22.101 Applied Nuclear Physics (F all 2006)

Lecture 2 (9/11/06) Schrödinger Wave Equation

References --

R. M. Eisberg, Fundamentals of Modern Physics (W iley & Sons, New York, 1961).

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

W ith this lecture we begin the discussi on of quantum m echanical description of nuclei. There are certain properties of a nucle us which can be desc ribed properly only by the use of quantum m e chanics. The ones whic h are m o st f a m iliar to all students are the energy levels of a nucleu s and the tra n sition s tha t can take p l a ce f r om one level to another. Other exam ples are the various t ypes of nuclear radiation which are som e times treated as particles and at other tim es as waves.

It is no t our goal in this s ubject to ta ke up the study of quantum m echanics as a topic by itself. On the other hand, we have no reasons to avoid using quantum m e cha n ics whenever it is the proper way to understand nu clear concepts and radi ation interactions. Indeed a serious student in 22.101 has little c hoice in deciding whether or not to learn quantum m e chanics. This is because th e concepts and term inologies in quantum m echanics are such in teg r al parts of n u clear c oncepts and the interaction of radiation with m a tter that som e knowledge of quantum m echanics is essential to having full comm and of the language of nuclear physic s. One can go further to say that it is inconceivable that any self -respecting nuclear engineering student will choose not to be exposed to this language, because in all likelihood this expe rience will com e in handy someday. The position we adopt throughout the term is to learn enough quantum m echa n ics to appreciate the funda m e ntal concepts of nuc lear physics, and let each student go beyond this level if he/she is in terested. W h at th is m eans is tha t we will not alw a ys deriv e th e basic equ a tions and expr essions that we will us e ; the stud ent is expected to work with them as postulate s when this happ ens (as al ways, with the p r iv ilege of reading up on the background m a terial on your own).

Waves and Particles

We review som e basic properties of waves a nd the concept of wave-particle duality. In class i cal m e chanics the equation for a one-dim ensional periodic disturbance  ( x , t ) is

 2  

 t 2

2  2 

c  x 2

( 2 . 1 )

which has as a general solution,

i ( kx   t )

 ( x , t )   o e

( 2 . 2 )

where   2   is the circular frequency, the linear frequency, and k is the wavenum b er related to the wavelength  by k  2  /  . If (2 .2) is to b e a solution of (2.1), then k and m u st satisf y the rela tion

  ck ( 2 . 3 )

So our solution has the form of a traveling wave with phase v e locity equa l to c, which we denote by v ph . In general the relation between frequency and wavenum ber is called the dispersion relation . W e will se e tha t dif f e rent kinds of partic les can b e re presented as waves which are characterized by different dispersion relations.

The solution (2.2) is called a plane w a ve. In three dim e nsions a plane wave is of

the f o rm exp( i k  r ) . It is a wave in space which we can vis u alize as a s e ries of plan es perpendicular to the wavevector k ; at any spatial point on a gi ven plane the phase of the wave is the sam e . That is to say, the perpendicular plan es are planes of constant phase . W h en we include th e tim e variation exp(  i  t ) , then exp[ i ( k  r   t )] becom e s a

traveling p l ane wave, m e aning that th e planes of constant phase are now moving in the direction along k at a speed of  / k , which is the ph as e veloc ity of the wave.

The wave equation (2.1) also adm its solution s of the f o rm

 ( x , t )  a o sin kx cos  t (2.4)

These are standing wave solution s . One can te ll a standing wave from a t r aveling wave by the behavior of the nodes, the spatial posit ions where the w a ve function is zero. Fo r a standing wave the nodes do not m o ve, or change with tim e, whereas for a traveling wave, (2.2), the no des are x n  ( n    t ) / k . Clearly the nodes are positions m oving in the +x direction with the velocity dx / dt   / k . W e will see below th at the choice between traveling and standing w a ve solutions depends on the physical solution of interest (w hich kind of problem one is solving). For the ca lculation of energy levels of a nucleus, the bound state problem , we will be con cerned with standing wave solutions. In contrast, for the discussion of scattering pr oblem (see th e later lectu r e on neutron-proton scattering) it will be m o re appropriate to consid er traveling wa ve solutions.

