate higher members of the series from the lower members. F~~~equation (173) it follows that (2k - 1)(2k - 3) (k I)!
+
... 1
+
['
&+,
- (k + l)k.xk-I 2(2k - 1)
(k lMk - 2) + 2.4,(2k - 1)(2k - 3)
and if this operation be repeated k result is
-
-
..
.]
times more, the
.
.
- 1)(2k - 3 ) . . .
(2k
1
(Zk)!
=
2k@k - 1)
['%
- Z ( 2 k - 1)
.+-'+
Integrating the right-hand side by parts, it becomes
2k(2k - 1)(2k - 2)(2k - 3 ) .xrx-, 2.4.(2k - 1)(2k - 3 )
. . .]
-1 )= 1 Zkk!
(176)
1 2 k 2 -)
-1 )
A
,, ,
as may be demonstrated by expanding the expression (x2 - 1 The ktb derivative of the function in ( 1 7 6 ) ' is therefore P a ( x ) , so that Pdx) =
dk(xa d+
1
2q .
dPr dP, %3.dx
-fl('-
dz
.dx
+f
(1
1
P - x')-'.ddx
dP 2. & ' dr
which is equal to zero, since 1 - x2 = 0 a t x = * 1, while the two integrals cancel. Consequently, if m +
k,
(177) ASSOCIATED LEGENDRE FUNCTIONS
This is known as Rodrigues' Formula. It may be shown by use Of this following relation is valid: f 1
~ b ( x ) ~ , ( r ) d= x 0. (k
z
that the
m)
(1781
We shall now consider equation ( 1 6 7 ) . In the preceding discussion this equation was solved for the case m = 0, and i t was found that the equation gives solutions which are physically significant if we put a2 =
That is, the Legendre polynomials form an orlhogonal system. This conclusion also follows, of course, from the deduction (see Appendix I, Chapter IV) that the different solutions of any S. equation form an orthogonal system. Furthermore, since, as may be deduced from ( 1 7 7 ) ,
J:l
2 {pr(X)/
the normalizing factor is
d ;i;{(l
+1
It follows from these relations that if m and m # k,
2F
=
+ k is even
ma
=
- G p / ~
=
(1
- x2)m/zY ( 1 -xZ)(m/z) + 1 .d-Y , - xl) + 1 -x ) dx ' dX dY ( 1 - xl)- - - ~ ( l ~ 3 m / 2 Y+ ( l - ~ ~ ( m / + z )1 .F dx z .
..
- 3
mzxa(l - xa)m/2
and also that
dY
,
1 2k+1
,.
The orthogonality relation (1781may also be derived by the following proof which is independent of Rodrigues' formula, and is of very general application in spherical harmonics. Since Pa(%) and P,(x) satisfy Legendre's equation, d dx d {(I
dx
+ k(k + 1)Px = 0
%/ +
- XZ)
+ l)P,
(,w
= 0
J:l~m
+ 1 ) - k(k + 1 ) 1-f 1 ~ 2 & dP &[(I - 29 $1 dx - f l p i
-
=
&[(I
- x2) d%]dx.
BYRRLY, chapter V; MACROBERT, chapter V.
Y' + ( 1 - x y m / Z ) + d'dx'
Substituting these relations into ( 1 8 1 ) and dividing by ( 1 - x ~ ) " / ~which , is not zero, except a t the limits, we: obtain the relation (1
-
d a~Y
- 2 (3m + l ) x~, ddxY +~( A - m)(k + m
+ 1 ) Y = 0. (182).
(i) (ii)
(i) byi p,~ andl (ii) and ~ subtracting ~ ~ l ~ ~ byipa ~ and then integrating, we obtain the relation, lm(m
(181)
- X+Y.
P ~ ( X ) P , ( Z ) ~= X f1 pl(x)p,(x)dx = 0
and
0
-m(l (1
and
g
+ { k ( k + 1)
-r2)$/
x
Then 2k
(170)
Let us introduce a new function, Y,d e h e d by the relation
dX (180)
+ 1)
If we substitute for a2 in equation ( 1 6 7 ) we obtain the
(179)
=
k(k
With this dzerential equation we shall compare the differential equation derived from ( 1 6 8 a ) or (168b) b y putting 2 = k(k 1 ) a n d X = P k ( x ) , since this func-~ tion satisfies the equation. The result is
+
2 + k(k + 1)Pt
dzPx ( 1 - 23- 2 x . dP
=
0
(183).
