Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 13 The Schrödinger Equation.
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Transcript of Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 13 The Schrödinger Equation.
Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid
Chapter 13 Chapter 13 The Schrödinger EquationThe Schrödinger Equation
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
ObjectivesObjectives
• Key concepts of operators, eigenfunctions, wave functions, and eigenvalues.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
OutlineOutline
1. What Determines If a System Needs to Be Described Using Quantum Mechanics?
2. Classical Waves and the Nondispersive Wave Equation
3. Waves Are Conveniently Represented as Complex Functions
4. Quantum Mechanical Waves and the Schrödinger Equation
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
OutlineOutline
1. Solving the Schrödinger Equation: Operators, Observables, Eigenfunctions, and Eigenvalues
2. The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal
3. The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set
4. Summing Up the New Concepts
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.1 What Determines If a System Needs to Be 13.1 What Determines If a System Needs to Be Described Using Described Using Quantum Mechanics? Quantum Mechanics?
• Particles and waves in quantum mechanics are not separate and distinct entities.
• Waves can show particle-like properties and particles can also show wave-like properties.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.1 What Determines If a System Needs to Be 13.1 What Determines If a System Needs to Be Described Using Described Using Quantum Mechanics? Quantum Mechanics?
• In a quantum mechanical system, only certain values of the energy are allowed, and such system has a discrete energy spectrum.
• Thus, Boltzmann distribution is used.
where n = number of atoms ε = energy
kTee
j
i
j
i jieg
g
n
n /
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.1Example 13.1
Consider a system of 1000 particles that can only have two energies, , with . The difference in the energy between these two values is . Assume that g1=g2=1.
a. Graph the number of particles, n1 and n2, in states as a function of . Explain your result.b. At what value of do 750 of the particles have the energy ?
21 and 12
21
21 and
/kT
/kT
1
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
We can write down the following two equations:
Solve these two equations for n2 and n1 to obtain
1000 and / 21/
12 nnenn kT
kT
kT
kT
en
e
en
/1
/
/
2
1
10001
1000
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
Part (b) is solved graphically. The parameter n1 is shown as a function of on an expanded scale on the right side of the preceding graphs, which shows that n1=750 for .
91.0/ kT
/kT
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.2 Classical Waves and the Nondispersive Wave 13.2 Classical Waves and the Nondispersive Wave EquationEquation
• 13.1 Transverse, Longitudinal, and Surface Waves
• A wave can be represented pictorially by • a succession of wave fronts, where the • amplitude has a maximum or minimum value.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.2 Classical Waves and the Nondispersive 13.2 Classical Waves and the Nondispersive Wave EquationWave Equation
• The wave amplitude ψ is:
• It is convenient to combine constants and variables to write the wave amplitude as
where k = 2πλ (wave vector) ω = 2πv (angular frequency)
T
txAtx
2sin,
wtkxAtx sin,
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.2 Classical Waves and the Nondispersive 13.2 Classical Waves and the Nondispersive Wave EquationWave Equation
• 13.2 Interference of Two Traveling Waves
• For wave propagation in a medium where frequencies have the same velocity (a nondispersive medium), we can write
2 2
2 2 2
, ,1x t x t
x v t
where v = velocity at which the wave propagates
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.2Example 13.2
The nondispersive wave equation in one dimension is given by
Show that the traveling wave is a solution of the nondispersive wave equation. How is the velocity of the wave related to k and w?
