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Chapter 15
Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid
Objectives
• Using the postulates to understand the
particle in the box (1-D, 2-D and 3-D)
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
1.
2.
3.
4.
The Free Particle
The Particle in a One-Dimensional Box
Two- and Three-Dimensional Boxes
Using the Postulates to Understand the
Particle in the Box and Vice Versa
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.1 The Free Particle
•
For free particle in a one-dimensional space on
which no forces are acting, the Schrödinger
equation is
d 2 x 
  E x 
2
dx
•
 x  is a function that can be differentiated
twice to return to the same function
 x   A e 2 mE / h x  A e ikx
2
where k 


  x   A e i 2 mE / h x  A e ikx
i

2
2


Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
2mE
h2
15.1 The Free Particle
•
If x is restricted to the interval  L  x  L then
the probability of finding the particle in an
interval of length dx can be calculated.
Px dx 
 * x  x dx
L
 * x  x dx
L
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd

A A e ikxe  ikx dx
L
A A  e ikxe ikx dx
L

dx
2L
15.2 The Particle in a One-Dimensional Box
•
15.1 The Classical Particle in a Box
•
When consider particle confined to a box in 1D, the potential is
V x   0, for a  x  0
V x   , for x  a, x  0
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.2 The Particle in a One-Dimensional Box
•
Consider the boundary condition satisfying 1-D,
 0  a  0
•
The acceptable wave functions must have the
form of
 nx 
, for n  1,2,3,4...
 a 
 n  A sin 
•
Thus the normalized eigenfunctions are
 n x  
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
2  nx 
sin 

a  a 
15.2 The Particle in a One-Dimensional Box
•
•
15.2 Energy Levels for the Particle in a
Box
15.3 Probability of Finding the Particle in
a Given Interval
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.1
From the formula given for the energy levels for
the particle in the box, En  h 2 n 2 / 8ma2 for n = 1, 2, 3,
4… , we can see that the spacing between adjacent
levels increases with n. This appears to indicate
that the energy spectrum does not become
continuous for large n, which must be the case for
the quantum mechanical result to be identical to
the classical result in the high-energy limit.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.1
A better way to look at the spacing between levels
is to form the ratio En1  En  / En . By forming this
ratio, we see that E / E becomes a smaller
fraction of the energy as n   .
This shows that the energy spectrum becomes
continuous for large n.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
We have,
En 1  En

En



h 2  n 12  n 2 / 8 ma 2
h 2 n 2 / 8 ma 2

2n  1

n2
which approaches zero as n   . Both the level
spacing and the energy increase with n, but the
energy increases faster (as n2), making the energy
spectrum appear to be continuous as n→∞
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.3 Two- and Three-Dimensional Boxes
•
•
1-D box is useful model system as it allows
focus to be on quantum mechanics instead of
mathematics.
For 3-D box, the potential energy is
V x, y, z   0 for 0  x  a; 0  y  b; 0  z  c   otherwise
•
Inside the box, the Schrödinger equation can
be written as
h2   2
2
2 
 2  2  2  x, y, z   E x, y, z 

2m  x y
z 
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.3 Two- and Three-Dimensional Boxes
•
The total energy eigenfunctions have the form
n yy
nxx
n z
 nx n y nz x, y, z   N sin
sin
sin z
a
b
c
•
And the total energy has the form
2
h 2  nx2 n y nz2 
E
 2 2
2

8m  a
b
c 
•
15.4 Eigenfunctions for the TwoDimensional Box
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.4 Using the Postulates to Understand the Particle in the
Box and Vice Versa
Postulate 1
The state of a quantum mechanical system is
completely specified by a wave function  ( x, t ) .
The probability that a particle will be found at time
t in a spatial interval of width dx centered at x0 is
given by  * ( x0 , t ) x0 , t dx .
• This postulate states that all information
obtained about the system is contained in the wave
function.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.2
Consider the function  x  c sin x / a  d sin 2x / a
a. Is  (x) an acceptable wave function for the
particle in the box?
b. Is  (x) an eigenfunction of the total energy
operator, ?
c. Is  (x) normalized?
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
a. If  (x) is to be an acceptable wave function, it
must satisfy the boundary conditions  (x) =0 at
x=0 and x=a. The first and second derivatives of
 (x) be well-behaved functions between x=0
must also
and x=a. This is the case for  (x) . We conclude
that  (x)
 x  c sin x / a  d sin 2x / a
is an acceptable wave function for the particle in
the box.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
b. Although  x  c sin x / a  d sin 2x / a
may be an
acceptable wave function, it need not be an
eigenfunction of a given operator. To see if  (x) is
 (x)
an eigenfunction
of the total energy operator, the
operator is applied to the function:
2 (x )
2 2


h2 d 
2

x
h
 


 x 
 2x  
x

 c sin  a   d sin 
 c sin    4d sin 
  
 
2 
2 
2m dx 
 a   2ma 
a
 a 
The result of this operation is not  (x) multiplied
by a constant. Therefore,  (x) is not an
eigenfunction of the total energy operator.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
c. To see if  (x) is normalized, the following
integral is evaluated:
2
 x (x )  2x 
0 c sin  a   d sin  a  dx
a

