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

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


DisciplinaFísico-química I6.510 materiais97.917 seguidores
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4 4 ) 
w h e r e [ A , B ] = A B - B A i s t h e c o m m u t a t o r o f A a n d B a n d t h e v e r t i c a l b a r s d e n o t e 
t h e a b s o l u t e v a l u e o f t h e i n t e g r a l . 
I f A a n d B c o m m u t e , t h e n t h e r i g h t s i d e o f E q u a t i o n 4 . 4 4 i s z e r o , s o C J a , C J b , o r 
b o t h c o u l d e q u a l z e r o s i m u l t a n e o u s l y . T h e r e i s n o r e s t r i c t i o n o n t h e u n c e r t a i n t i e s i n t h e 
m e a s u r e m e n t s o f a a n d b . I f , o n t h e o t h e r h a n d , A a n d B d o n o t c o m m u t e , t h e n t h e r i g h t 
s i d e o f E q u a t i o n 4 . 4 4 w i l l n o t e q u a l z e r o . T h u s , t h e r e i s a r e c i p r o c a l r e l a t i o n b e t w e e n 
O " a a n d C J b ; o n e c a n a p p r o a c h z e r o o n l y i f t h e o t h e r a p p r o a c h e s i n f i n i t y . T h e r e f o r e , b o t h 
a a n d b c a n n o t b e m e a s u r e d s i m u l t a n e o u s l y t o a r b i t r a r y p r e c i s i o n . 
L e t ' s c o n s i d e r a s a n e x a m p l e , t h e s i m u l t a n e o u s m e a s u r e m e n t o f t h e m o m e n t u m 
a n d p o s i t i o n o f a p a r t i c l e , s o t h a t A = P x a n d B = X i n E q u a t i o n 4 . 4 4 . E q u a t i o n 4 . 4 2 
t e l l s u s t h a t [ P x , X ] = - i n i , a n d s o E q u a t i o n 4 . 4 4 g i v e s 
O " p ( J x ~ ~I/ 1 / f * ( x ) ( - i h i ) l j f ( x ) d x l 
1 n 
> - 1 - i n I > -
- 2 - 2 
( 4 . 4 5 ) 
E q u a t i o n 4 . 4 5 i s t h e u s u a l e x p r e s s i o n g i v e n f o r t h e U n c e r t a i n t y P r i n c i p l e f o r m o m e n t u m 
a n d p o s i t i o n . I f C J P i s m a d e t o b e s m a l l , t h e n O " x i s n e c e s s a r i l y l a r g e , a n d i f O " x i s m a d e 
t o b e s m a l l , t h e n C J i s n e c e s s a r i l y l a r g e . T h u s , t h e m o m e n t u m a n d p o s i t i o n c a n n o t b e 
p 
m e a s u r e d s i m u l t a n e o u s l y t o a r b i t r a r y p r e c i s i o n . 
T h u s , w e s e e t h a t t h e r e i s a n i n t i m a t e c o n n e c t i o n b e t w e e n c o m m u t i n g o p e r a t o r s 
a n d t h e U n c e r t a i n t y P r i n c i p l e . I f t w o o p e r a t o r s A a n d B c o m m u t e , t h e n a a n d b c a n b e 
m e a s u r e d s i m u l t a n e o u s l y t o a n y p r e c i s i o n . I f t w o o p e r a t o r s A a n d B d o n o t c o m m u t e , 
t h e n a a n d b c a n n o t b e m e a s u r e d s i m u l t a n e o u s l y t o a r b i t r a r y p r e c i s i o n . 
1 3 3 
134 
Problems 
4-1. Which of the following candidates for wave functions are normalizable over the indicated 
intervals? 
2 
a. e-x 12 ( -00, 00) b. ex (0, 00) c. e;e (0, 2rr) d. sinhx (0, oo) 
e. xe-x (0, oo) 
Normalize those that can be normalized. Are the others suitable wave functions? 
4-2. Which of the following wave functions are normalized over the indicated two-dimensional 
intervals? 
O:sx<oo 
O:Sy<oo 
( 
4 ) 112 rrx rry 
c. ab sin -;; sin b 
Normalize those that aren't. 
b. e-(x+y)/2 O:sx<oo 
O:sy<oo 
4-3. Why does 1/1*1/1 have to be everywhere real, nonnegative, finite, and of definite value? 
4-4. In this problem, we will prove that the form of the SchrOdinger equation imposes the 
condition that the first derivative of a wave function be continuous. The Schrodinger 
equation is 
d 21/J 2m 
- 2 + - 2 [E- V(x)]l/J(x) = 0 dx h 
If we integrate both sides from a - E to a + E, where a is an arbitrary value of x and E is 
infinitesimally small, then we have 
dl/11 2m 1a+< 
- = - 2 [V(x)- E]l/J(x)dx dx x=a-< h a-< 
Now show that d 1/1/ dx is continuous if V (x) is continuous. 