2 2
2 2 2
, ,1x t x t
x v t
wtkxAtx sin,
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
The nondispersive wave equation in one dimension is given by
Show that the traveling wave is a solution of the nondispersive wave equation. How is the velocity of the wave related to k and w?
wtkxAtx sin,
2 2
2 2 2
, ,1x t x t
x v t
1 2( , ) cos( ) ( ) ( )!!!x t A t kx x t
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.2Example 13.2
We have
kwv
wtkxAv
w
t
wtkxA
t
tx
vx
tx
/ gives results two theseEquating
sinsin
,1,
22
2
2
2
2
2
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.3 Waves Are Conveniently Represented as 13.3 Waves Are Conveniently Represented as Complex Complex Functions Functions
• It is easier to work with the whole complex function knowing as we can extract the real part of wave function.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
1 2( , ) ( ) ( ) sin )cos(x t x t A kx t
Standing (stationary) waveStanding (stationary) wave
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.3Example 13.3
a. Express the complex number 4+4i in the form .b. Express the complex number in the form a+ib.
2/33 ie
ire
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
a. The magnitude of 4+4i is The phase is given by
Therefore, 4+4i can be written
42
1cosor
2
1
24
4cos 1
244444 2/1 ii
4/24 ie
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
b. Using the relation can be written
iii 3032
3sin
2
3cos3
2/33 , sincosexp ii eiie
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.2 Some Common Waves (Traveling waves)13.2 Some Common Waves (Traveling waves)
0 0, cos 2 cosr r
r tr t kr t
T
, sin 2
sin( )
x tx t A
T
A t kx
Planar wave
Spherical wave
Cylindrical wave
0, cosr
r t kr t
2 2
2 2
, ,x t x t
t x
2 , ( , )1 ( )2 2
r t r trr r rv t
2
2
2
2
2 , ( , ) ( , )22 2
2( , ) ( ( , ))2 2
r t r t r tr rrv t
r r t r r trv t
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.4 Quantum Mechanical Waves and the 13.4 Quantum Mechanical Waves and the Schrödinger Schrödinger Equation Equation
• The time-independent Schrödinger equation in one dimension is
• It used to study the stationary states of quantum mechanical systems.
xExxVdx
xd
m
h
2
22
2
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.4 Quantum Mechanical Waves and the 13.4 Quantum Mechanical Waves and the Schrödinger Schrödinger Equation Equation
• An analogous quantum mechanical form of time-dependent classical nondispersive wave equation is the time-dependent Schrödinger equation, given as
where V(x,t) = potential energy function• This equation relates the temporal and
spatial derivatives of ψ(x,t) and applied in systems where energy changes with time.
txtxVx
tx
m
h
t
txih ,,
,
2
,2
22
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.4 Quantum Mechanical Waves and the 13.4 Quantum Mechanical Waves and the Schrödinger Schrödinger Equation Equation
• For stationary states of a quantum mechanical system, we have
• Since , we can show that that wave functions whose energy is independent of time have the form of
txEt
txih ,
,
txtx E/h-ie ,
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.5 Solving the Schrödinger Equation: Operators, 13.5 Solving the Schrödinger Equation: Operators, Observables, Observables, Eigenfunctions, and Eigenvalues Eigenfunctions, and Eigenvalues
• We waould need to use operators, observables, eigenfunctions, and eigenvalues for quantum mechanical wave equation.
• The time-independent Schrödinger equation is an eigenvalue equation for the total energy, E
where {} = total energy operator or• It can be simplified as
xExxVxm
hnnn
2
22
2
H
nnn EH ˆ
n n nO
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Eigenequation, eigenfunction, Eigenequation, eigenfunction, eigenvalueeigenvalue
n n nO
The effect of an operator acting on its eigenfunction is the same asa number multiplied with that eigenfunction.
There may be an infinite number of eiegenfunctions and eigenvalues.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.5Example 13.5
Consider the operators . Is the function an eigenfunction of these operators? If so, what are the eigenvalues? Note that A, B, and k are real numbers.
ikxikx BeAex
22 / and / dxddxd
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
To test if a function is an eigenfunction of an operator, we carry out the operation and see if the result is the same function multiplied by a constant:
In this case, the result is not multiplied by a constant, so is not an eigenfunction of the operator d/dx unless either A or B is zero.
ikxikxikxikxikxikx
BeAeikikBikAedx
BeAed
x
x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
This equation shows that is an eigenfunction of the operator with the eigenvalue k2.