 2(x ) x   d * d sin 2  2x   cd * c * d sin  x  sin  2x dx
  c * c sin
 a
 a 
 a   a 
0 
a
2
2
 x 
 2x 
 x   2x 


  c sin 2  dx   d sin 2 
dx

cd
*

c
*
d
sin

  sin 
dx

 a
 a 
 a  a 
0
0
0
a
a
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
a
Solution
Using the standard integral  sin 2bydy  y / 2  1 / 4b sin by
and recognizing that the third
2
 2 2(x
2
 )x 
2  2x  
c
sin

d
sin
 

 dx
0 
 a
 a 
asin 2  sin 0
asin 4  sin 0  a 2
2 a
2 a
2
 c  (x)

d


c

d

 2
 2
4
8
2
a

Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd

Solution
Therefore,  (x) is not normalized, but the function
 (x)
2
 x 
 2
c
sin

d
sin
 


a
 a
 a



is normalized for the condition that c 2  d 2  1
 (x)
Note that a superposition wave function has a more
complicated dependence on time than does an
eigenfunction of the total energy operator.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
For instance,  (x) for the wave function under
consideration is given by
2  iE / h  x 
iE

(x
)
  x, t  
ce
sin

ce


a 
 a
1
 (x)
2
/h
 2x 
sin 
   x  f t 
 a 
This wave function cannot be written as a product of a
function of x and a function of t. Therefore, it is not a
standing wave and does not describe a state
whose properties are, in general, independent of time.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.4 Using the Postulates to Understand the Particle in the
Box and Vice Versa
• 15.5 Acceptable Wave Functions for the
Particle in a Box
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.3
What is the probability, P, of finding the particle in
the central third of the box if it is in its ground
state?
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
For the ground state,  1 x   2 / a sin x / a  . From
the postulate, P is the sum of all the probabilities of
finding the particle in intervals of width dx within
the central third of the box. This probability is given
by the integral
2
P
a
 x 
a / 3sin  a dx
2a / 3
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
2
Solution
Solving this integral,
2 a a
P  
a  6 4
4
2 

 sin
 sin
  0.609
3
3 

Although we cannot predict the outcome of a single
measurement, we can predict that for 60.9% of a
large number of individual measurements, the
particle is found in the central third of the box.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.4 Using the Postulates to Understand the Particle in the
Box and Vice Versa
Postulate 3
In any single measurement of the observable that
corresponds to the operator  , the only values
that will ever be measured are the eigenvalues of
that operator.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.4 Using the Postulates to Understand the Particle in the
Box and Vice Versa
Postulate 4
If the system is in a state described by the wave
function  x, t  , and the value of the observable a is
measured once each on many identically prepared
systems, the average value of all of these
measurements is given by

a 
ˆ  x, t dx



*
x
,
t
A



 * x, t  x, t dx

Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
15.4 Using the Postulates to Understand the Particle in the
Box and Vice Versa
• 15.6 Expectation Values for E, p, and x for
a Superposition Wave Function
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.4
Assume that a particle is confined to a box of
length a, and that the system wave function is
 x   2 / a sin x / a 
a. Is this state an eigenfunction of the position
operator?
b. Calculate the average value of the position that
would be obtained for a large number of
measurements. Explain your result.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.4
a. The position operator
xˆ  x . Because ,
x x   2 / a sin x / a   c x 
where c is a constant, the wave function is not an
eigenfunction of the position operator.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.4
b. The expectation value is calculated using the
fourth postulate:
x 
2
2   x 
2   x 
 x 
sin
x
sin
dx

x sin   dx
 
  


a 0   a 
a 0   a 
 a
a
a
Using the standard integral
a
2
x 2 cos 2bx x sin 2bx
0 x(sin bx) dx  4  8b 2  4b
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 15.4
We have
a

 2x 
 2x  
 x sin 

 2 cos
 a2  a
2 x
2  a 2 a 2
a 
a 


x 


   2  0   2  
2
a 4
a  4 8
  
 
 8  2
4


8 


a


a

0
The average position is midway in the box. This is
exactly what we would expect, because the particle
is equally likely to be in each half of the box.
Chapter 15: Using Quantum Mechanics on Simple Systems
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd