ELEMENTS of the QUANTUM THEORY* V-B. THE RIGID ROTATOR (CONCLUDED)

SAUL DUSHMAN

Research Laboratory, General Electric Co., Schenectady, New York

GEOiMETRICAL INTERPRETATION OF SURFACE

SPHERICAL HARMONICS?

L ET US consider now the geometrical interpreta- tion of the Legendre functions and tesseral har- monics which have been discussed in the pre-

vious sections. The function

(A cos mrl + B sin mq)PT(cos 9) (192)

represents a surface spherical harmonic of kth degree and mth order. If m = 0, the function has the form P a (cos O), which is Legendre's coefficient of the first kind, of degree k. This function is a polynomial of degree k and therefore has k distinct zero points be- tween cos 0 = -1 and cos 0 = +l. As shown in the curves in Figure 23 and Figure 24 these "nodes" are arranged symmetrically about cos 0 = 0, i. e., 0 = a/2. Hence, on a sphere with the origin as center, the fuuc-. tion P a (cos 0) becomes 0 on k different circles, which as shown in Figure 26 correspond to different degrees of

-6 and mu = a-6. On a sphere, as shown in Figure 27, this corresponds to m great circles through the pole 0 = 0 (circles of "longitude"), distributed symmetri- cally, so that the angle between the planes of any two consecutive circles is equal to r/m.

The factor sin" 0 is equal to zero only a t 9 = 0 and 0 = P. The differential coefficient dmPa(x)/dxm is represented by a function which is the mth derivative of a polynomial of degree k. Thus the highest power of x bas the value k-m, and the function has k-m

6=0 a I

8=n FIGURE 26.-GEOMETRICAL ILLUSTRITION OF ZONAL HARMONICS

"latitude," &at is, they poles at 0 = 0 and 0 = a. These circles are symmetrical with respect to the "equatorial" circle, and if k is odd, the latter is one of the set of circles for which Pa (cos 0) = 0. Further- more, as shown by the plots of the functions, since the value of any function PR (COS 0) exhibits k- 1 loops, there are 2(k- 1) circles parallel to the nodal circles a t which the function has the same absolute value. I t is . for this reason that the Lecrendre coefficients of zero order are known as Zonal ~a;monics. The point 0 = o

, ~ ~ ~ ~ ~ T ~ ~ k - ~ ~ ~ ~ ~ ~ ~ ~ FIGURE 28:-GEOMETRICAL

ILLUSTRATION OF SECTORIAL is desimated the Pole, and the diameter t8rough the HARMONICS pole, the Axis of the zonal harmonic.

For greater than and less than k, the functions zeros on circles with 0 = 0 as pole, which are arranged

are represented by the expression, like the corresponding circles in the case of the zonal harmonics. Since the two sets of circles intersect

dmPk(cos 8) ( A cos m? + B sin mq) sinm8. orthogonally, these harmonics are designated Tesseral

d"'(cos 8) Harmoniss. Fiaure 27 shows the model sauares corre- .*

This may be written in the form sponding to these harmonics. For m = k, the differential coefficient becomes a

-2sin(mv + 6).sinm o . ~ ~ ~ ~ ' ~ ~ ~ ') (193) constant factor, atid the spherical harmonic is of the dm(cos 8) form

where A = sin 6, and B = cos 6. It vanishes for m? = d m . sin(kq + 8)sinV (191)

* T h i s is the sixth of a series of articles presenting a more AS pointed out already this vanishes on k great detailed and extended treatment of the subject matter covered circles passing through the points 0 = 0 and 0 = a, the in Dr. Dushman's contrihutian to the symposium on Moderniz- ing the Course in General Chemistry conducted by the Division between the of any two consecutive of Chemical ducati ion at the eighty-eighth meeting of the being a/k. Since the sphere is thus divided up into American Chemical Society, Cleveland, Ohlo, September 12, 2k sectors, as shown in Figure 28, these functions are 1934. The author reserves the right to publication in book form.

t This section is taken to a great extent from the discussion in known as Sestorial T. M. MACRORERT'S "Spherical harmonics," p p 131-2. For any given value of 7, equation (104) is the same

485

as (191), and it is evident that as the value of k is increased, the function tends to assume appreciable values in an increasingly narrower region symmetrical about the equatorial plane. The interpretation of this result from a physical point of view is considered in the following section.

