physical chemistry 2 nd edition thomas engel, philip reid chapter 15 using quantum mechanics on...
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Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid
Chapter 15 Chapter 15 Using Quantum Mechanics on Simple SystemsUsing Quantum Mechanics on Simple Systems
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
ObjectivesObjectives
• Using the postulates to understand the particle in the box (1-D, 2-D and 3-D)
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
OutlineOutline
1. The Free Particle2. The Particle in a One-Dimensional Box3. Two- and Three-Dimensional Boxes4. Using the Postulates to Understand the
Particle in the Box and Vice Versa
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.1 The Free Particle15.1 The Free Particle
• For free particle in a one-dimensional space on which no forces are acting, the Schrödinger equation is
• is a function that can be differentiated twice to return to the same function
xEdx
xd
2
2
x
ikxxhmEi
ikxxhmEi
eAeAx
eAeAx
2
2
/2
/2
2
22
h
mEk
where
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.1 The Free Particle15.1 The Free Particle
• If x is restricted to the interval then the probability of finding the particle in an interval of length dx can be calculated.
LxL
L
dx
dxeeAA
dxeeAA
dxxx
dxxxdxxP L
L
ikxikx
ikxikx
L
L
2*
*
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.2 The Particle in a One-Dimensional Box15.2 The Particle in a One-Dimensional Box
• 15.1 The Classical Particle in a Box
• When consider particle confined to a box in 1-D, the potential is
0,for x ,
0for ,0
xaxV
xaxV
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.2 The Particle in a One-Dimensional Box15.2 The Particle in a One-Dimensional Box
• Consider the boundary condition satisfying 1-D,
• The acceptable wave functions must have the form of
• Thus the normalized eigenfunctions are
...4,3,2,1for ,sin
na
xnAn
a
xn
axn
sin2
00 a
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.2 The Particle in a One-Dimensional Box15.2 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
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.1Example 15.1
From the formula given for the energy levels for the particle in the box, 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.
222 8/ manhEn
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.1Example 15.1
A better way to look at the spacing between levels is to form the ratio . By forming this ratio, we see that becomes a smaller fraction of the energy as .
This shows that the energy spectrum becomes continuous for large n.
nnn EEE /1
EE /n
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
We have,
which approaches zero as . 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→∞
28/
8/11 12222
2222
n
n
E
EEmanh
mannh
n
nn
n
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.3 Two- and Three-Dimensional Boxes15.3 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
• Inside the box, the Schrödinger equation can be written as
otherwise 0 ;0 ;0for 0,, czbyaxzyxV
zyxEzyxzyxm
h,,,,
2 2
2
2
2
2
22
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.3 Two- and Three-Dimensional Boxes15.3 Two- and Three-Dimensional Boxes
• The total energy eigenfunctions have the form
• And the total energy has the form
• 15.4 Eigenfunctions for the Two- Dimensional Box
c
zn
b
yn
a
xnNzyx zyx
nnn zyx
sinsinsin,,
2
2
2
2
2
22
8 c
n
b
n
a
n
m
hE zyx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.4 Using the Postulates to Understand the Particle in 15.4 Using the Postulates to Understand the Particle in the the Box and Vice Versa Box and Vice Versa
Postulate 1The state of a quantum mechanical system is completely specified by a wave function . The probability that a particle will be found at time t in a spatial interval of width dx centered at x0 is given by .
• This postulate states that all information obtained about the system is contained in the wave function.
dxtxtx ,),(* 00
),( tx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.2Example 15.2
Consider the functiona. Is an acceptable wave function for the particle in the box?b. Is an eigenfunction of the total energy operator, ?c. Is normalized?
)(x axdaxcx /2sin/sin
)(x
)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
a. If is to be an acceptable wave function, it must satisfy the boundary conditions =0 at x=0 and x=a. The first and second derivatives of must also be well-behaved functions between x=0 and x=a. This is the case for . We conclude that
is an acceptable wave function for the particle in the box.
axdaxcx /2sin/sin
)(x)(x
)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
b. Although may be an acceptable wave function, it need not be an eigenfunction of a given operator. To see if is an eigenfunction of the total energy operator, the operator is applied to the function:
The result of this operation is not multiplied by a constant. Therefore, is not an eigenfunction of the total energy operator.
axdaxcx /2sin/sin
a
xd
a
xc
ma
h
a
xdc
dx
d
m
hax 2
sin4sin2
2sinsin
2 2
22
2
22
)(x
)(x)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
c. To see if is normalized, the following integral is evaluated:
dxa
x
a
xdccddx
a
xddx
a
xc
dxa
x
a
xdccd
a
xdd
a
xcc
dxa
xd
a
xc
aaa
a
a
2sinsin**
2sinsin
2sinsin**
2sin*sin*
2sinsin
0
2
2
0
2
2
0
0
22
2
0
)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
Using the standard integral and recognizing that the third
bybybydy sin4/12/sin 2
2222
2
0
2222
28
0sin4sin
24
0sin2sin
2
2sinsin
dcaaa
daa
c
dxa
xd
a
xc
a
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
Therefore, is not normalized, but the function
is normalized for the condition thatNote that a superposition wave function has a more complicated dependence on time than does an eigenfunction of the total energy operator.
122 dc
ad
a
xc
a
2sinsin
2
)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
)(x
)(x
For instance, for the wave function under consideration is given by
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 statewhose properties are, in general, independent of time.
tfxa
xce
a
xce
atx hiEhiE
2
sinsin2
, // 21
)(x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.4 Using the Postulates to Understand the Particle in 15.4 Using the Postulates to Understand the Particle in the the Box and Vice Versa Box and Vice Versa
• 15.5 Acceptable Wave Functions for the Particle in a Box
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.3Example 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?
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
For the ground state, . 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
dxa
x
aP
a
a
3/2
3/
2sin2
axax /sin/21
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
SolutionSolution
Solving this integral,
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.
609.03
2sin
3
4sin
46
2
aa
aP
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.4 Using the Postulates to Understand the Particle in 15.4 Using the Postulates to Understand the Particle in the the Box and Vice Versa Box and Vice Versa
Postulate 3In 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.
A
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.4 Using the Postulates to Understand the Particle in 15.4 Using the Postulates to Understand the Particle in the the Box and Vice Versa Box and Vice Versa
Postulate 4If the system is in a state described by the wave function , 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
dxtxtx
dxtxAtx
a
,,*
,ˆ,*
tx,
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
15.4 Using the Postulates to Understand the Particle in 15.4 Using the Postulates to Understand the Particle in the the Box and Vice Versa Box and Vice Versa
• 15.6 Expectation Values for E, p, and x for a Superposition Wave Function
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.4Example 15.4
Assume that a particle is confined to a box of length a, and that the system wave function is
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.
axax /sin/2
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.4Example 15.4
a. The position operator . Because ,
where c is a constant, the wave function is not an eigenfunction of the position operator.
xcaxaxx /sin/2
xx ˆ
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.4Example 15.4
b. The expectation value is calculated using the fourth postulate:
Using the standard integral
dxa
xx
adx
a
xx
a
x
ax
aa 2
00
sin2
sinsin2
b
bxx
b
bxxdxbxx
a
4
2sin
8
2cos
4)(sin
2
22
0
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 15: Using Quantum Mechanics on Simple Systems
Example 15.4Example 15.4
We have
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.
280
84
2
4
2sin
8
2cos
4
22
2
2
22
0
2
2 aaaa
aa
ax
x
a
ax
x
ax
a