22 / dxd
x
xkBeAekBikAeikdx
BeAed ikxikxikxikxikxikx
2222
2
2
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Matrix as Quantum Mechanical Matrix as Quantum Mechanical Operator Operator
n n nO
1 0 1 0 0 1 0, , ,
0 1 0 1 1 0 0
i
i
1 0 0 1 0 0 0 1 0 0 0
0 1 0 , 0 0 0 , 1 0 1 , 0
0 0 1 0 0 1 0 1 0 0 0
i
i i
i
0 1 0 1
1 0 1 0k k
k k k kk k
a a
b b
0 1 0 0 1 0
1 0 1 1 0 1
0 1 0 0 1 0
k k
k k k k k k
k k
a a
b b
c c
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Eigenvalues and eigenvector of the Matrix Operator Eigenvalues and eigenvector of the Matrix Operator
n n nO
21 2
1 1 2 2
1 1 2 2
0 1 0 1
1 0 1 0
0 1 0 10 det 0
1 0 1 0
(0 ) 1 0 1 1, 1.
0 1 0 11 , 1
1 0 1 0
k kk k k k
k k
k k k
k k k
k k
a a
b b
a
b
a a a a
b b b b
1 1 2 21 2
1 1 2 2
1 21 1 2 2
1 2
1 11 1 2 22 2
1 12 21 2
1 1 2 21 11 22 2
1: 1:
Using normality condition: 1, 1
, .
1, , 1, .
b a b a
a b a b
a aa b a b
b b
a b a b
a a
b b
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
ExerciseExercise
n n nO
0 0
0 0
? ?
k kk k k k
k k
kk
k
a ai i
b bi i
a
b
不記得矩陣對角線化的同學請閱讀下述投影片或看線性代數的書:http://140.117.34.2/faculty/phy/sw_ding/teaching/chem-math1/cm07.ppt
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Review of Orthogonal decomposition Review of Orthogonal decomposition of vectors and functions:of vectors and functions:
1 2 3
Vector in 3D space:
Orthogonality: 0, 0, 0,
Normality: 1, 1, 1.
Denoting = , ,
1 if
0 if
x y z
j k jk
A A A
j k
j k
V i j k
i j i k j k
i i j j k k
i i j i k i
i i
1 1 2 2 3 31
1 2 1 3 2 3 1
1 1 2 2 3 3 1 1
Vector in nD space: ...
Orthogonality: 0, 0,... 0,..., 0.
Normality: 1, 1, 1,..., , 1.
1 if
0 if
n
n n j jj
n n
n n n n
j k jk
A A A A A
j k
j k
V i i i i i
i i i i i i i i
i i i i i i i i i i
i i
0
0
{ , 0,1,2,3... } * = ,
, *
Extend to functions of continuous variables:
{ ( ), 0,1, 2,3... } * ( ) ( )= ,
( ) ,
m n m nm
n n n nn
m n m nm
n nn
m
a a
x m dx x x
x a
* ( ) ( )n na dx x x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.6 The Eigenfunctions of a Quantum Mechanical 13.6 The Eigenfunctions of a Quantum Mechanical OperatorOperator Are Orthogonal Are Orthogonal
• Orthogonality is a concept of vector space.
• 3-D Cartesian coordinate space is defined by
• In function space, the analogous expression that defines orthogonality is
0 zyzxyx
2* ( ) ( ) | ( ) | 1i i idx x x dx x
0 if
* ( ) ( )1 if i j ij
i jdx x x
i j
Orthogonormality:
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.6Example 13.6
Show graphically that sin x and cos 3x are orthogonal functions. Also show graphically that 1for 0sinsin
mndxnxmx
sin sin 0 for mx nx dx n m
12 sin sin 1 for integers.nx nx dx n
sin cos 0 for any ,mx nx dx n m
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
The functions are shown in the following graphs. The vertical axes have been offset to avoid overlap and the horizontal line indicates the zero for each plot.Because the functions are periodic, we can draw conclusions about their behaviour in an infinite interval by considering their behaviour in any interval that is an integral multiple of the period.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
The integral of these functions equals the sum of the areas between the curves and the zero line. Areas above and below the line contribute with positive and negative signs, respectively, and indicate that and . By similar means, we could show that any two functions of the type sin mx and sin nx or cos mx and cos nx are orthogonal unless n=m. Are the functions cos mx and sin mx(m=n) orthogonal?