THE PHYSICAL SIGNIFICANCE OF THE CHARACTERISTIC PUNCTIONS

We may now consider the significance of the some- what tedious calculations and seemingly complicated results that have been derived in the previous sections.

The problem to be solved is the following. Given a diatomic molecule, what will be the possible energy states and modes of rotation for such a molecule? The problem first originated because of the observa- tions on the temperature variation of the specific heat of diatomic gases. In order to account for the increase in specific heat with temperature, it was found neces- sary to assume that in addition to kinetic energy of translational motion, diatomic molecules also possess an energy due to rotation about an axis of symmetry pass- ing through the center of gravity.

On the basis of ordinary mechanical concepts, the method of determining the energy states of such a system for the case of rotation in a plane, was as fol- lows :

Refemng to equation (155) it is evident that for ro- tation in a plane, e = dO/dt = 0, and sin 0 = 1. Hence, this equation becomes

which shows that ?j = d?/dt = a constant, since ro is a constant of the molecule. Hence, the angular velocity is constant; that is, the molecule rotates with constant angular velocity about an axis perpendicular to the line joining the two atoms.

The angular momentum is derived by means of the definition which states that the momentum correspond- ing to the motion with respect t$ any coordinate, g, is given by f i = b E / b i , where E is in general a function of two or more coordinates and the corresponding ve-

P locities. Thus, in the case where E = 5(w: + w; + v;),

p. = b E / b , = P%

Hence it follows that

In order to, determine the discrete series of values of the energy states the so-called Wilson-Sommerfeld quantizing condition was introduced, according to which

that is,

For the mse of rotation in space (the case desig- nated as the rigid rotator with free axis) the classical theory also led to the result expressed in the last equa- tion, since the mass p must always describe an orbit which consists of a meat circle on the surface of a sphere . of radius ro.

But observations on band soectra. in which the lines constitutine individual bands are due to transition - between states differing in amounts of rotational en- ergy, showed that this result was not quite satisfac- tory. On the basis of the S. equation, as shown in equation (159) the same result is deduced, if i t is as- sumed that the molecule is capable of rotation in only one plane. The result deduced in equation (171) is, however, in very good agreement with observations on band spectra. Hence, we conclude that diatomic molecules possess two modes of motion about their center of gravity, one in which there is a rotation in the plane containing the axis of the molecule, about an axis of symmetry a t right angles to this plane, and an- other which corresponds to a precession of the axis of the molecule about the axis of symmetry.

Now it is the essence of the S. equation that it starts with this Physical model and then, instead of discussing the consequences to be deduced from this model by classical mechanics, considers a partial differential equation which is derived from the physical model by a definite mathematical procedure, and which repre- sents mathematically the propagation of a wave motion. We abandon, as it were, the concrete, tangible model of a dumb-bell shaped mass rotating about an axis and consider instead the nature and properties of certain wave patterns obtained by solving the partial differen- tial equation.

In the previous sections it b s found that correspond- ing to each energy state of quantum number k, as de- fined by equation (171), there are 2k+l characteristic functions which represent 2k + 1 different possible modes of vibration. The next question to be consid- ered is this. What is the physical interpretation of these functions which we recognized as Legendre poly- nomials of the first kind?

As in the case of the characteristic functions, 4 ("amplitude" functions), deduced in solving the prob- lem of the linear harmonic oscillator, we assign physical interpretations to 43 or @. In the case of the rigid rotator in three dimensions, the function 4 has the value

1 . p; (x )dx -.c~m~.- ( x = cos 8) & N '

where N is the normalizing factor for the particular Le~endre polynomial. This is determined from the where m = 0, 1, 2, 3, etc.

