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


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2 
= - - \ 7
2 
+ V ( x , y , z ) 
2 m 
A n g u l a r m o m e n t u m 
L x = Y P z - Z P y 
L 
X 
- i 1 i (Y~- z~) 
a z a y 
L Y = z p x - x p z 
i 
- i 1 i (z~- x~) 
y 
a x a z 
L z = x p y - Y P x 
i 
- i 1 i (x~- y~) 
z 
a y a x 
b e t h e f r e q u e n c y o f r o t a t i o n ( c y c l e s p e r s e c o n d ) . T h e s p e e e d o f t h e p a r t i c l e , t h e n , i s 
v = 2 n r v r o t = r w r o t ' w h e r e w r o t = 2 n v r o t h a s t h e u n i t s o f r a d i a n s p e r s e c o n d a n d i s 
c a l l e d t h e a n g u l a r s p e e d . T h e k i n e t i c e n e r g y o f t h e r e v o l v i n g p a r t i c l e i s 
K = l m v
2 
= l m r
2
w
2 
= . ! . J w
2 
2 2 2 
( 4 . 3 ) 
1 1 9 
120 Chapter 4 I Some Postulates and General Principles of Quantum Mechanics 
where the quantity I = mr2 is the moment of inertia. By comparing the first and last 
expressions for the kinetic energy in Equation 4.3, we can make the correspondences 
w ++ v and I ++ m, where wand I are angular quantities and v and mare linear quan-
tities. According to this correspondence, there should be a quantity I w corresponding 
to the linear momentum mv, and in fact the quantity L, defined by 
L = Iw = (mr2 ) (~) = mvr (4.4) 
is called the angular momentum and is a fundamental quantity associated with rotating 
systems, just as linear momentum is a fundamental quantity in linear systems. 
Kinetic energy can be written in terms of momentum. For a linear system, we have 
mv2 (mv) 2 p 2 
K=-=--=-
2 2m 2m 
(4.5) 
and for rotating systems, 
Iw2 (Iw) 2 L2 
K-------
- 2 - 2I - 2I (4.6) 
The correspondences between linear systems and rotating systems are given in 
Table 4.2. 
We learned in MathChapter C that the angular momentum of a particle is actually 
a vector quantity defined by L = r x p, where r is its position from a fixed point 
and p = mv is its momentum (Figure C.8). Figure C.8 shows that the direction of L is 
perpendicular to the plane formed by rand p. The components ofL are (Equation C.l8) 
TABlE 4.2 
Lx = YPz- ZPy 
L = zp -xp y X Z 
Lz = xpy- YPx 
The correspondences between linear systems and rotating systems. 
Linear motion 
Mass (m) 
Speed (v) 
Momentum (p = mv) 
Angular motion 
Moment of inertia (/) 
Angular speed ( w) 
Angular momentum (L = I w) 
(4.7) 
( 
mv2 p2) Kinetic energy K = - = -
2 2m ( 
[w
2 
L
2
) Rotational kinetic energy K = - = -
2 21 
4 - 2 . 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 R e p r e s e n t C l a s s i c a l - M e c h a n i c a l V a r i a b l e s 
N o t e t h a t t h e a n g u l a r m o m e n t u m o p e r a t o r s g i v e n i n T a b l e 4 . 1 c a n b e o b t a i n e d f r o m 
E q u a t i o n 4 . 7 b y l e t t i n g t h e l i n e a r m o m e n t a , P x ' P y ' a n d p z a s s u m e t h e i r o p e r a t o r 
e q u i v a l e n t s . 
