[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)
1279 pág.

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


DisciplinaFísico-química I6.478 materiais97.625 seguidores
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p e r p e n d i c u l a r t o t h e y a x i s . 
C - 6 . F i n d t h e a n g l e b e t w e e n t h e t w o v e c t o r s A = - i + 2 j + k a n d B = 3 i - j + 2 k . 
C - 7 . D e t e r m i n e C = A x B g i v e n t h a t A = - i + 2 j + k a n d B = 3 i - j + 2 k . W h a t i s B x A 
e q u a l t o ? 
C - 8 . S h o w t h a t A x A = 0 . 
C - 9 . U s i n g E q u a t i o n s C . 1 4 , p r o v e t h a t A x B i s g i v e n b y E q u a t i o n C . l 5 . 
C - 1 0 . S h o w t h a t I L l = m v r f o r c i r c u l a r m o t i o n . 
C - 1 1 . S h o w t h a t 
a n d 
d d A d B 
- ( A · B ) = - - B + A · -
d t d t d t 
d d A d B 
- ( A X B ) = - X B + A X -
d t d t d t 
C - 1 2 . U s i n g t h e r e s u l t s o f P r o b l e m C - 1 1 , p r o v e t h a t 
A X d 2 A = ! ! . _ ( A X d A ) 
d t
2 
d t d t 
C - 1 3 . I n v e c t o r n o t a t i o n , N e w t o n ' s e q u a t i o n s f o r a s i n g l e p a r t i c l e a r e 
d
2
r 
m -
2 
= F ( x , y , z ) 
d t 
B y o p e r a t i n g o n t h i s e q u a t i o n f r o m t h e l e f t b y r x a n d u s i n g t h e r e s u l t o f P r o b l e m C - 1 2 , 
s h o w t h a t 
d L 
d t = r x F 
w h e r e L = m r x d r j d t = r x m d r j d t = r x m v = r x p . T h i s i s t h e f o r m o f N e w t o n ' s 
e q u a t i o n s f o r a r o t a t i n g s y s t e m . N o t i c e t h a t d L / d t = 0 , o r t h a t a n g u l a r m o m e n t u m i s 
c o n s e r v e d i f r x F = 0 . C a n y o u i d e n t i f y r x F ? 
1 1 3 
I 
Werner Heisenberg was born on December 5, 1901 in Duisburg, Germany, grew up in 
Munich, and died in 1976. In 1923, Heisenberg received his Ph.D. in physics from the 
University of Munich. He then spent a year as an assistant to Max Born at the University 
of Gottingen and three years with Niels Bohr in Copenhagen. He was chair of theoretical 
physics at the University of Leipzig from 1927 to 1941, the youngest to have received such 
an appointment. Because of a deep loyalty to Germany, Heisenberg opted to stay in Germany 
when the Nazis came to power. During World War II, he was in charge of German research 
on the atomic bomb. After the war, he was named director of the Max Planck Institute for 
Physics, where he strove to rebuild German science. Heisenberg developed one of the first 
formulations of quantum mechanics, but it was based on matrix algebra, which was less easy 
to use than the wave equation of Schrodinger. The two formulations, however, were later 
shown to be equivalent. His Uncertainty Principle, which he published in 1927, illuminates 
a fundamental principle of nature involving the measurement and observation of physical 
quantities. Heisenberg was awarded the 1932 Nobel Prize for physics in 1933 "for the creation 
of quantum mechanics." His role in Nazi Germany is somewhat clouded, prompting one author 
(David Cassidy) to title his biography of Heisenberg Uncertainty (W.H. Freeman, 1993). 