'

It follows from (193, since fi, is a constant, that, regtion -

A m . . . . . . IPXcos 0) J 2 sin 8 ..... A,x . P:(cos 8) 1% sin 8 ..... A 2 ~ . (P;(cos 0) l a sin 8 . . . . Ass.. ( P : ( c o s 0) I * sin 8

For comparison there has also been plotted, as curve Aoo in Figure 30, the function

( l / N 3 (Po(cos 0) )%sin 0 = ( 1 / 2 ) sin 8.

Since

1 ( 1 / 2 ) sin B ~ O = 1

and the total area under each of the curves shown in Figure 29 and Figure 30 is eqwl to 1 , the average ordi- nate is given by l/?r = 0.318. This has been indicated by the straight line BB in the two figures.

II

- 6 10

F I ~ ~ E %.-PLOTS OF TEE ZONAL DISTRIBUTION FUNCTIONS C o n n ~ s ~ o m r ~ o TO k = 3 .S

.8

( 1 1 ~ ~ ) f - 1 (px ( z ) ) z ~ % = - ( I / N ) s f (Pr(cos 8) )"Sin 0d0 =

.7

& t ( ~ ? ( ~ ~ ~ 0 ) ) W n 0d8 = 1 (197)

Hence (1/N)2[Pr(cos 8)12 sin fld9 is regarded as representing the probability of locating the particle in the region on the surface of a sphere lying between the zonal circles 0 and fl + dfl, while (1/2a)~'"~~'"" d q = .4 (1/2a)dq, represents the probability of occurrence in the meridian section located between the angles q and ,3 q + dq. Evidently this probability is independent of 9. .2

From equation (149b) i t follows that the element of area, dA, on the surface of a sphere of unit radius is given by

d.4 = sin 8d8dq * 0 , f / o 0

and the probability of locating the particle in this area - 0 a t the angles 9 and q is FIGURE 30.-P~o~s oa THE ZONAL DISTRIBUTION FUNCTIONS

CORRE~PONDINO TO k = 3

1 P d A = m ( P , (COS 0 ) )*sin 8dOdq (198)

1f now we compare two zones of equal widths, a t the angles 91 and 82, it is evident that the areas of the two

1 Hence P = -- (PF (cos 8) 1 corresponds to a prob- zones will be 2 s sin fl,dfl and 2a sin &dB, where dB is the

2?rNz width of each zone, and the radius of the sphere is ability per unit area or "probability-density." taken as unity. Hence the relative values of the prob-

From values of the normalized polynomials as func- ability-density as given by P in equation .(198), and tions of 8, such as illustrated in the plots shown in Fig- plotted in Figure 31 and Figure 32, are quite different ure 25, it is possible to calculate both P and P sin 0, from the values P sin fl shown in Figure 29 and Figure and results obtained in this manner are illustrated by 30. In the former, the distance from the center to any the plots shown in Figures 29, 30, 31, and 32. The point on the curve gives the relative value of P at first two figures give values of P sin 9. The designa- the corresponding value of 9. In Figure 31 the func- tions on the curves and the corresponding aormlized tion [P,O(cos 0) ]Z/N2 has been plottcd and should be functions are as follows: compared with the curve Aoa in Figure 29, while the

plot in Figure 32 which corre- sponds to (Pi (cos 8) ) %/NZ is to be compared with thecurve Aa3 in Figure 30.

Intermsof the modelof adi- atomic molecule these curves indicate that the axis of the molecule will tend to be oriented with respect to the axis of symmetry in those di- rections for which P is a maxi- mum. This interpretation is most readily evident from the plot Als in Figure 30 and the corresponding plot in Figure 32.. In this case there is a relativelynarrowregion about the value 8 = s /2 for which p is a maximum. As the value of k is increased (keep- ing m = k), the width of this region decreases rapidly. That is, in the rotational states of higher energy con- tent, the molecule will. tend more and more to rotate

Hence, in terms of the angles and corresponding angular momenta, the so-called Hamiltonian form of expression for the energy, E, becomes

If the same rule were followed as that used for con- verting an expression for E in terms of rectangular co- ordinates and their corresponding momenta into a S. equation (see Chapter 11), the resulting differential equation would be of the form

that is,

which obviously is not identical with equation (l56), and is not the correct form of S. equation to represent the particular problem.