Suppose now that V(x) is not continuous at x =a, as in 
------------------L-----------------\u2022x 
a 
Show that 
dl/11 2m 
- = - 2 ['-'; + V,- 2£]1/J(a)E dx x=a-< h 
P r o b l e m s 
s o t h a t d 1 j ; / d x i s c o n t i n u o u s e v e n i f V ( x ) h a s a f i n i t e d i s c o n t i n u i t y . W h a t i f V ( x ) h a s a n 
i n f i n i t e d i s c o n t i n u i t y , a s i n t h e p r o b l e m o f a p a r t i c l e i n a b o x ? A r e t h e f i r s t d e r i v a t i v e s o f 
t h e w a v e f u n c t i o n s c o n t i n u o u s a t t h e b o u n d a r i e s o f t h e b o x ? 
4 - 5 . D e t e r m i n e w h e t h e r t h e f o l l o w i n g f u n c t i o n s a r e a c c e p t a b l e o r n o t a s s t a t e f u n c t i o n s o v e r 
t h e i n d i c a t e d i n t e r v a l s . 
1 
a . ( 0 , o o ) b . e -
2
x s i n h x ( 0 , o o ) 
X 
C . e - x C O S X ( 0 , 0 0 ) 
d . e x ( - o o , o o ) 
4 - 6 . C a l c u l a t e t h e v a l u e s o f O ' i = ( £
2
) - ( £ )
2 
f o r a p a r t i c l e i n a b o x i n t h e s t a t e d e s c r i b e d b y 
(
6 3 0 )
1 1 2 
1 / J ( x ) = ~ x
2
( a - x )
2 
O : = : x : = : a 
4 - 7 . C o n s i d e r a f r e e p a r t i c l e c o n s t r a i n e d t o m o v e o v e r t h e r e c t a n g u l a r r e g i o n 0 : : : x : : : a , 
0 : : : y : : : b . T h e e n e r g y e i g e n f u n c t i o n s o f t h i s s y s t e m a r e 
( 
4 )
1
/
2 
n n x n n y 
1 j ; ( x , y ) = - s i n _ x _ _ s i n _ Y _ 
n x , n y a b a b 
T h e H a m i l t o n i a n o p e r a t o r f o r t h i s s y s t e m i s 
, n 2 ( a 2 a
2 
) 
H = - 2 m a x
2 
+ a i 
S h o w t h a t i f t h e s y s t e m i s i n o n e o f i t s e i g e n s t a t e s , t h e n 
( J ' i = ( £ 2 ) - ( £ ) 2 = 0 
4 - 8 . T h e m o m e n t u m o p e r a t o r i n t w o d i m e n s i o n s i s 
, ( a a ) 
p = - i n i a x + j a y 
n x = 1 , 2 , 3 , 
n Y 1 , 2 , 3 , 
U s i n g t h e w a v e f u n c t i o n g i v e n i n P r o b l e m 4 - - 7 , c a l c u l a t e t h e v a l u e o f ( p ) a n d t h e n 
( ] ' } = ( p 2 ) - ( p ) 2 
C o m p a r e y o u r r e s u l t w i t h 0 ' } i n t h e o n e - d i m e n s i o n a l c a s e . 
4 - 9 . S u p p o s e t h a t a p a r t i c l e i n a t w o - d i m e n s i o n a l b o x ( c f . P r o b l e m 4 - - 7 ) i s i n t h e s t a t e 
3 0 
1 / J ( x , y ) = 5 5 l f 2 x ( a - x ) y ( b - y ) 
( a b ) 
S h o w t h a t 1 j ; ( x , y ) i s n o r m a l i z e d , a n d t h e n c a l c u l a t e t h e v a l u e o f ( E ) a s s o c i a t e d w i t h t h e 
s t a t e d e s c r i b e d b y 1 / J ( x , y ) . 
1 3 5 
136 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics 
4-1 0. Show that 
1/lo(x) = n-114e-x2 !2 
1/lt (x) = (4/n)114xe_x2/2 
1/J2(x) = (4n)-11\2x2 - 1)e-x
212 
are orthonormal over the interval -oo < x < oo. 
4-11. Show that the polynomials 
satisfy the orthogonality relation 
where o1n is the Kroenecker delta (Equation 4.30). 
4-12. Show that the set of functions (2/a) 112 cos(nnxja), n = 0, 1, 2, ... is orthonormal over 
the interval 0 :S x :S a. 
4-13. Prove that if o is the Kroenecker delta 
nm 
0 - { 1 
nm- 0 
then 
00 
&quot;co =c 
.L....t n nm m 
n=l 
and 
&quot;&quot;abo =&quot;ab L....t .L....t n m nm L....t n n 
These results will be used later. 
4-14. Determine whether or not the following pairs of operators commute. 
A s 
(a) d 
dx 
d 2 d 
-+2-
dx2 dx 
(b) X d 
dx 
(c) SQR SQRT 
2 d d2 (d) X-
dx2 dx 
4-15. In ordinary