[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 materiais97.886 seguidores
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b l e m o f a p a r t i c l e i n 
a t h r e e - d i m e n s i o n a l b o x r e d u c e s t o t h r e e o n e - d i m e n s i o n a l p r o b l e m s . T h i s i s n o a c c i -
d e n t . I t i s a d i r e c t r e s u l t o f t h e f a c t t h a t 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 a p a r t i c l e i n a 
t h r e e - d i m e n s i o n a l b o x i s a s u m o f t h r e e i n d e p e n d e n t t e r m s 
w h e r e 
h 2 a 2 
H x 
2 m a x
2 
A A A A 
H = H x + H Y + H z 
h 2 a 2 
H Y = - 2 m a l 
h 2 a 2 
H 
z 
2 m a z
2 
I n s u c h a c a s e , w e s a y t h a t t h e H a m i l t o n i a n o p e r a t o r i s s e p a r a b l e . 
T h u s , w e s e e t h a t i f f i i s s e p a r a b l e , t h a t i s , i f f i c a n b e w r i t t e n a s t h e s u m o f t e r m s 
i n v o l v i n g i n d e p e n d e n t c o o r d i n a t e s , s a y 
f i = H
1
( s ) + H
2
( w ) ( 3 . 6 1 ) 
w h e r e s a n d w a r e t h e i n d e p e n d e n t c o o r d i n a t e s , t h e n t h e e i g e n f u n c t i o n s o f f i a r e g i v e n 
b y t h e p r o d u c t s o f t h e e i g e n f u n c t i o n s o f H
1 
a n d H
2
, 
w h e r e 
1 / l n m ( s , w ) = < P n ( s ) c p m ( w ) 
H
1 
( s ) ¢ n ( s ) = E n i f J n ( s ) 
H
2
( w ) c p m ( w ) = E m c p m ( w ) 
a n d E n m , t h e e i g e n v a l u e s o f f i , a r e t h e s u m s o f t h e e i g e n v a l u e s o f H
1 
a n d H
2
, 
E n m = E n + E m 
( 3 . 6 2 ) 
( 3 . 6 3 ) 
( 3 . 6 4 ) 
T h i s i m p o r t a n t r e s u l t p r o v i d e s a s i g n i f i c a n t s i m p l i f i c a t i o n b e c a u s e i t r e d u c e s t h e o r i g i n a l 
p r o b l e m t o s e v e r a l s i m p l e r p r o b l e m s . 
W e h a v e u s e d t h e s i m p l e c a s e o f a p a r t i c l e i n a b o x t o i l l u s t r a t e s o m e o f t h e g e n e r a l 
p r i n c i p l e s a n d r e s u l t s o f q u a n t u m m e c h a n i c s . I n C h a p t e r 4 , w e p r e s e n t a n d d i s c u s s a 
s e t o f p o s t u l a t e s t h a t w e u s e t h r o u g h o u t t h e r e m a i n d e r o f t h i s b o o k . 
96 
Problems 
3-1. Evaluate g = A f, where A and f are given below: 
A f 
(a) SQRT 
(b) 
d3 
-+x3 
dx 3 
e-ax 
(c) 11 dx x3- 2x + 3 
a2 a2 a2 
x
3/z4 (d) -+-+-
ax
2 
al az2 
3-2. Determine whether the following operators are linear or nonlinear: 
a. Af(x) = SQRf(x) [square f(x)] 
b. Af(x) = j*(x) [form the complex conjugate of f(x)] 
c. Af(x) = 0 [multiply j(x) by zero] 
d. Af(x) = [f(xW 1 [take the reciprocal of f(x)] 
e. Af(x) = f(O) [evaluate f(x) at x = 0] 
f. Af(x) = ln f(x) [take the logarithm of f(x)] 
3-3. In each case, show that f(x) is an eigenfunction of the operator given. Find the eigenvalue. 
A f(x) 
(a) 
d2 
dx2 
cos wx 
(b) d 
dt 
eiwt 
(c) d
2 d 
-+2-+3 
dx2 dx 
eOIX 
a 
x2e6y (d) 
ay 
3-4. Show that (cos ax)(cos by)(cos cz) is an eigenfunction of the operator, 
a2 a2 a2 
V2=-+-+-
ax2 al az2 
which is called the Laplacian operator. 