FIGURE 31.-PROBABILITY DENSITY FUNCTION COR- RESPONDLNG TO ASSOCIATED LEOENDRE FUNCTION, Pz (cos 8)

about an axis of symmetry at right angles to the axis of the molecule.

ANGULAR MOMENTUM FOR MOTION OF ROTATOR F~cune 32.-PROBABILITY DENSITY FUNCTION CORRESPOND-

In Chapter 11, it was shown that in case of motion ING TO ASSOCIATED LEGENDRE FUNCTION, P: (COS R)

along a coordinate x, the corresponding momentum is obtained by solving the equation We will now try to deducea ~ l e for calculating the

h dd angular momenta with resiiect to 8 and 7 for the case of p a = - . - 2ri d z (199) a rotating body, on the basis of wave mechanics. As a

first step we consider the relation which exists according If the result of performing the operation on the to ordinary mechanics between the angular momentun

right-hand side of this equation is of the form a+, with respect to the z-axis (the axis from which 8 is where or is a constant, the conelusion is drawn that this measured) and the linear momenta, p, and p,, with re- will be the value of the momentyn observed in any spect to the x- and y-axes, respectively. In order to experiment arranged for this purpose when the particle simplify the calculation: we shall assume that the mo- is in the state designated by the eigenfunction +. tion occurs in the XOY plane only (see Figure 22a), so

In the case of angular momenta equation (199) is not that = 0 and 8 = n/2. applicable, as appears from the following consideration. In that case we have the following relations, In the case of the rigid rotator in space, the equation

= , cos n ; = -, sin + cosrl dr for the kinetic energy in terms of the angles 8 and 7

(i)

and their corresponding angular velocities is given, ac- Y = ' Sin ?; = ' cos vd? + dr (ii)

cording to (155b), by Hence

E = &(p + sinv.$3_ r d y - ydz = (xr cos n + yr sin n)dv + (z sin 7 - y cos ?)dr 2 = radrl (iit")

~~t the angular momenta are given, in accordance since x2 + y2 = r2, and the coefficient of dr is equal to

with equation (195), by zero. But

aE p - - = p& = I i - b9 pr2dil/dt = praq = M,,

and where Ma denotes the angular momentum with respect to the z-axis. Therefore we have the relation

" bE ; p, = - = #r; sinab.li = I sinPB .li

bq Mz = P(XY - YX) = zA - YP=

Now in deriving the S. equation from the expression circumstances it is most convenient to calculate the for the energy in terms of rectangular coordinates and square of the resultant angular momentum vector, M2, associated momenta, we set which is defined thus:

h a p h a M' = M: + Mi + M: = - Zri A = 2z ' G'

and as an operator, this is defined by the relation We therefore conclude that in quantum mechanics

we are justified in assuming that M, may be used as an operator, which is defined by the relation

(4 Introducing spherical coordinates, and proceeding as

us assume that we have any function which in the case of the single variable, q, it may be shown that is a function of the coijrdinates q and r, or of the co- the differential equation for the determination of MZ ordinates x and y. We have the following relations of the form, between the differential coefficients:

a+ I E - a ~ - - dx &F dy 4n2 ~ [ ~ . b ( s i n e . z ) + m . ~ ] + ~ z + = ~ sm @ a9 (20%) a? ax dn ay dn

and That is, MZ as an operator has the form aF - =?!.dr+E.d_y. dr 3% dr ay ddr h4 1 a ?I 1 a

M¶ = - 4r2 -[,.-.(sin@s) s m 0 a 9 +-.- h2

stn' 8 ?Inz] = - G Ti. * Substituting from (i) and (ii) it follows that where 0 is an operator of the same type as the La-

a -=x. - -9 . - =

( ) placian operator VZ. In fact equation (150) may be 3s ay 3% written in the form

