22.101 Applied Nuclear Physics (F all 2006)

Lecture 4 (9/18/06)

Bound States in Three Dimensions – Orbital Angular Momentum

References --

R. L. Liboff, Introductory Quantum Mechanics (Holden Day, New York, 1980).

L. I. Schiff, Quantum Mechanics , M c Graw-Hill, New York, 1955).

P. M. Morse and H. Feshbach, Me tho d s of Theore tica l Physic s (McGraw- Hill, New York, 1953).

We will now extend the bound-state calcu la tion to three-d i mensional sys t em s. The problem we want to solve is essentially the sam e as before, excep t that we wish to determ ine the bound-state energy levels and co rresponding wave functions for a particle in a th ree-d i m e nsional spherical well potential. Although th is is a three-dim e nsional potential, w e can tak e advantage of its symm etry in angular space and reduce the calculation to an equation still involving onl y one variable, the ra dial dista n ce between the par tic le position and the or ig in. In other words, the s phe r i cal po tentia l is s t ill a function of one variable,

V ( r )   V o r < r o

= 0 otherwise

(4.1)

Here r is the radial position of the particle re la tive to the o r igin. Any potentia l tha t is a function only of r, the magnitude of the position r and not the position vector itself, is called a cen tral-f or ce potentia l. As we will s ee, th is f o rm of the potential m a kes the solution of the Schrödinger wave equation part icularly sim p le. For a system where th e potential or interaction energy has no a ngular dependence, one can reform ulate the problem by factorizing the wave function into a com ponent that involves only the radial coordinate and another com ponent that involves only the angular coordinates. The wave equation is then reduced to a system of uncoupled one-dim ensional equations, each describing a radial com p onent of the wave function. As to the justification for using a

central-force potential for our discussion, th is w ill depend on which properties of the nucleus we wish to study.

We again begin with the tim e-independent wave equation

 ħ 2 2 

  2 m   V ( r )   ( r )  E  ( r ) (4.2)

 

Since the potential function has spherical symm et ry, it is natural for us to carry out the analysis in the spherical coordinate system rather than the Cartesian system. A position vector r then is specified by the radial coordinate r and two angular coordinates,  and

, the polar and azim u thal angles respectively, see Fig. 1. In this coordinate system

Fig. 1. The spherical co ordinate sys t em . A point in space is located by th e radial coordinate r, and polar and azim u thal angles  and .

the Laplacian operator  2 is of the form

2 2 1

  L 2 

  D r

 r 2  ħ 2 

(4.3)

where D 2 is an operato r inv o lving only the radial coordinate,

D 2  1

r r 2

  2  

 r    r 

( 4 . 4 )

and the operator L 2 involves only the a ngular coordinates,

 L 2  1  

sin 

   1  2

ħ 2 sin     

    sin 2    2 (4.5)

In term s of these oper a to rs th e wave equation (4.2) becom e s

 2 2 

  2 m D r  2 mr 2  V ( r )   ( r  )  E  ( r  ) (4.6)

 

For any potential V(r) th e angular variation of  is always determ ined by the operator L 2 /2m r 2 . Therefore one can study the operator L 2 separ a te ly and then use its prop er tie s to sim p lify the solution of (4.6). This need s to be done only once, since the angular varia tion is independent of whatever form one takes for V(r). It tu rns out that L 2 is v e r y well known (it is the square of L which is th e ang u lar m o m e ntum operator ) ; it is the operator that describes the angular motion of a free particle in three-dim e nsional space.

We first summarize the basic properties of L 2 before discussing any physical inte rpre tatio n . It can b e show n that the eigenfunction of L 2 are the sph e rical harm oni cs functions, Y m (  ,  ) ,

L 2 Y m (  ,  )  ħ 2 ℓ ( ℓ  1 ) Y m (  ,  ) (4.7)

where

ℓ ℓ

  1 / 2

Y m (  ,  )    P m (cos  ) e im  (4.8)

and

 

P m (  ) 

 

( 1   2 ) m / 2 d ℓ  m

ℓ

(  2

 1 ) ℓ (4.9)

ℓ 2 ℓ ℓ ! d  ℓ  m

with   cos  . The function P m (  ) is called the as s o ciated Leg e ndre polyn om ials, which are in turn expressible in term s of Legendre polynom ials P ℓ (  ) ,

P m (  )  ( 1   )

m / 2 d m

P ℓ (  ) ( 4 . 1 0 )

with P o (x) = 1, P 1 (x) = x , P 2 (x) = (3x 2 – 1)/2, P 3 (x) = (5x 3 -3x ) /2, e t c. Spe c ial f unction s like Y m and P m are quite exten s ively d i scu ssed in s t and a rd tex t s [see, for exam ple,

ℓ ℓ

Schiff, p.70] and reference books on m a the m a tical functions [ M ore and Feshbach, p. 1264]. For our purposes it is sufficient to regard them as well known and tabulated quantities like sines and cosines, an d when ever the need aris es we will in voke their special properties as given in the m a them atical handbooks.