Our inter e st in the p r oper ties of wave s lie s in th e fact th at th e quantum m e chanical description of a nucleus is ba sed on the wave representation of the nucle us. It was f i rst postulated by deBroglie (1924) that one can a ssociate a particle of m o mentum p and total energy E with a group of waves (wave packet) which are characterized by a wavelength

 and a frequency , with the relation

  h / p ( 2 . 5 )

  E / h ( 2 . 6 )

Moreover, the m o tion of the particle is govern ed by the propagation of the wave packet. This statem ent is th e ess e nce of partic le-wave duality, a concept which we adopt throughout our study of nuclear physics [see, for exam ple, Eisberg, chap 6].

It is im portant to distinguish between a single wave and a group of wave s. This distinction is seen m o st sim p ly by consideri ng a group of two waves of slightly different wavelengths and frequencies. Suppo se we take as the wave packet

 ( x , t )   1 ( x , t )   2 ( x , t ) (2.7)

with

 1 ( x , t )  sin( kx   t ) ( 2 . 8 )

 2 ( x , t )  sin[ k ( k  dk ) x  (   d  ) t ] (2.9)

Using the id entity

sin A  sin B  2 cos[( A  B ) / 2 ] sin[( A  B ) / 2 ] (2.10)

we rewrite  ( x , t ) as

 ( x , t )  2 cos[( dks  d  t ) / 2 ] sin{[( 2 k  dk ) x  ( 2   d  )] t / 2 }

 2 cos[( dkx  d  t ) / 2 ] sin( kx   t ) ( 2 . 1 1 )

In this approxim ation, term s of higher order in dk / k or d  /  are dropped. Eq. (2.11) shows two oscillations, o n e is th e wave p acket o s cillating in s p ace with a period of

2  / k , while its a m plitude os ci llates with a period of 2  / dk (see Fig. 1 ) . Notice th at the la tte r osc illation h as its

Figure by MIT OCW.

Fig. 1 . Spatial variation of a su m of t w o waves of slightly different frequencies and wavenum b ers showing the wave packet m oves w ith ve locity w which is distin ct f r om the propagation (group) velo city g of the am plitude [from Eisberg, p. 144].

own propagation velocity, d  / dk . This velocity is in f a ct the speed with which the associa t ed p a rticle is m o ving. Thus we identif y

g  d  / dk ( 2 . 1 2 )

as the group velocity. T h e group velocity s hould not be confused with the propagation velocity of the wave packet, ca lled the phase velocity, given by

w    E / p  c

(2.13)

Here m o is the rest m a ss of the partic le and c th e speed of lig ht. W e see the wave packet moves with a velocity greater th an c, whereas th e associated p a rticle speed is neces sarily less than c. This m eans the phase velocity is greater than or equa l to the group velocity.

In this cla ss we will be d ealing with three kinds o f partic les w hose wave repres entations are of in teres t . Thes e are (i) nucleons or nuclides which can be treated as non-relativistic particles for our purposes, (ii) electrons and positrons which usually should be treated as relativistic particles sin ce their energies tend to be com p arable or greater th an the res t -m ass energy, an d (iii) pho to ns which are fully relativ istic sin ce th ey have zero rest-m ass energy. For a non-rela tivis tic particle of m a ss m m o ving with mom e ntum p, the associated wavevector k is p  ħ k . Its kinetic energy is T =

p 2 / 2 m  ħ 2 k 2 / 2 m . This is the “particle view”. The corresponding “wave view ” would have the m o m e ntum ma gnitude p = h/  , with   2  / k , and energy (usually denoted as E rathe r tha n T) as h   ħ  . The wavevector, or its m a gn itude, the wavenum b er k, is a useful variable for the discussion of particle scattering s i nce in a b eam of such particles the only en ergies are k i n e tic, and both m o m e ntum and energy can be specified by giving

k . For electro m agnetic waves, the as so ci ated par tic le, the pho ton, has m o m e ntum p ,

which is also given by ħ k , but its energ y is E  ħ ck  ħ p . Com p aring these two cases we see tha t the disper sio n rela tion is   ħ k 2 / 2 m f o r a non-re lativis tic pa r tic le, and

  ck for a photon. The group velocity, according to9 (2.12), is v g  ħ k / m  p / m and

v g  c , respec tive l y, which is c onsisten t with our intu itive notion ab out par tic le speed and the universal speed of a photon.