Hence we conclude, by comparing (iv) with (v), that as an operator

h a M ' - 2ui bn (201) Equation (203a) may therefore be written in the form

Thus, in the case of the rigid rotator with fixed axis, h' M* + - *)+ = 0 the normalized function, as given by equation (165) is ( 4nP

(2036)

Z*m = *&/+ Similarly equation (156) may be written as

Therefore, in order to determine whether the angular hz momentum has a definite value, we consider the equa- ( M I + *)!, =- D

That is, M, operating on the function Z, yields as a result a constant multiplied by 2. Hence, we conclude that an experiment arranged to determi5e the magni- tude and sign of the angular momentum would lead to a value *mh/(2r) , depending on the relation between the direction of rotation and that of the perturbing field. In this case the result deduced by the operator method is identical with that deduced by ordinary mechanics, when the quantizing principle, equation (196), is intro- duced. As stated previously, the observations on band spectra show that it is not correct to treat the problem of a rotating molecule as one in only two di- mensions. I t is therefore necessary to determine what the form of the operator must be for the case of a rotator in three dimensions.

This case is somewhat more complicated because, as is evident from equation (200), the angular momentum terms for 6 and q do not enter into the expression for E in the same manner. I t turns out* that under these -

*This is dixussed more fully by E. C. KEDIBLE, Phys. RN., Supplement, 1, 157 (1929); also see J. FRENKEL'S "Wellenme- chanik" pp. 24&53 (1929 edition).

The solution of the latter, as shown already, is

Hence the solution of (203b) must.also be the same. That is,

M.II = k(k + l ) h 2 4r2

(205a)

while, as deduced in equation (202)

Equation (205a) leads to the conclusion that the total angular momentum of the rotating system may be desig-

h nated by a vector M, whose magnitude is 2a -.\/k( k+ 1)

-

and that the comfionent of this vector along the z-axis is given by mh/2a. In terms of a unit vector of magni- tude h/ (2r ) , the total angular momentum and the com- ponent of this vector along the z-axis are therefore 4 ' ) and m , respectively.

These results are specially significant in connection with the problem of the hydrogen atom, which is dis- cussed in the following chapter.

APPENDIX I11 are known as Fourier's series. The nossibilitv of ex-

EXPANSION OF AN ARBITRARY FUNCTION IN TERMS OF AN

ORTHOGONAL SYSTEM OF FUNCTIONS

In subsequent discussions we shall have occasion to make use of the very important property of orthogonal normalized functions which is expressed in the form

where 4, and $, are any two eigenfunctions of the system, N is the normalizing factor, dv is the element of volume, area, or length, and the integration is ex- tended over the whole region in which the functions are applicable.

As has been mentioned previously, the simplest type of normalized orthogonal expressions are the trigo- nometric functions

(I/&) sin m0 and (l/v%) cos n0,

for which the limits are 0 and 2r, so that 2 r

cos me cas nod0 =

pressing any arbitrary function of 0 in terms of such a series may be illustrated by the following examples.

(1) It is required to develop the function f(0) = 0 as a sine series for the region 0 = 0 to 0 = a. We have

E = ol sin 8 + ar sin 28 + . . . . . + a . sin nE + . . . Let us multiply each side of this equation by sin nod0

and integrate between 0 and a. Then

JwOsin nod0 = a l l s i n Esin nEd0 + . . . . + a , k r sinP ned0 + . . . Because of equation (i) all the terms on the right-

hand side, except the one involving sinznO, vanish. Hence we obtain a relation for determining a,, which is of the form

Now d(R cos no) = - n o . sin nEd0 + cos n0 - do

Hence

1 * 1 . - J1 0 sin n8dE = - n - . 0 cos n lo + ;li sm a

since sin n0 = 0 for 0 = 0 and 0 = a, and cos nr =

(- 1)". In a similar manner all the other coefficients, a,, a,

etc., may be determined and the required development bas the form :.