It is cle a r from (4.7) tha t Y m (  ,  ) is an eigenfunction of L 2 with corresponding eigenvalue ℓ ( ℓ  1 ) ħ 2 . Since the angular m o mentum of the particle, like its energy, is quantized, the index ℓ can take on only positive integral values or zero,

ℓ = 0, 1, 2, 3, …

Sim ilarly, the index m can have integral values from - ℓ to ℓ ,

m = - ℓ , - ℓ +1, …, -1, 0, 1, …, ℓ -1, ℓ

For a given ℓ , there can be 2 ℓ +1 values of m . The significance o f m can be seen from the property of L z , the projection of the orbital angular m o m e ntum vector L along a certain direction in space (in th e absence of any external field, this cho i ce is up to the observer). F o llowing co nvention we will choos e th is directio n to be alon g the z-axis of our coordinate system , in which case the operator L z has the representation ,

L   i ħ  /   , and its eignefunctions are also Y m (  ,  ) , with eigenvalues m ħ . The

z ℓ

indices ℓ and m are called quantum numbers. Sinc e the angu lar space is tw o-dim e nsional

(corresponding to two degrees of freedom ) , it is to be expected that two quantum num b ers will em erge from our analysis. By the sam e token we should expect three quantum numbers in our description of three- dimensional system s. W e s hould regard the particle as e x isting in va rious s t ates wh ich are specified by a unique set of quantum num b ers, each one is associated with a cer tain orbital angular m o m e ntum which has a definite m a gnitude and orientation with respec t to our chosen direc tion along the z-axis. The particular angular mom e ntum st ate is described by the function Y m (  ,  ) with ℓ

known as the orbital angular momentum quantum number , and m the magnetic quantum number . It is useful to k eep in m i nd that Y m (  ,  ) is ac tua lly a rathe r s i m p le f unction f o r low order in dices. For exam ple, the first four spherical harmonics are:

Y 0  1 /

, Y  1  3 / 8  e  i  sin  , Y 0 

cos  , Y 1  3 / 8  e i  sin 

0 1 1 1

Two other properties of the spherical ha rm onics are worth m e ntioning. F i rst is that { Y m (  ,  ) }, w ith ℓ = 0, 1, 2, … and  ℓ  m  ℓ , is a com p lete s e t of functions in the space of 0     and 0    2  in the sense that any arbitrary function of  and

can be represented by an expansion in these functions. Another property is orthonorm a lity,

 2 

  ℓ ℓ '

ℓℓ '

mm '

sin  d 

0 0

d  Y m * (  ,  ) Y m ' (  ,  )   

(4.11)

where  ℓℓ ' deno tes th e K r onecker de lta f unction ; it is unity w h en the tw o subs crip ts ar e equal, oth e r w ise th e function is zero.

Returning to the wave equa tion (4.6) we look for a soluti on as an expansion of the

wave function in spherical harm oni cs series,

 ( r  )   R ( r ) Y m (  ,  ) ( 4 . 1 2 )

ℓ ℓ

ℓ , m

Because of (4.7) the L 2 o p erato r in (4 .6) can be replaced by th e factor ℓ ( ℓ  1 ) ħ 2 . In view of (4.11) we can elim inate the angular p a rt of the problem by m u ltiplying th e wave equation by the com p lex conjugate of a spheri cal harm onic and integr atin g over all so lid

angles (recall an elem ent of solid ang l e is sin  d  d  ), obtaining

 ħ 2 2 ℓ ( ℓ  1 ) ħ 2 

  2 m D r  2 mr 2  V ( r )  R ℓ ( r )  ER ℓ ( r ) (4.13)

This is an equation in one va riable, the radial coordinate r, although we are treating a three-dim e nsional problem . We can m a ke this e quation look lik e a one-dim e nsional problem by transform i ng the dependent variable R ℓ . Define the radial function

u ℓ ( r )  rR ℓ ( r ) ( 4 . 1 4 )

Inser ting this into (4.13 ) we get

ħ 2 d 2 u ( r )  ℓ ( ℓ  1 ) ħ 2 

 ℓ  

2 m dr 2 

2 mr 2  V ( r ) 

u ℓ ( r )  Eu ℓ ( r ) (4.15)

W e will call (4.15) the radial wave equation . It is the basic st arting point of three- dim e nsional problem s involving a particle inte racting with a centra l potential field.

