[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)
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[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)


DisciplinaFísico-química I6.512 materiais97.933 seguidores
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R C t a b l e s ) 
x 2 n e - a x d x = -
1
0 0 
2 1 · 3 · 5 · · · ( 2 n - l ) ( r r ) I / 2 
0 2 n + l a n a 
n 2 : 1 
a n d 
r o o 2 d ~ 
l o x 2 n + l e - O I X X = 2 a n + l 
w h e r e n ! i s n f a c t o r i a l , o r n ! = n ( n - l ) ( n - 2 ) · · · ( 1 ) . 
B - 7 . U s e t h e M a x w e l l - B o l t z m a n n d i s t r i b u t i o n i n P r o b l e m B - 6 t o d e t e r m i n e t h e a v e r a g e k i n e t i c 
e n e r g y o f a g a s - p h a s e m o l e c u l e a s a f u n c t i o n o f t e m p e r a t u r e . T h e n e c e s s a r y i n t e g r a l i s g i v e n 
i n P r o b l e m B - 6 . 
7 1 
I 
Erwin Schrodinger was born in Vienna, Austr-ia, on August 12, 1887, and died there in 1961. 
He received his Ph.D. in theoretical physics in 1910 from the University of Vienna. He then 
held a number of positions in Germany and in 1927 succeeded Max Planck at the University 
of Berlin at Planck's request. SchrOdinger left Berlin ip 1933 because of his opposition to Hitler 
and Nazi policies and eventually moved to the University of Graz in Austria in 1936. After the 
invasion of Austria by Germany, he was forcibly removed from his professorship in 1936. 
He then moved to the Institute of Advanced Studies, which was created for him, at the University 
College, Dublin, Ireland. He remained there for 17 years and then retired to his native Austria. 
Schrodinger shared the Nobel Prize for physics with Paul Dirac in 1933 for the "discovery of 
new productive forms of atomic theory." Schrodinger rejected the probabilistic interpretation 
of the wave equation, which led to serious disagreement with Max Born, but they remained 
warm friends in spite of their scientific disagreement. Schrodinger preferred to work alone, and 
so no school developed around him, as it did for several other developers of quantum mechanics. 
His influential book, What is Life?, caused a number of physicists to become interested in 
biology. His personal life, which was rather unconventional, has been engagingly related by 
Walter Moore in his book, Schrodinger (Cambridge University Press, 1989). 
C H A P T E R 
3 
T h e S c h r o d i n g e r E q u a t i o n 
a n d a P a r t i c l e I n a B o x 
T h e S c h r o d i n g e r e q u a t i o n i s o u r f u n d a m e n t a l e q u a t i o n o f q u a n t u m m e c h a n i c s . T h e 
s o l u t i o n s t o t h e S c h r o d i n g e r e q u a t i o n a r e c a l l e d w a v e f u n c t i o n s . W e w i l l s e e t h a t a 
w a v e f u n c t i o n g i v e s a c o m p l e t e d e s c r i p t i o n o f a n y s y s t e m . I n t h i s c h a p t e r , w e p r e s e n t 
a n d d i s c u s s t h e v e r s i o n o f t h e S c h r o d i n g e r e q u a t i o n t h a t d o e s n o t c o n t a i n t i m e a s a 
v a r i a b l e . S o l u t i o n s t o t h e t i m e - i n d e p e n d e n t S c h r o d i n g e r e q u a t i o n a r e c a l l e d s t a t i o n a r y -
s t a t e w a v e f u n c t i o n s b e c a u s e t h e y a r e i n d e p e n d e n t o f t i m e . M a n y p r o b l e m s o f i n t e r e s t 
t o c h e m i s t s c a n b e t r e a t e d b y u s i n g o n l y s t a t i o n a r y - s t a t e w a v e f u n c t i o n s . W e d o n o t 
c o n s i d e r a n y t i m e d e p e n d e n c e u n t i l C h a p t e r 1 3 , w h e r e w e d i s c u s s m o l e c u l a r s p e c -
t r o s c o p y . 