Figure 33 shows in the left-hand series of plots, the straight line y = 0 from 0 = 0 to 9 = a, and the cuc- cessive approximations to this line which are obtained by taking

y, = 2 sin E R = 2 sin 8 - sin 28 y, = 2 sin 0 - sin 20 + ( 2 / 3 ) sin 30.

It will be observed that while the series does not converge very rapidly, th' curves gradually approxi- mate y = 0 more and more closely, with increase in number of terms.

(2) It is desired to express xZ as a cosine function for the range, x = -c to x = c.

RGURE 33.-ILLUSTRATING FOURIER'S SERIES ANALYSIS ?mc We introduce a new variable, z = 7, so that

"

By means of this relation i t becomes possible to develop any given function of 0 in terms of the sines or z = -a for x = -c, and z = a for x = c. Then, of the cosines of multiples of 0. Series involving only these trigonometric functions, that is, series of the form r2 = (c.~) ' = a. + a, cos a + a* cos 2a + . . . . (in)

il

oo + a, cos z + a+ cos 2x + . . . . . + a , cos nx + . . . . b, sin x. + b2 sin 22 + . . . . . f bm sin m r + . . . . Multiplying each side by cos nzdz and integrating

OCTOBER, 1935

between the limits, all coefficients on the right-hand - w

f ( r ) = a. + C a, cos nr + C b. sin nr 1 1

side, except a. J:v cosZ nzdz, vanish, and we obtain where

the relation for determining a,, of the form

that is,

Now d(zP sin n z ) = na' cos nadz + 22 sin nzda d(a cos n z ) = -ne sin nzdz + cos nzde.

Hence

JI 8% cos nzda = ZI n . sinnz]l r - l*- a sin nzdz

' e sm nzdz (since sin nr = 0). =-is-. .

Therefore

The coefficient aa is determined from the relation

$ J:- asdB = J: adz, since c& o = 1, r

That is,

The possibility of obtaining such developments of an arbitrary function depends, evidently, upon the existence of the orthogonality relation expressed by

(') equation (i).* The same type of reasoning may he applied to develop an arbitrary function of x between the limits 0 and m in terms of a series of Hermitian or Laguerre polynomials.

Thus if f(x) is a function which tends to vanish for x = t m we can obtain the coefficients a, in the series

from the relation,

Consequently, the development for x2 is of the form

The curves on the right-hand side of Figure 33 show the parabola y = x2 at the top, and the straight line y = cZ/3, which is evidently,the average value of x2 over the range 0 to c. The other curves correspond to the expressions:

As shown by the plot of y,, this expression corre- sponds fairly closely to y = xZ for the range x =

*0.75.c and by using more terms, the range over which the series represents the parabolic function may be made to approach x = +=c very satisfactorily.

More generally, any function f(x) can be expressed within a definite range of value of x in the form

Similarly, f(0), an arbitraryt function of 0, may he represented in the range a > 0 > 0 by the series of Legendre coeffcients of zero order, in the form

where .

Even if the integral in equation (ix) or (x) cannot be calculated by direct integration, i t can always be evaluated by plotting the integrand (that is, the ex- pression to be integrated) as a function of x or 9 and determining the area under. the plot graphically. However, in most cases, i t is possible for the ex- perienced mathematician to develop a convergent series for the integral, by means of which its actual value may be determined. -

* The treatises bv W. E. BYHPILY AND MACROBERT are the most comprehensive on >his topic. A more elementary discussion of ~o&ier's series is given by L. I(. INCERSOLL AND 0. J. ZOBEL, "An Introduction to the Mathematical Theory af Heat Con- duction." Ginn and Comoanv. Boston. 1913. which eives numer- ~ ~~~~ . - ~ ~ . . . ous applications of these series. ~n elem&tary &cussion of Fourier's series and spherical harmonics is also given by D. H ~ P H R E Y , whose treatise is mentioned in the references cited in Chapter 11. An excellent discussion of Fourier's series with applications is given in R. A. HOUSTON'S "An Introduction to Mathematical Physics," also mentioned in the list of references.

t "Arbitrary" in the sense that it is possible to plot the func- tion graphically.