The Schrödinger Wave Equation

The Schrödinger equation is the fundam ental equati on governing the deBroglie wave with which we asso ciate a particle. The wave or wave function is to be tre a ted a s a space- and tim e -dependent quantity,  ( r , t ) . One does not derive the Schrödinger equation in the sam e sense that Newton’s equation of m o tion, F  m a , is postulated, and not derived. The wave equation is th erefor e a po stulate which one accep ts when stud ying physical phenom e na using quantum m echanics. Of course one can give system atic motivations to suggest w hy such an equa tion is valid [see Eisberg, chap 7 f o r a developm ent]. Such discussions would be beyo nd the scope of this class. W e now write down the Schrödinger equation in its tim e-dependent form for a particle in a potential field V(r),

  ( r , t )

 ħ 2 2 

i ħ  t

   2 m 

 V ( r )   ( r , t ) (2.14)



Notice th at the quantity in the bra c ke t is th e Ham ilton i an H of the system . H is a m a the m atical operator w hose physical m eaning is the total energy. Thus it consists of the kinetic part p 2 /2m and the potential p a rt V(r). Th e appearance of the Laplacian operator

 2 is to be exp ected, s i n ce the par tic le m o m e ntum p is an operator in configuration space, with p   i ħ  . The m o mentum operator is th erefo r e a firs t-o r d e r differential

operato r in c onfiguration space. By defi ning the Ham iltonian H as the op erato r

2

H     V ( r ) ( 2 . 1 5 )

2 m

we can express the tim e-dependent S c hr ödinger equation in op erator notation,

i ħ   ( r , t )  H  ( r , t ) ( 2 . 1 6 )

 t

As a side rem a rk (2.14) is valid only for a non -re lativis tic pa r tic le, wherea s (2.16) is m o re genera l if H is lef t unsp e cif i ed. This m eans that one can use a rela tiv istic e xpression f o r H, then (2.16) would lead to an equation fi rst derived by Dirac. The Dirac equation is what one should consider if the particle were an electron. C o m p ared to the class i cal wave equation, (2.1), which relates the second spatial derivative of the wave function to the second -o rder tim e derivative, the tim e-de pendent Schrödinger wa ve equation, (2.14) or (2.16), is seen to r e la te the spa tia l derivative of the wave function to the firs t -orde r tim e derivative. This is a significant distinct ion, although we have no need to go into it further in this class. Among the im plications is the fact that the clas sical wave is real and m easurable (for exam ple, an elastic string o r an electrom agnetic wave), whereas th e Schrödinger wave function is complex, and therefore not measurable . To ascrib e physical m e aning to the wave function one n eeds to consider the probability density defined as  * ( r , t )  ( r , t ) , where  * ( r , t ) is the com p lex conjugate of the wave function.

Al m o st all o u r discussio n s are con c erne d with the tim e-independent form of the Schrödinger equation. T h is is obtained by cons id ering a p e rio d ic solu tion to (2.16) of the for m

 ( r , t )   ( r ) e iEt / ħ ( 2 . 1 7 )

where E is a constan t (so on to be ide n tif ied as th e tota l ene r g y ). Inse rtin g this so lution into (2.16 ) g i ves th e tim e-inde pendent Schrödinger equation,

H  ( r )  E  ( r ) ( 2 . 1 8 )

W e see that (2.18) has th e for m of an eige nvalue problem with H being a linear operator, E the eigenvalue, and  ( r ) the eigenfunction.