3-5. Write out the operator A2 for A = 
d2 
a. -2 
dx 
d 
b. -+x 
dx 
Hint: Be sure to include f (x) before carrying out the operations. 
d 2 d 
c. - 2 -2x-+1 dx dx 
P r o b l e m s 
3 - 6 . I n S e c t i o n 3 - 5 , w e a p p l i e d t h e e q u a t i o n s f o r a p a r t i c l e i n a b o x t o t h e n e l e c t r o n s i n 
b u t a d i e n e . T h i s s i m p l e m o d e l i s c a l l e d t h e f r e e - e l e c t r o n m o d e l . U s i n g t h e s a m e a r g u m e n t , 
s h o w t h a t t h e l e n g t h o f h e x a t r i e n e c a n b e e s t i m a t e d t o b e 8 6 7 p m . S h o w t h a t t h e f i r s t 
e l e c t r o n i c t r a n s i t i o n i s p r e d i c t e d t o o c c u r a t 2 . 8 x 1 0
4 
c m -
1
\u2022 ( R e m e m b e r t h a t h e x a t r i e n e 
h a s s i x n e l e c t r o n s . ) 
3 - 7 . P r o v e t h a t i f 1 / J ( x ) i s a s o l u t i o n t o t h e S c h r O d i n g e r e q u a t i o n , t h e n a n y c o n s t a n t t i m e s 1 / J ( x ) 
i s a l s o a s o l u t i o n . 
3 - 8 . S h o w t h a t t h e p r o b a b i l i t y a s s o c i a t e d w i t h t h e s t a t e 1 / J n f o r a p a r t i c l e i n a o n e - d i m e n s i o n a l 
b o x o f l e n g t h a o b e y s t h e f o l l o w i n g r e l a t i o n s h i p s : 
-
4 
P r o b ( O : : : : = x : : : : = a / 4 ) = P r o b ( 3 a j 4 : : : : = x : : : : = a ) 
4 
a n d 
P r o b ( a / 4 : : : : = x : : : : = a / 2 ) = P r o b ( a / 2 : : : : = x : : : : = 3 a j 4 ) 
1 
4 
n - l 
( - 1 ) &quot; &quot; 2 
2 n n 
n - l 
1 ( - 1 ) &quot; &quot; 2 
- + - - -
4 2 n n 
n e v e n 
n o d d 
n e v e n 
n o d d 
3 - 9 . W h a t a r e t h e u n i t s , i f a n y , f o r 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 i n a o n e - d i m e n s i o n a l b o x ? 
3 - 1 0 . U s i n g a t a b l e o f i n t e g r a l s , s h o w t h a t 
a n d 
1
a · 2 n n x a 
s m - - d x = -
o a 2 
1
a . 
2 
n n x a
2 
x s m - - d x = -
o a 4 
x
2 
s i n
2 
- - d x = - - - - - - 2 n n 
1
a n n x ( a ) 3 ( 4 n
3
n
3 
) 
0 
a 2 n n 3 
A l l t h e s e i n t e g r a l s c a n b e e v a l u a t e d f r o m 
1
a n n x 
I ( f J ) = e f l x s i n
2 
- - d x 
o a 
S h o w t h a t t h e a b o v e i n t e g r a l s a r e g i v e n b y I ( 0 ) , I ' ( 0 ) , a n d I &quot; ( 0 ) , r e s p e c t i v e l y , w h e r e t h e 
p r i m e s d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o f 3 . U s i n g a t a b l e o f i n t e g r a l s , e v a l u a t e I ( f 3 ) 
a n d t h e n t h e a b o v e t h r e e i n t e g r a l s b y d i f f e r e n t i a t i o n . 
3 - 1 1 . S h o w t h a t 
a 
( x ) = 2 
f o r a l l t h e s t a t e s o f a p a r t i c l e i n a b o x . I s t h i s r e s u l t p h y s i c a l l y r e a s o n a b l e ? 
3 - 1 2 . S h o w t h a t ( p ) = 0 f o r a l l s t a t e s o f a o n e - d i m e n s i o n a l b o x o f l e n g t h a . 
9 7 
98 Chapter 3 I The Schri:idinger Equation and a Particle In a Box 
3-13. Show that 
for a particle in a box is less than a, the width of the box, for any value of n. If ux is the 
uncertainty in the position of the particle, could ux ever be larger than a? 
3-14. Using the trigonometric identity 
sin 2() = 2 sin () cos () 
show that 
sin -- cos --dx = 0 1a nnx nnx o a a 
3-15. Prove that 
n jO 
3-16. Using the trigonometric identity 
. . 1 1 
sm a sm f3 = 2 cos(a - {3) - 2 cos(a + {3) 
show that the particle-in-a-box wave functions (Equations 3.27) satisfy the relation 
m jn 
(The asterisk in this case is superfluous because the functions are real.) If a set of functions 
satisfies the above integral condition, we say that the set is orthogonal and, in particular, 
that 1/f m (x) is orthogonal to 1/f/x ). If, in addition, the functions are normalized, then we 
say that the set is orthonormal. 
3-17. Prove that the set of functions 
n = 0, ±1, ±2, ... 
is orthonormal (cf. Problem 3-16) over the interval-a :S x :S a. A compact way to express 
orthonormality in the 1/fn is to write 
The symbol8mn is called a Kroenecker delta and is defined by 
8