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

DisciplinaFísico-química I6.478 materiais97.625 seguidores
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```algebra, (P + Q)(P- Q) = P2 - Q2 . Expand (P + Q)(P- (!).Under
what conditions do we find the same result as in the case of ordinary algebra?
P r o b l e m s
4 - 1 6 . E v a l u a t e t h e c o m m u t a t o r [ A , B ] , w h e r e A a n d B a r e g i v e n b e l o w .
A
B
d 2
( a )
d x
2
X
d d
( b )
- - x
- + x
d x d x
( c )
1 x d x
d
d x
d 2
d
( d )
- - x
- + x z
d x
2
d x
4 - 1 7 . R e f e r r i n g t o T a b l e 4 . 1 f o r t h e o p e r a t o r e x p r e s s i o n s f o r a n g u l a r m o m e n t u m , s h o w t h a t
[ i , i ] = i h i
X y Z
[ i , i ] = i h i
Y Z X
a n d
[ i , i J = i h i
Z X y
( D o y o u s e e a p a t t e r n h e r e t o h e l p r e m e m b e r t h e s e c o m m u t a t i o n r e l a t i o n s ? ) W h a t d o
t h e s e e x p r e s s i o n s s a y a b o u t t h e a b i l i t y t o m e a s u r e t h e c o m p o n e n t s o f a n g u l a r m o m e n t u m
s i m u l t a n e o u s l y ?
4 - 1 8 . D e f i n i n g
i z = i z + i z + i z
X J Z
s h o w t h a t i
2
c o m m u t e s w i t h e a c h c o m p o n e n t s e p a r a t e l y . W h a t d o e s t h i s r e s u l t t e l l y o u
a b o u t t h e a b i l i t y t o m e a s u r e t h e s q u a r e o f t h e t o t a l a n g u l a r m o m e n t u m a n d i t s c o m p o n e n t s
s i m u l t a n e o u s l y ?
4 - 1 9 . I n C h a p t e r 6 w e w i l l u s e t h e o p e r a t o r s
i + = i + i i
X y
a n d
L = i x - i f y
S h o w t h a t
i i = i
2
- i
2
+ h i
+ - z z
[ i z ' i + ] = h i +
a n d t h a t
[ i , i ] = - h i
z - -
1 3 7
138 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics
4-20. Consider a particle in a two-dimensional box. Determine [X, P], [X, P], [Y, P ], and y X y [Y,PJ
4-21. Can the position and total angular momentum of any electron be measured simultaneously
to arbitrary precision?
4-22. Can the angular momentum and kinetic energy of a particle be measured simultaneously
to arbitrary precision?
4-23. Using the result of Problem 4-20, what are the &quot;uncertainty relationships&quot; /'<,.x/'<,.pY and
&quot;&quot;'-Y&quot;&quot;'-Px equal to?
4-24. We can define functions of operators through their Taylor series (MathChapter 1). For
example, we define the operator exp (S) by
Under what conditions does the equality
hold?
4-25. In this chapter, we learned that if ljfn is an eigenfunction of the time-independent
Schrodinger equation, then
Show that if 1/1 m and 1/1 n are both stationary states of ii, then the state
satisfies the time-dependent Schrodinger equation.
4-26. Starting with
(x) =I \ll*(x, t)x\ll(x, t)dx
and the time-dependent SchrOdinger equation, show that
d(x) I i A A
-- = \11*-(Hx- xH)\IIdx
dt 1i
Given that
show that
A A 1i2 d 1i2 i A i1i A
Hx- xH = -2-- = ---P = --P
2m dx m 1i x m x
P r o b l e m s
F i n a l l y , s u b s t i t u t e t h i s r e s u l t i n t o t h e e q u a t i o n f o r d ( x ) j d t t o s h o w t h a t
I n t e r p r e t t h i s r e s u l t .
m d ( x ) = ( P )
d t
4 - 2 7 . G e n e r a l i z e t h e r e s u l t o f P r o b l e m 4 - 2 6 a n d s h o w t h a t i f F i s a n y d y n a m i c a l q u a n t i t y , t h e n
/ / _ _ ,
d ( F ) =flli*~(HF- F H ) I l i d x
d t
U s e t h i s e q u a t i o n t o s h o w t h a t
d ( P ) = ( - d V )
d t d x
I n t e r p r e t t h i s r e s u l t . T h i s l a s t e q u a t i o n i s k n o w n a s E h r e n f e s t ' s t h e o r e m .