C H A P T E R 4 
S o m e P o s t u l a t e s a n d G e n e r a l P r i n c i p l e s 
o f Q u a n t u m M e c h a n i c s 
U p t o n o w , w e h a v e m a d e a n u m b e r o f c o n j e c t u r e s c o n c e r n i n g t h e f o r m u l a t i o n o f 
q u a n t u m m e c h a n i c s . F o r e x a m p l e , w e h a v e b e e n l e d t o v i e w t h e v a r i a b l e s o f c l a s s i c a l 
m e c h a n i c s a s r e p r e s e n t e d i n q u a n t u m m e c h a n i c s b y o p e r a t o r s . T h e s e o p e r a t e o n w a v e 
f u n c t i o n s t o g i v e t h e a v e r a g e o r e x p e c t e d r e s u l t s o f m e a s u r e m e n t s . I n t h i s c h a p t e r , w e 
f o r m a l i z e t h e v a r i o u s c o n j e c t u r e s w e m a d e i n C h a p t e r 3 a s a s e t o f p o s t u l a t e s a n d t h e n 
d i s c u s s s o m e g e n e r a l t h e o r e m s t h a t f o l l o w f r o m t h e s e p o s t u l a t e s . T h i s f o r m a l i z a t i o n 
i s s i m i l a r t o s p e c i f y i n g a s e t o f a x i o m s i n g e o m e t r y a n d t h e n l o g i c a l l y d e d u c i n g t h e 
c o n s e q u e n c e s o f t h e s e a x i o m s . T h e u l t i m a t e t e s t o f w h e t h e r t h e a x i o m s o r p o s t u l a t e s 
a r e s e n s i b l e i s t o c o m p a r e t h e e n d r e s u l t s w i t h e x p e r i m e n t a l d a t a . H e r e w e p r e s e n t a 
f a i r l y e l e m e n t a r y s e t o f p o s t u l a t e s t h a t w i l l s u f f i c e f o r a l l t h e s y s t e m s w e d i s c u s s i n t h i s 
b o o k a n d f o r a l m o s t a l l s y s t e m s o f i n t e r e s t i n c h e m i s t r y . 
4 - 1 . T h e S t a t e o f a S y s t e m I s C o m p l e t e l y S p e c i f i e d b y i t s 
W a v e F u n c t i o n 
C l a s s i c a l m e c h a n i c s d e a l s w i t h q u a n t i t i e s c a l l e d d y n a m i c a l v a r i a b l e s , s u c h a s p o s i t i o n , 
m o m e n t u m , a n g u l a r m o m e n t u m , a n d e n e r g y . A m e a s u r a b l e d y n a m i c a l v a r i a b l e i s 
c a l l e d a n o b s e r v a b l e . T h e c l a s s i c a l - m e c h a n i c a l s t a t e o f a p a r t i c l e a t a n y p a r t i c u l a r 
t i m e i s s p e c i f i e d c o m p l e t e l y b y t h e t h r e e p o s i t i o n c o o r d i n a t e s ( x , y , z ) a n d t h e t h r e e 
m o m e n t a ( p , p , p ) o r v e l o c i t i e s ( v , v , v ) a t t h a t t i m e . T h e t i m e e v o l u t i o n o f t h e 
X Y Z X y Z 
s y s t e m i s g o v e r n e d b y N e w t o n ' s e q u a t i o n s , 
d
2
x d
2
y d
2
z 
m -
2 
= F ( x , y , z ) , m -
2 
= F ( x , y , z ) , m -
2 
= F ( x , y , z ) ( 4 . 1 ) 
d t X d t y d t z 
w h e r e F , F , a n d F a r e t h e c o m p o n e n t s o f t h e f o r c e , F ( x , y , z ) . N e w t o n ' s e q u a t i o n s , 
X y Z 
a l o n g w i t h t h e i n i t i a l p o s i t i o n a n d m o m e n t u m o f a p a r t i c l e , g i v e u s x ( t ) , y ( t ) , a n d z ( t ) , 
w h i c h d e s c r i b e t h e p o s i t i o n o f t h e p a r t i c l e a s a f u n c t i o n o f t i m e . T h e t h r e e - d i m e n s i o n a l 
p a t h d e s c r i b e d b y x ( t ) , y ( t ) , a n d z ( t ) i s c a l l e d t h e t r a j e c t o r y o f t h e p a r t i c l e . T h e 1 1 5 
116 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics 
trajectory of a particle offers a complete description of the state of the particle. Classical 
mechanics provides a method for calculating the trajectory of a particle in terms of the 
forces acting upon the particle through Newton's equations, Equations 4.1. 
Newton's equations plus the forces involved enable us to deduce the entire history 
and predict the entire future behavior of the particle. We should suspect immediately 
that such predictions are not possible in quantum mechanics because the Uncertainty 
Principle tells us that we cannot specify or determine the position and momentum 
of a particle simultaneously to any desired precision. The Uncertainty Principle is of 
no practical importance for macroscopic bodies (see Example 1-10), and so classical 
mechanics is a perfectly adequate prescription for macroscopic bodies. For very small 
bodies, such as electrons, atoms, and molecules, however, the