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Chapter 6

ANGULAR MOMENTUM

Particle in a RingConsider a variant of the one-dimensional particle in a box problem in whichthe x-axis is bent into a ring of radius R. We can write the same Schrodingerequation

h2

2md2(x)

dx2 = E (x) (1)

There are no boundary conditions in this case since the x-axis closes uponitself. A more appropriate independent variable for this problem is theangular position on the ring given by, = x/R . The Schr odinger equationwould then read

h2

2mR 2d2()

d2 = E () (2)

The kinetic energy of a body rotating in the xy-plane can be expressed as

E = L2z2I

(3)

where I = mR 2 is the moment of inertia and Lz , the z-component of angular

momentum. (Since L = r p , if r and p lie in the xy-plane, L points in thez-direction.) The structure of Eq (2) suggests that this angular-momentumoperator is given by

Lz = ih

(4)

This result will follow from a more general derivation in the following Sec-tion. The Schrodinger equation (2) can now be written more compactlyas

() + m2() = 0 (5)

wherem2 2IE/ h2 (6)

(Please do not confuse this variable m with the mass of the particle!) Pos-sible solutions to (5) are

() = const e im (7)

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In order for this wavefunction to be physically acceptable, it must be single-valued . Since increased by any multiple of 2 represents the same pointon the ring, we must have

( + 2 ) = () (8)

and thereforeeim (+2 ) = eim (9)

This requires thate2im = 1 (10)

which is true only is m is an integer:

m = 0 , 1, 2 . . . (11)Using (6), this gives the quantized energy values

E m = h2

2I m2 (12)

In contrast to the particle in a box, the eigenfunctions corresponding to + mand m [cf. Eq (7)] are linearly independent, so both must be accepted.Therefore all eigenvalues, except E 0 , are two-fold (or doubly) degenerate.The eigenfunctions can all be written in the form const eim , with m al-lowed to take either positive and negative values (or 0), as in Eq (10). Thenormalized eigenfunctions are

m () = 1 2 e

im (13)

and can be veried to satisfy the normalization condition containing thecomplex conjugate

2

0m () m () d = 1 (14)

where we have noted that m () = (2 ) 1/ 2 e im . The mutual orthogo-

nality of the functions (13) also follows easily, for

2

0m m () d =

12

2

0ei (m m ) d

= 12

2

0[cos(m m ) + i sin(m m )]d = 0

for m = m (14)

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The solutions (12) are also eigenfunctions of the angular momentumoperator (4), with

Lz m () = mh m (), m = 0 , 1, 2 . . . (16)This is a instance of a fundamental result in quantum mechanics, that anymeasured component of orbital angular momentum is restricted to integralmultiples of h. The Bohr theory of the hydrogen atom, to be discussed inthe next Chapter, can be derived from this principle alone.

Free Electron Model for Aromatic Molecules

The benzene molecule consists of a ring of six carbon atoms around whichsix delocalized pi-electrons can circulate. A variant of the FEM for rings pre-

dicts the ground-state electron conguration which we can write as 1 2

24

,as shown here:

Figure 1. Free electron modelfor benzene. Dotted arrow showsthe lowest-energy excitation.

The enhanced stability the benzene molecule can be attributed to the com-plete shells of -electron orbitals, analogous to the way that noble gas elec-tron congurations achieve their stability. Naphthalene, apart from thecentral CC bond, can be modeled as a ring containing 10 electrons in thenext closed-shell conguration 1 2 2 4 34 . These molecules fulll H uckels4N +2 rule for aromatic stability. The molecules cyclobutadiene (1 2 22)and cyclooctatetraene (1 2 24 32), even though they consist of rings withalternating single and double bonds, do not exhibit aromatic stability sincethey contain partially-lled orbitals.

The longest wavelength absorption in the benzene spectrum can be

estimated according to this model ashc

= E 2 E 1 = h2

2mR 2(22 12)

The ring radius R can be approximated by the CC distance in benzene,1.39 A. We predict 210 nm, whereas the experimental absorption hasmax 268 nm.

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Spherical Polar Coordinates

The motion of a free particle on the surface of a sphere will involve com-ponents of angular momentum in three-dimensional space. Spherical polarcoordinates provide the most convenient description for this and relatedproblems with spherical symmetry. The position of an arbitrary point r isdescribed by three coordinates r, , , as shown in Fig. 2.

Figure 2. Sphericalpolar coordinates.

These are connected to cartesian coordinates by the relations

x = r sin cos y = r sin sin

z = r cos (16)The radial variable r represents the distance from r to the origin, or thelength of the vector r :

r = x2 + y2 + z2 (18)The coordinate is the angle between the vector r and the z-axis, similarto latitude in geography, but with = 0 and = corresponding to theNorth and South Poles, respectively. The angle describes the rotation of r about the z-axis, running from 0 to 2 , similar to geographic longitude.