A c c o r d i n g t o P o s t u l a t e 2 , a l l 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 a r e l i n e a r . T h e r e i s 
a n i m p o r t a n t p r o p e r t y o f l i n e a r o p e r a t o r s t h a t w e h a v e n o t d i s c u s s e d y e t . C o n s i d e r a n 
e i g e n v a l u e p r o b l e m w i t h a t w o - f o l d d e g e n e r a c y ; t h a t i s , c o n s i d e r t h e t w o e q u a t i o n s 
A ¢
1 
= a ¢
1 
a n d A ¢
2 
= a ¢
2 
B o t h ¢
1 
a n d ¢
2 
h a v e t h e s a m e e i g e n v a l u e a . I f t h i s i s t h e c a s e , t h e n a n y l i n e a r c o m b i -
n a t i o n o f ¢
1 
a n d ¢
2
, s a y c
1
¢
1 
+ c
2
¢
2
, i s a n e i g e n f u n c t i o n o f A . T h e p r o o f r e l i e s o n t h e 
l i n e a r p r o p e r t y o f A ( S e c t i o n 3 - 2 ) : 
A ( c 1 ¢ 1 + c 2 ¢ 2 ) = c 1 A ¢ 1 + c 2 A c p 2 
= c
1
a c p
1 
+ c
2
a c p
2 
= a ( c
1
c p
1 
+ c
2
c p
2
) 
E X A M P L E 4 - 3 
C o n s i d e r t h e e i g e n v a l u e p r o b l e m 
d z c f > ( ¢ ) = - m z c f > ( ¢ ) 
~ 
w h e r e m i s a r e a l ( n o t i m a g i n a r y n o r c o m p l e x ) n u m b e r . T h e t w o e i g e n f u n c t i o n s o f 
A = d
2 
/ d ¢
2 
a r e 
c p m ( ¢ ) = e i m ¢ 
a n d 
c p - m ( ¢ ) = e - i m ¢ 
W e c a n e a s i l y s h o w t h a t e a c h o f t h e s e e i g e n f u n c t i o n s h a s t h e e i g e n v a l u e - m
2
\u2022 S h o w t h a t 
a n y l i n e a r c o m b i n a t i o n o f c t > m ( ¢ ) a n d c t > - m ( ¢ ) i s a l s o a n e i g e n f u n c t i o n o f A = d
2 
/ d ¢
2
. 
S O L U T I O N : 
d 2 d 2 e i m ¢ d 2 e - i m ¢ 
( 1 m ¢ + - 1 m ¢ ) _ + 
d ¢ 2 c i e c 2 e - c l d ¢ 2 c2~ 
= - c l m 2 e i m ¢ - c 2 m 2 e - i m ¢ 
= - m 2 ( c l e i m ¢ + c 2 e - i m ¢ ) 
E x a m p l e 4 - 3 h e l p s s h o w t h a t t h i s r e s u l t i s d i r e c t l y d u e t o t h e l i n e a r p r o p e r t y o f 
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 . A l t h o u g h w e h a v e c o n s i d e r e d o n l y a t w o - f o l d d e g e n -
e r a c y , t h e r e s u l t i s e a s i l y g e n e r a l i z e d . W e w i l l u s e t h i s p r o p e r t y o f l i n e a r o p e r a t o r s 
w h e n w e d i s c u s s t h e h y d r o g e n a t o m i n C h a p t e r 6 . 
1 2 1 
122 
4-3. Observable Quantities Must Be Eigenvalues of Quantum 
Mechanical Operators 
We now present our third postulate: 
Postulate 3 
In any measurement of the observable associated with the operator A, the 
only values that will ever be observed are the eigenvalues an' which satisfy the 
eigenvalue equation 
A,,, =a 1/f 
'f'n n n (4.8) 
Thus, in any experiment designed to measure the observable corresponding to A, the 
only values we find areal' a2 , \u2022\u2022\u2022 corresponding to the states 1/f" o/2 , \u2022\u2022.\u2022 No other 
values will ever be observed. 
As a specific example, consider the measurement of the energy. The operator 
corresponding to the energy is the Hamiltonian operator, and its eigenvalue equation is 
(4.9) 
This is just the Schrodinger equation. The solution of this equation gives the 1/fn and En. 
For the case of a particle in a box, En = n2h2 j8ma2 (Equation 3.21). Postulate 3 says 
that if we measure the energy of a particle in a box, we will find one of these energies 
and no others. 
According to Postulate 1, wave functions have a probabilistic interpretation, and 
so we can use them to calculate average values of physical quantities. Recall from 
Section 3-7 that we argued that the average position of a particle in a box is given by 
(for all n) (4.10) 
This leads us to our fourth postulate. 
Postulate 4 
If a system is in a state described by a normalized ;vave function 1/f, then the 
average value of the observable corresponding to A is given by 
(a) = J 1/f* A 1/fdx (4.11) 
all space 
4 - 3 . 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 E i g e n v a l u e s o f 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 
E X A M P L E 4 - 4 
W e w i l l l e a r n i n t h e n e x t c h a p t e r t h a t a g o o d a p p r o x i m a t e w a v e f u n c t i o n f o r t h e 
v i b r a t i o n a l p r o p e r t i e s o f a d i a t o m i c m o l e c u l e i n i t s l o w e s t q u a n t u m s t a t e i s 
( 
C i ) 1 / 4 2 
1 / l o ( x ) = ; e - a x / 2 
- O O < X < O O