~-28. T h e f a c t t h a t e i g e n v a l u e s , w h i c h c o r r e s p o n d t o p h y s i c a l l y o b s e r v a b l e q u a n t i t i e s , m u s t
b e r e a l i m p o s e s a c e r t a i n c o n d i t i o n o n q u a n t u m - m e c h a n i c a l o p e r a t o r s . T o s e e w h a t t h i s
c o n d i t i o n i s , s t a r t w i t h
A l f r = a l f r
( 1 )
w h e r e A a n d 1 / r m a y b e c o m p l e x , b u t a m u s t b e r e a l . M u l t i p l y E q u a t i o n 1 f r o m t h e l e f t b y
1 / r * a n d t h e n i n t e g r a t e t o o b t a i n
J 1 / r * A l f r d r = a J 1 / r * l f r d r = a
( 2 )
N o w t a k e t h e c o m p l e x c o n j u g a t e o f E q u a t i o n 1 , m u l t i p l y f r o m t h e l e f t b y l f r , a n d t h e n
i n t e g r a t e t o o b t a i n
J 1 / r A * l f r * d r = a * = a
E q u a t e t h e l e f t s i d e s o f E q u a t i o n s 2 a n d 3 t o g i v e
I 1 / r * A l f r d r = J 1 / r . A * l / r * d r
- 0
- : ; : : : ; ;
( 3 )
t \ e V v r l ; { i c Y I ' l c-p4~&quot;'l-c¥
&quot;A~ e · r , V Y t V &quot; ' w t :
; \ · r . - I Y q . e -~ (
4
)
T h i s i s t h e c o n d i t i o n t h a t a n o p e r a t o r m u s t s a t i s f y i f i t s e i g e n v a l u e s a r e t o b e r e a l . S u c h
o p e r a t o r s a r e c a l l e d H e r m i t i a n o p e r a t o r s . --~ : : = : : : ; : : : : : : : :
4 - 2 9 . I n t h i s p r o b l e m , w e w i l l p r o v e t h a t n o t o n l y a r e t h e e i g e n v a l u e s o f H e r m i t i a n o p e r a t o r s
r e a l b u t t h a t t h e i r e i g e n f u n c t i o n s a r e o r t h o g o n a l . C o n s i d e r t h e t w o e i g e n v a l u e e q u a t i o n s
A , , , = a 1 / r a n d A 1 / r = a 1 / r
'~&quot;n n n m m m
M u l t i p l y t h e f i r s t e q u a t i o n b y 1/r~ a n d i n t e g r a t e ; t h e n t a k e t h e c o m p l e x c o n j u g a t e o f t h e
s e c o n d , m u l t i p l y b y 1 j r n , a n d i n t e g r a t e . S u b t r a c t t h e t w o r e s u l t i n g e q u a t i o n s f r o m e a c h o t h e r
t o g e t
/ _ : 1/r~Alfrndx- / _ : 1/rn.A*lfr~dx = ( a n - a ; ) / _ : 1/r~lfrndx
1 3 9
&quot; &quot; ' r
y o . e J
140 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics
Because A is Hermitian, the left side is zero, and so
Discuss the two possibilities n = m and n =f- m. Show that an =a;, which is just another
proof that the eigenvalues are real. When n =f- m, show that
m =f- n
if the system is nondegenerate. Are 1/J m and 1/Jn necessarily orthogonal if they are degenerate?
4-30. All the operators in Table 4.1 are Hermitian. In this problem, we show how to determine
if an operator is Hermitian. Consider the operator A = d I dx. If A is Hermitian, it will
satisfy Equation 4 of Problem 4--28. Substitute A = d I dx into Equation 4 and integrate by
parts to obtain
1
00 dl/J 00 100 do/*
1/1*-dx = [ 1/1*1/1[- 1/J-dx
-oo dx -oo -oo dx
For a wave function to be normalizable, it must vanish at infinity, so the first term on the
right side is zero. Therefore, we have
100 d 100 d 1/J*-1/Jdx =- 1/J-1/J*dx -oo dx _00 dx
For an arbitrary function 1/J (x ), d I dx does not satisfy Equation 4 of Problem 4-28, so it is
not Hermitian.
4-31. Following the procedure in Problem 4--30, show that the momentum operator is Hermi-
tian.
4-32. Specify which of the following operators are Hermitian: idldx, d 2 ldx2 , and id2 ldx2 .
Assume that -oo < x < oo and that the functions on which these operators operate are
appropriately well behaved at infinity.
Problems 4-33 through 4-38 examine systems with piece-wise constant potentials.
4-33. Consider a particle moving in the potential energy
V(x)```