We observe that (4.15) is actually a sy stem of uncoupled equ a tions, on e for each fixed value of the orbital angul ar m o m e ntum qua ntum number ℓ . W ith ref e rence to the wave equation in one dim e nsion, the extra term involving ℓ ( ℓ  1 ) in (4.15) rep r es ents th e contr i bution to the po ten tia l f i eld due to the c e ntr i f ugal m o tion of the particle. The 1 / r 2 dependence m a kes the effect particularly im portant near the o r igin ; in o t h e r words, centrigfugal motion gives rise to a ba rrie r which tends to keep the par tic le away f r om the

origin. Th is effect is of course abs e nt in th e cas e of ℓ = 0, a state of zero orbital angular mom e ntum , as one would expect. T h e first few ℓ states usually are the only ones of interest in o u r discus sio n (becaus e they tend to have the lowest ener gies); they are given special spectroscopic designations,

notation: s, p, d, f, g, h , …

ℓ = 0, 1, 2, 3, 4, 5 , …

where t h e fi rst four l e t t e rs st and for ‘sharp’, ‘prin c ipal’, ‘d iffuse’, and ‘fun dam e ntal’ respec tive l y. Af ter f the letters are a s signed in alp h abetical o r der, as in h, i, j, … The wave function describing the state of orbital angular m o m e ntum ℓ is often called the ℓ th partial wave ,

 ( r  )  R ( r ) Y m (  ) ( 4 . 1 6 )

ℓ ℓ ℓ

Notice th at in the cas e of s-wave the wave function is spherically symm etric since Y 0 is

independent of  and .

Interpre tation of Orbita l Angular Momentum

In classical m echanics, the angular mom e nt um of a particle in m o tion is defined as the vector product, L  r  p , where r is th e particle po sition and p its line a r

mom e ntum . L is directed along the axis of rotation (right-hand rule), as shown in Fig. 2.

Fig. 2. Angular m o m e ntum of a particle at position r m oving with linear mom e ntum p (clas s ic al de f i nition ) .

L is called an axial or pseudovector in contrast to r and p , which are polar vectors. Under inversion, r   r , and p   p , but L  L . Quantum mechanically, L 2 is an ope r a tor

with eigenvalues and eigenfunctions given in (4.7). Thus the m a gnitude of L is

ħ ℓ ( ℓ  1 ) , with ℓ = 0, 1, 2, …being the orbital a ngular m o mentum quantum num b er.

We can specify the m a gnitude and one Ca rtesian com ponent (u sually called the z-

com ponent) of L by specifying ℓ and m, an example is shown in Fig. 3. What about the x- and y-components? They are undeterm in ed, in that they cannot be observed

Fig. 3 . The ℓ ( ℓ  1 ) = 5 projections along the z-axis of an orbital angular m o m e ntum with

ℓ = 2. Magnitude of L is 6 ħ .

sim u ltaneously with the observation of L 2 and L z . Another useful inte rpretation is to look at the energ y conservation equa tion in te rm s of radial and tan g entia l m o tions. By this we m ean that the tota l ene r g y can be written a s

2

E  2 m ( v r  v t )  V  2 mv r  2 m r 2  V ( 4 . 1 7 )

where the decom position into radial and tangent ial velocities is depicted in Fig. 4. Eq.(4.17) can be com p ared with the radial wave equation (4.15).

Fig. 4 . Decom posing the velocity vect or of a particle at position r into radial and tangential com ponents.

Thus far we have confined our discussions of the wave equation to its solution in spherical coordinates. T h ere are situations wh ere it will be more appropriate to work in another coordinate system. As a si mple exam ple of a bound-state problem , we can consider the system of a fr ee particle contained in a c ubical box of di m e nsion L along each sid e . In this case it is clearly more conven i ent to write the wave equ a tion in Cartesian coordinates,

ħ 2   2  2  2 

2 m   x 2  y 2  z 2  ( xyz ) E ( xyz ) (4.17)

 

0 <x, y, z < L. The boundary conditions are  = 0 whenever x , y, or z is 0 or L. Since both the equation and the boundary conditions are separable in the three coordinates, the solution is of the product for m ,

 ( xyz )   ( x )  ( y )  ( z )

x y z

= ( 2 / L ) 3 / 2 sin( n  x / L ) sin( n  y / L ) sin( n  z / L ) (4.18)

where n x , n y , n z are positive integers (excluding zero ), and th e energy beco m e s a sum of three contributions,