I n t h i s c h a p t e r , w e p r e s e n t t h e t i m e - i n d e p e n d e n t S c h r o d i n g e r e q u a t i o n a n d t h e n a p -
p l y i t t o a f r e e p a r t i c l e o f m a s s m t h a t i s r e s t r i c t e d t o l i e a l o n g a o n e - d i m e n s i o n a l i n t e r v a l 
o f l e n g t h a . T h i s s y s t e m i s c a l l e d a p a r t i c l e i n a b o x a n d t h e c a l c u l a t i o n o f i t s p r o p e r t i e s 
i s a s t a n d a r d i n t r o d u c t o r y p r o b l e m i n q u a n t u m m e c h a n i c s . T h e p a r t i c l e - i n - a - b o x p r o b -
l e m i s s i m p l e , y e t v e r y i n s t r u c t i v e . I n t h e c o u r s e o f d i s c u s s i n g t h i s p r o b l e m , w e w i l l 
i n t r o d u c e t h e p r o b a b i l i s t i c i n t e r p r e t a t i o n o f w a v e f u n c t i o n s . W e u s e t h i s i n t e r p r e t a t i o n 
t o i l l u s t r a t e t h e a p p l i c a t i o n o f t h e U n c e r t a i n t y P r i n c i p l e t o a p a r t i c l e i n a b o x . 
3 - 1 . T h e S c h r o d i n g e r E q u a t i o n I s t h e E q u a t i o n f o r F i n d i n g t h e W a v e 
F u n c t i o n o f a P a r t i c l e 
W e c a n n o t d e r i v e t h e S c h r o d i n g e r e q u a t i o n a n y m o r e t h a n w e c a n d e r i v e N e w t o n ' s l a w s , 
a n d N e w t o n ' s s e c o n d l a w , f = m a , i n p a r t i c u l a r . W e s h a l l r e g a r d t h e S c h r o d i n g e r 
e q u a t i o n t o b e a f u n d a m e n t a l p o s t u l a t e , o r a x i o m , o f q u a n t u m m e c h a n i c s , j u s t a s 
N e w t o n ' s l a w s a r e f u n d a m e n t a l p o s t u l a t e s o f c l a s s i c a l m e c h a n i c s . E v e n t h o u g h w e 
c a n n o t d e r i v e t h e S c h r o d i n g e r e q u a t i o n , w e c a n a t l e a s t s h o w t h a t i t i s p l a u s i b l e a n d 
p e r h a p s e v e n t r a c e S c h r o d i n g e r ' s o r i g i n a l l i n e o f t h o u g h t . W e f i n i s h e d C h a p t e r 1 w i t h 
a d i s c u s s i o n o f m a t t e r w a v e s , a r g u i n g t h a t m a t t e r h a s w a v e l i k e c h a r a c t e r i n a d d i t i o n t o 
i t s o b v i o u s p a r t i c l e l i k e c h a r a c t e r . A s o n e s t o r y g o e s , a t a m e e t i n g a t w h i c h t h i s n e w 7 3 
74 Chapter 3 I The Schri:idinger Equation and a Particle In a Box 
idea of matter waves was being discussed, someone mentioned that if indeed matter 
does possess wavelike properties, then there must be some sort of wave equation that 
governs them. 
Let us start with the classical one-dimensional wave equation for simplicity: 
32u 1 32u 
ax2 - v2 """iif2 (3.1) 
We have seen in Chapter 2 that Equation 3.1 can be solved by the method of separation 
of variables and that u(x, t) can be written as the product of a function of x and a 
harmonic or sinusoidal function of time. We will express the temporal part as cos wt 
(cf. Equation 2.25) and write u(x, t) as 
u(x, t) = 1/f(x) coswt (3.2) 
Because 1/1 (x) is the spatial factor of the amplitude u (x, t ), we will call1/f (x) the spatial 
amplitude of the wave. If we substitute Equation 3.2 into Equation 3.1, we obtain an 
equation for the spatial amplitude 1/f(x), 
(3.3) 
Using the fact that w = 2nv and that vA = v, Equation 3.3 becomes 
d 21/f 4n2 
dx2 + yl/l(x) = 0 (3.4) 
We now introduce the idea of de Broglie matter waves into Equation 3.4_ Tb~_total 
~~---
energy of a particle is the sum of its kinetic energy an.d its potential energy, 
p2 
E =- + V(x) 
2m 
(3.5) 
where p = m v is the momentum of the particle and V (x) is its potential energy. If we 
solve Eqw1tion3~5 for the mo~~ntum p, we find 
p = {2m[E- V(x)]} 112 (3.6) 
According to the de Broglie formula, 
h h 
A - - - --------,-= 
- p - {2m[E- V(x)]} 112 
Substituting this into Equation 3.4, we find 
d 2 1/l 2m 
dx2 + r;z[E- V(x)]l/l(x) = 0 (3.7) 
where h (called h bar)= hj2n. 
~---·- --.. -----·--- ···---··-~---~· 
3 - 2 . C l a s s i c a l - M e c h a n i c a