If we now differentiate this m times, it can be shown~ that the resulting equation is identical with ( 1 8 2 ) and. leads to the result y=
(1
- X+/B
=
d"P .k dzm
(184)
The function X thus obtained is known as the associated Legendre Function of degree k and order m, of thefcrst kind, and is designated thus
Since the differential coefficient becomes zero for m > k, it follows that m can have only the series of integral values m = 0, 1,2, . . . k. Thus corresponding to any value of k, there are (k 1) Legendre functions which satisfy equation (167) and also (2k 1) fuuctions, Z,, which satisfy equation (163), corresponding to m = 0, *l, *2, ... * k . Since
+
+
lar to that given in the previous case. As Condon and Morse describe it: "this is the degeneracy of random space orientation in a centrally symmetric field, and gives the multiplicity into which the terms are split when a non-symmetric perturbing field removes the degeneracy." Thus in the presence of magnetic or electrostatic fields this degeneracy may be completely removed, because such fields introduce perturbing effects. In a subsequent chapter these phenomena will he discussed more fully. The functions PT(x) satisfy the condition for orthogonality of the form
(1 - x*)m/2 = sin"8
the associated Legendre function may also be written in the form* p;(x)
= sin*
.dmPs(%) d?P
(186)
The functions
where obviously m must not exceed either j or k. This may be deduced by an argument quite similar to that used for demonstrating the orthogonal nature of the Legendre polynomials of zero order. It may also be shown that
Hence the normalizing factor for the tesseral harmonic given in equations (187a) or (187b)i s given by*
are known as Tesseral Harmonics of the m-th order and k-th degree. In terms of exponentials, the functions are
and as stated already, for any given value of k, there are 1)functions which satisfy the differential epuatwns (2k (163) and (164). It will be observed that the only condition attached to m in the solution of (163) was that it must have an integral value (including 0). The condition that m cannot exceed k was derived from the subsequent deduction based on the fact that the eigenfunctions which satisfy equation (164) are of the type PT ( 4 . It was also deduced that the eigenvalues+mrresponding to the different energy states are given by
+
Thus it follows that for any given energy state, corre1) eigensfionding to a given value of k, there are (2k functwm. In the case of the rigid rotator with fixed axis, i t was found that for each energy state there are two possible eigenfunctions. The order of degeneracy is therefore two in that case. But in the c u e of the rigid rotator with free axis we find that the order of degeneracy is 2k 1. The physical interpretation is simi-
+
+
*The notation given is that used by Byerly, MacRobert, and mmt of the authorities. On the other hand, Courant and Hilbert, as well as Condon and Morse. use the notation,
Finally, i t is of interest to give the expressions for some of the associated Legendre functions correspouding to the expressions for the functions of zero order which were given in equations (174). The first ten of these polynomials (including the f g ~ t i o u sof zero order) are as follows: ..
where x = cos 8. Since dkPk(x)/d2 = C,a constant, it follows that the functions P:(x), for which the order and degree are identical, possess no nodes, but pass through a maximum a t x = 0 ( i . e., 8 = r/2) as is evident from the relation
Figure 25t shows plots of the four normalized func-
* It is becoming customary in treatises on quantum mechanics to designate the sqvarc of the expressions on the right-hand side of (180) and (190) or the inlegrals in (179) and (189) by N. For instance, H. BETHE,in "Handbuch der Physik." Vol. 24, Part 1. uses this notation, which may give rise to some confusion for the reader who consults that discussion. t CONDONAND MORSE, p. 55.
tions P:(cos B), P:(cos 8), Pi(cos 8), and Pi(cos 8) as functions of 8. The normalization factors, as deduced from the equation (190) are N = 2/24/7, 2/240/7, and respectively. The actual relations for the four mrmalized functions, and the designations for the corresponding curves, are Fa = PO.(cos 8) = 0.936 (5 cos' 8 - 3 cos 8) F,. = Pi (cos 8) = 0.810 sin 8 (5 cos" - 1) F- = P: (cos 8) = 2.563 skia 9 .cos B Fat= P: (cos 8 ) = 1.046 sina 8
2.0 1.5
4%
/. 0 0.5
%L
0 -0.5
- /. 0 -1.5
-2.0
0'
30'
60D 90'
120'
150' /BO'
In each case the nodal points were obtained by solving FIW, PLOTS mE NORMALIZED LEGENDRE FUNCTIONS the equations PF(cos 8) = 0, while the points of maximum values were obtained by solving the equations . d e ( c o s 8)/d8 = 0, where m = 0, 1, 2, or 3.