The volume element in spherical polar coordinates is given by

d = r 2 sin drdd,r {0, }, {0, }, {0, 2} (19)

and represented graphically by the distorted cube in Fig. 3.

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again introducing the moment of inertia I = MR 2 . The variables and can be separated in Eq (22) after multiplying through by sin 2 . If we write

Y (, ) = ( )() (24)

and follow the procedure used for the three-dimensional box, we nd thatdependence on alone occurs in the term

()()

= const (25)

This is identical in form to Eq (5), with the constant equal to m2 , and wecan write down the analogous solutions

m () = 12 eim , m = 0 , 1, 2 . . . (26)Substituting (24) into (22) and cancelling the functions ( ), we obtain anordinary differential equation for ( )

1sin

dd

sin dd

m2

sin2 + () = 0 (27)

Consulting our friendly neighborhood mathematician, we learn that thesingle-valued, nite solutions to (27) are known as associated Legendre func-tions . The parameters and m are restricted to the values

= ( + 1) , = 0 , 1, 2 . . . (28)

whilem = 0 , 1, 2 . . . (2 +1 values) (29)

Putting (28) into (23), the allowed energy levels for a particle on a sphereare found to be

E = h2

2I ( + 1) (30)

Since the energy is independent of the second quantum number m, the levels(30) are (2 +1)-fold degenerate.

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The spherical harmonics constitute an orthonormal set satisfying theintegral relations

0 2

0Y

m(, )Y m (, ) sin dd = mm (31)

The following table lists the spherical harmonics through = 2, which willbe sufficient for our purposes.

Spherical Harmonics Y m (, )

Y 00 = 1

4

1/ 2

Y 10 = 34

1/ 2

cos

Y 1 1 = 34

1/ 2

sin e i

Y 20 = 516

1/ 2

(3 cos2 1)

Y 2 1 = 158

1/ 2

cos sin e i

Y 2 2 = 1532

1/ 2

sin2 e 2i

A graphical representation of these functions is given in Fig. 4. Surfaces of constant absolute value are drawn, positive where green and negative wherered. Other colors represent complex values.

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Figure 4. Contours of spherical harmonics.

Theory of Angular Momentum

Generalization of the energy-angular momentum relation (3) to three di-mensions gives

E =

L2

2I (32)Thus from (21)-(23) we can identify the operator for the square of totalangular momentum

L2 = h2 1sin

sin

+ 1sin2

2

2 (33)

By (28) and (29), the functions Y (, ) are simultaneous eigenfunctions of L2 and Lz such that

L2Y m (, ) = ( + 1) h2 Y m (, )

and Lz Y m (, ) = m h Y m (, ) (34)

But the Y m (, ) are not eigenfunctions of either Lx and Ly (unless = 0).Note that the magnitude of the total angular momentum ( + 1) h is

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greater than its maximum observable component in any direction, namelyh. The quantum-mechanical behavior of the angular momentum and its

components can be represented by a vector model, illustrated in Fig. 5. Theangular momentum vector L , with magnitude

( + 1) h, can be pictured

as precessing about the z-axis, with its z-component Lz constant. Thecomponents Lx and Ly uctuate in the course of precession, correspondingto the fact that the system is not in an eigenstate of either. There are 2 +1different allowed values for Lz , with eigenvalues m h (m = 0 , 1, 2 . . . )equally spaced between + h and h .

=

=

l =

= -

= -

=

Figure 5. Vector modelfor angular momentum,showing the case = 2.

This discreteness in the allowed directions of the angular momentum vec-tor is called space quantization . The existence of simultaneous eigenstatesof L2 and any one component, conventionally Lz , is consistent with the

commutation relations derived in Chap. 4:[Lx , Ly ] = ihLz et cyc (4.43)

and[L2 , Lz ] = 0 (4.44)

Electron Spin

The electron, as well as certain other fundamental particles, possesses anintrinsic angular momentum or spin , in addition to its orbital angular mo-mentum. These two types of angular momentum are analogous to the dailyand annual motions, respectively, of the Earth around the Sun. To dis-tinguish the spin angular momentum from the orbital, we designate thequantum numbers as s and ms , in place of and m. For the electron, thequantum number s always has the value 12 , while ms can have one of two

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values, 12 . The electron is said to be an elementary particle of spin 12 .The proton and neutron also have spin 12 and belong to the classication of particles called fermions , which are govened by the Pauli exclusion princi-ple. Other particles, including the photon, have integer values of spin and

are classied as bosons . These do not obey the Pauli principle, so that anarbitrary number can occupy the same quantum state. A complete theoryof spin requires relativistic quantum mechanics. For our purposes, it is suf-cient to recognize the two possible internal states of the electron, whichcan be called spin up and spin down. These are designated, respectively,by and as factors in the electron wavefunction. Spins play an essentialrole in determining the possible electronic states of atoms and molecules.

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