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

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```consequences of the
Uncertainty Principle are far from negligible and the classical-mechanical picture is
not valid. This leads us to our first postulate of quantum mechanics:
Postulate 1
The state of a quantum-mechanical system is completely specified by a function
1/1 (x) that depends upon the coordinate of the particle. All possible information
about the system can be derived from 1/1 (x ). This function, called the wave
function or the state function, has the important property that 1/J* (x) 1/1 (x )dx is
the probability that the particle lies in the interval dx, located at the position x.
In Postulate 1 we have assumed, for simplicity, that only one coordinate is needed
to specify the position of a particle, as in the case of a particle in a one-dimensional
box. In three dimensions, we would have that 1/J*(x, y, z)o/(x, y, z)dxdydz is the
probability that the particle described by 1/1 (x, y, z) lies in the volume element dxdydz
located at the point (x, y, z). To keep the notation as simple as possible, we will express
most of the equations to come in one dimension.
If there is more than one particle, say two, then 1jl*(x1' x2)1jl(x1, x 2)dx1dx2 is the
probability that particle llies in the interval dx1 located at x1, and that particle 2 lies in
the interval dx2 located at x2 \u2022 Postulate l says that the state of a quantum-mechanical
system such as two electrons is completely specified by this function and that nothing
else is required.
Because the square of the wave function has a probabilistic interpretation, it must
satisfy certain physical requirements. The total probabilty of finding a particle some-
where must be unity, thus
J 1jl*(x)1jl(x)dx = 1 (4.2)
all space
The notation &quot;all space&quot; here means that we integrate over all possible values of x.
We have expressed Equation 4.2 for a one-dimensional system; for two- or three-
dimensional systems, Equation 4.2 would be a double or a triple integral. Wave func-
tions that satisfy Equation 4.2 are said to be normalized.
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
E X A M P L E 4 - 1
T h e w a v e f u n c t i o n s f o r a p a r t i c l e r e s t r i c t e d t o l i e i n a r e c t a n g u l a r r e g i o n o f l e n g t h s a
a n d b ( 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 ) a r e
(
4 ) 1 / 2 n : r r x n : r r y
1 / f ( x , y ) = - s i n _ x _ _ s i n _ Y _
&quot; x &quot; y a b a b
n x = 1 , 2 , . . .
n y = 1 , 2 , . . .
S h o w t h a t t h e s e w a v e f u n c t i o n s a r e n o r m a l i z e d .
S O L U T I O N : W e w i s h t o s h o w t h a t
1 Q 1 b d x d y l / f * ( x , y ) l / f ( x , y ) =
4 1 &quot; 1 b n : r r x n : r r y
- d x d y s i n
2
_ x _ _ s i n
2
_ Y _ = 1
a b
0 0
a b
o : : : : : x : : : : : a
O : : S y : : S b
T h i s d o u b l e i n t e g r a l a c t u a l l y f a c t o r s i n t o a p r o d u c t o f t w o s i n g l e i n t e g r a l s :
4 1 &quot; . n : r r x 1 b . n : r r y ?
- d x s m
2
_ x _ _ d y s m
2
_ Y _ = ' = 1
a b
0
a
0
b
E q u a t i o n 3 . 2 6 s h o w s t h a t t h e f i r s t i n t e g r a l i s e q u a l t o a / 2 a n d t h a t t h e s e c o n d i s e q u a l
t o b j 2 , s o t h a t w e h a v e
4 a b
- · - · - = 1
a b 2 2
a n d t h u s t h e a b o v e w a v e f u n c t i o n s a r e n o r m a l i z e d .
E v e n i f t h e i n t e g r a l i n E q u a t i o n 4 . 2 e q u a l s s o m e c o n s t a n t A = f . 1 , w e c a n d i v i d e
1 j J ( x ) b y A
1
/
2
t o m a k e i t n o r m a l i z e d . O n t h e o t h e r h a n d , i f t h e i n t e g r a l d i v e r g e s ( i . e . g o e s
t o i n f i n i t y ) , n o r m a l i z i n g 1 j J ( x ) i s n o t p o s s i b l e , a n d i t i s n o t a c c e p t a b l e a s a s t a t e f u n c t i o n
( s e e E x a m p l e 4 - 2 b ) . F u n c t i o n s t h a t c a n b e n o r m a l i z e d a r e s a i d t o b e n o r m a l i z a b l e .