E

x y z

 E  E  E

x y z

( ħ  ) 2  2 

2  2 

= n x

2 mL 2

n y n z

( 4 . 1 9 )

We see that the wave functions and corres ponding energy levels ar e specified by the set of three quantum numbers (n x , n x , n z ). W h ile each state of the system is describ e d by a unique set of quantum num bers, there can be m o re than on e s t ate at a particular energ y level. W h enever th is hap p ens, the lev e l is s a id to be degenerate . For example, (112),

(121) , and (211) are three different states, but they are all at the sam e energy, so the level at 6 ( ħ  ) 2 / 2 mL 2 is tr iply d e genera te. The concept of degeneracy is useful in our later discussion of the nuclear shell m odel where one has to determ ine how many nucleons can be put into a certain energy level. In Fi g. 6 we show the energy level diagram for a particle in a cubical box. Another way to display the info rm ation is through a table, such as Table I.

Fig. 6 . Bound states of a particle in a cubical box of width L.

Table I . The first few energy levels of a par ticle in a cubical box which correspond to Fig. 6.

n x n y n z

2 mL 2

( ħ  ) 2 E

degeneracy

1	1	1	3		1
1	1	2	6		3
1	2	1
2	1	1
1	2	2, …	9		3
1	1	3, …	11		3
2	2	2	12		1

The energy unit is seen to be  E  ( ħ  ) 2 / 2 mL 2 . W e can use th is expression to estim ate the m a gnitude of the energy levels f o r el ectrons in an atom , for which m = 9.1 x10 -2 8 gm and L ~ 3 x 10 -8 cm , and for nucleons in a nucleus, for which m = 1.6x10 -24 gm and L ~ 5F. The energies com e out to be ~30 ev a nd 6 Mev respectively, va lues wh ich ar e typic a l in atom ic and nuclea r ph ysics. Notic e that if an electron were in a nucleus, then it would have energies of the order 10 10 ev !

In closing this section w e note that B ohr had put forth the “correspondence principle” w h ich states that quantum m echan ical results will approach th e class i cal results when the quantum num b ers are large. T hus we have

 2  2 sin 2 ( n  x / L )  1 ( 4 . 2 0 )

n L L

n  

What this means is that the probability of finding a particle anywhe re in the box is 1/L, i.e., one has a uniform di stribution, see Fig. 7.

Fig. 7 . The behavior of sin 2 nx in the lim it of larg e n.

Parity

Parity is a s y m m etry prope rty of the wave function asso cia t e d with the in version

operation. This operation is one where the position vector r is reflected through the origin (see F i g. 1), so r   r . For physical system s which are not subjected to an extern al vector field, we e xpect these system s will rem a in the sam e under an invers ion

operation, or the Ham iltonian is invariant under inversion. If  ( r ) is a solu tion to the wave equation, then applying the inversion operation we get

H  (  r )  E  (  r ) ( 4 . 2 1 )

which shows that  (  r ) is also a solution. A genera l solution is therefore obtained by adding or subtracting the two solutions,

H   ( r )   (  r )   E   ( r )   (  r )  ( 4 . 2 2 )

Since the function   ( r )   ( r )   (  r ) is manifestly invarian t under inversion, it is said to h a ve positiv e par i ty, or its pa rity, denoted by the sym b ol , is +1. Sim ilarly,

  ( r )   ( r )   (  r ) changes sign under inversion, so it has negative parity, or = -1.

The significance of (4.22) is that a phys ical solution of our quantum m e c h anical description should have definite parity; this is the condition we have previously im posed on our solutions in solving the wave equation (see Lec3). Notice that there are function s which do not have definite parity, for exam ple, Asinkx + Bcoskx. This is the reason that we take eith er the s i ne f unction o r th e cosine f u n c tion f o r th e inte rior solu tion in Le c3. In general, one can accep t a solution as a linea r co mbination o f individual solution s all having the s a m e parity. A linear co m b ination of solutions with different parities has no definite parity, and is therefore unacceptable.

In spherical coordinates, the inversion operation of changing r to – r is equ i valen t to changing the polar angle  to    , and the azim uthal angle to    . The effect

of the transf orm a tion on the spherical harm onic function Y m (  ,  ) ~ e im  P m (  ) is

ℓ ℓ

e im   e im  e im   (  1 ) m e im 

P m (  )  (  1 ) ℓ  m P m (  )

ℓ ℓ

so the par ity of Y m (  ,  ) is (  1 ) ℓ . In other words, the parity of a state with a definite orbital angular m o m e ntum is even if ℓ is even, and odd if ℓ is odd. All eigenfunctions of the Ha m iltonian with a sphe rically symmetric potential are therefore either even or odd

in par ity.