It is instruc t ive to poin t o u t a ce rta i n s i m ilarity be tween the Schrödinger equation and the class i cal wave eq uation when the la tter in corporates the concept o f deBroglie waves. To show this we f i rst write th e three-dim e nsional generalization (2.1) as

 2  ( r , t )  2 2

 t 2

v ph   ( r , t ) ( 2 . 1 9 )

and use (2.13),

v ph

 E  E p

( 2 . 2 0 )

For period ic solution s ,  ( r , t )   ( r ) e iEt / ħ , we see that one is led immediately to (2.19). Notice th at the connectio n between th e class i cal wave equation and the Sch r ödinger equation is p o ssible on ly in te rm s of the tim e-independent form of the equations. As m e ntioned above, the tw o equations, in their ti m e -dependent form s, differ in im portant ways, consequently different properties have to be ascribed to the classical wave function and the Schrödinger wave function.

Bound (E < 0) and Scatterig (E > 0) State Solutions

Following our previous statem ent about th e different types of wave solutions, we can ask what types of solutions to the Schrödi nger equation are of in terest. To answer this ques tion we consider (2.18) in on e dim e nsion f o r the sake of illustratio n . W r iting o u t the equa tion explic itly, we have

d 2  ( x )  

dx 2

k 2  ( x ) ( 2 . 2 1 )

where k 2  2 m [ E  V ( x )] / ħ 2 . In g e neral k 2 is a function of x because o f the potential energy V(x), but for piecewise constant potentia l functions such as a rectan gular well or barrier, we can write a separate equation fo r each region where V(x) is co nstant, and thereby tre a t k 2 as a constant in (2.21 ) . A ge nera l solution to ( 2 .21) is th en

 ( x )  Ae ikx  Be  ikx ( 2 . 2 2 )

where A and B are constants to b e determ ined by appropriate boundary co nditions. N o w suppose we are dealing with fin ite-range potenti als so that V ( x )  0 as x   , then k

becom e s ( 2 mE / ħ 2 ) 1 / 2 . For E > 0, k is real and  , as given by (2.17), is seen to have the for m of traveling plane waves. On the other hand, if E < 0, k  i  is im aginary, then

  e   x e  i  t , and the solution has the for m of a st a nding wave. W h at this m eans is that for the description of scatte ring problem s one should use positive-energy s o lutions (th e se are called scattering states), while for bound-state calculati ons one should work with negative-energy solution s. Fig. 2 illu stra tes the b e havior of the two types of solutions.

The condition at infinity, x   , is tha t  is a plane wave in the scattering problem ,

and an exponentially decaying function in the bound-state problem . In other words, outside the potential (the ex terior region) the scattering state should be a plane wave representing the presence of an incom i ng or outgoing pa rticle, while the bound state should be represented by an exponentially dam p e d wave signifying the localization of the particle inside the poten tia l we ll. In side th e pote n tia l (th e interior reg i on) both solu tio ns are seen to be oscillatory, with the shorter period corres ponding to higher kinetic energy T = E – V.

Fig. 2. Traveling and standing wave functions as solutions to scattering and bound-state problem s respectively.

There are general properties of  which we require for either problem . The s e arise from the fact that we are seeking physic al solutions to the wa ve equation, and that

 ( r ) 2 d 3 r has the inte rpretation of being the p r ob ability of f i nding the p a rticle in an

elem ent of volum e d3r about r . In v i ew of (2.17) we see

 ( r , t ) 2   ( r ) 2 , which m eans

that we are dealing with stat ionary solutions. S i nce a ti m e -independent potential cannot create or destroy particle s, the norm a liza tion con d ition

 d 3 r  ( r ) 2  1 (2.23)

can be applied to the bound-state solutions with integration lim its extending to infinity. For scatter i n g solution s o n e needs to sp ecify an arbitrary but finite vo lum e  for t h e norm a lization of a plane wave. This turns out no t to be a dif f i culty ; in any calcu lation all physical results will be f ound to be independent of  . Other properties of  or 

which can be invoked as conditions for the solu tions to be physically m eaningful are:

(i)	f i nite eve r y w here
(ii)	single-valued and continuous everyw here
(iii)	first derivative continuous
(iv)	  0	when	V  

C ondition ( i ii) is equ i valent to the sta t em ent th at the particle current m u st be continuous everywhere. The curren t is related to the wave function by the expression

j ( r )  ħ [   ( r )   ( r )   ( r )    ( r )] (2.24) 2 mi

This is a generally useful relation whic h can be derived directly from (2.18).