O n l y n o r m a l i z a b l e f u n c t i o n s a r e a c c e p t a b l e a s s t a t e f u n c t i o n s . F u r t h e r m o r e , f o r 1 j J ( x )
t o b e a p h y s i c a l l y a c c e p t a b l e w a v e f u n c t i o n , i t a n d i t s f i r s t d e r i v a t i v e m u s t b e s i n g l e -
v a l u e d , c o n t i n u o u s , a n d f i n i t e ( c f . P r o b l e m 4 - 4 ) . W e s u m m a r i z e t h e s e r e q u i r e m e n t s b y
s a y i n g t h a t 1 j J ( x ) m u s t b e w e l l b e h a v e d .
E X A M P L E 4 - 2
D e t e r m i n e w h e t h e r e a c h o f t h e f o l l o w i n g f u n c t i o n s i s a c c e p t a b l e o r n o t a s a s t a t e
f u n c t i o n 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 :
a .
e - x
( 0 , o o )
b .
e - x
( - o o , o o )
c .
s i n -
1
x
( - 1 , 1 )
d .
e - l x l
( - o o , o o )
1 1 7
118 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics
SOLUTION:
a. acceptable; e-x is single-valued, continuous, finite, and normalizable over
the interval (0, oo).
b. Not acceptable; e-x cannot be normalized over the interval ( -oo, oo)
because e-x diverges as x --* -oo.
c. Not acceptable; sin- 1 xis a multivalued function. For example,
\u2022 _ 1 n n n sm 1 = -, - + 2n, - + 4n, etc
2 2 2
d. Not acceptable; the first derivative of e-lxl is not continuous at x = 0.
4-2. Quantum-Mechanical Operators Represent
Classical-Mechanical Variables
In Chapter 3, we concluded that classical mechanical quantities are represented by
linear operators in quantum mechanics. We now formalize this conclusion by our next
postulate.
Postulate 2
To every observable in classical mechanics there corresponds a linear operator
in quantum mechanics.
We have seen some examples of the correspondence between observables and operators
in Chapter 3. These correspondences are listed in Table 4.1.
The only new entry in Table 4.1 is that for the angular momentum. Although we
discussed angular momentum briefly in MathChapter C, we will discuss it more fully
here. Linear momentum is given by mv and is usually denoted by the symbol p. Now
consider a particle rotating in a plane about a fixed center as in Figure 4.1. Let vrot
FIGURE 4.1
The rotation of a single particle about a fixed
point.
T A B L E 4 . 1
C l a s s i c a l - m e c h a n i c a l o b s e r v a b l e s a n d t h e i r c o r r e s p o n d i n g 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 .
O b s e r v a b l e O p e r a t o r
N a m e S y m b o l
S y m b o l O p e r a t i o n
P o s i t i o n
X
X
M u l t i p l y b y x
r
R
M u l t i p l y b y r
a
M o m e n t u m
P x
p
- i n -
X
a x
p
p
· ( a . a a )
- z 1 i I - + J - + k -
a x a y a z
1 i 2 a z
K i n e t i c e n e r g y K
K
- 2 m a x
2
X
X
K K .
T i z ( a z a z a z )
- 2 m a x
2
+ a l + a z
2
1 i 2
= - - \ 7 2
2 m
P o t e n t i a l e n e r g y V ( x )
v ( x )
M u l t i p l y b y V ( x )
V ( x , y , z ) v ( x , . Y . z )
M u l t i p l y b y V ( x , y , z )
T o t a l e n e r g y E
H
r z z ( a z a z a z )
- 2 m a x
2
+ a l + a z
2
+ V ( x , y , z )
1 i```