03cossin
dxxmx
0sinsin
dxxx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.6 The Eigenfunctions of a Quantum Mechanical 13.6 The Eigenfunctions of a Quantum Mechanical OperatorOperator Are Orthogonal Are Orthogonal
• 3-D system importance to us is the atom. • Atomic wave functions are best described • by spherical coordinates.
0 if * ( ) ( )
1 if
0 if * ( , , ) ( , , )
1 if
0 if sin * ( , , ) ( , , )
1 if
i j ij
i j ij
V
i j ij
V
i jdx x x
i j
i jdxdydz x y z x y z
i j
i jr drd d r r
i j
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Example 13.8Example 13.8
Normalize the function over the interval
Solution:Volume element in spherical coordinates is , thus
re
20 ;0 ;0 r
ddrdr sin2
14
1sin
0
222
0
22
0
2
0
2
drerN
drerddN
r
r
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
SolutionSolution
Using the standard integral ,
we obtainThe normalized wave function is
Note that the integration of any function involving r, even if it does not explicitly involve , requires integration over all three variables.
re
re
1
integer positive a is ,0 /!0
1 naandxex naxn
1
that so 12
!24
32 NN
or
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.7 The Eigenfunctions of a Quantum Mechanical 13.7 The Eigenfunctions of a Quantum Mechanical Operator Operator Form a Complete Set Form a Complete Set
• The eigenfunctions of a quantum mechanical operator form a complete set.
• This means that any well-behaved wave function, f (x) can be expanded in the eigenfunctions of any of the quantum mechanical operators.
• 13.4 Expanding Functions in Fourier Series
0
( ) , * ( )n n n nn
x a a dx x
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
13.7 The Eigenfunctions of a Quantum Mechanical 13.7 The Eigenfunctions of a Quantum Mechanical Operator Operator Form a Complete Set Form a Complete Set
• Fourier series graphs
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Fourier seriesFourier series
cos nx
0,1,2,n sin nx
1,2,3,n
cos cos 0mx nxdx
m n sin sin 0mx nxdx
m n
cos sin 0mx nxdx
,all m n
2 , 0cos cos
, 0
if nnx nxdx
if n
sin sinnx nxdx
0if n
01 2 3
1 2 3
0
1
( ) cos cos 2 cos32
sin sin 2 sin 3
( cos sin )2 n n
n
af x a x a x a x
b x b x b x
aa nx b nx
1( )cosna f x nxdx
1( )sinnb f x nxdx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
The function f(x) (blue line) is approximated by the summation of sine functions (red line):
01 2 3
1 2 3
0
1
( ) cos cos 2 cos32
sin sin 2 sin 3
( cos sin )2 n n
n
af x a x a x a x
b x b x b x
aa nx b nx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
Fourier Transforms (FT)Fourier Transforms (FT)
0
( ) ( ) cos ( )sinf x u y xy y xy dy
1
( ) ( ) cosu y f x xydx
1( ) ( )siny f x xydx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
FT in exponential formFT in exponential form
1cos ( )
2ixy ixyxy e e
1sin ( )
2ixy ixyxy e e
1( ) ( ) ( )
2w y u y i y
( ) ( ) ixyf x w y e dy
1( ) ( )
2ixyw y f x e dx
0
( ) ( ) cos ( )sinf x u y xy y xy dy
1
( ) ( ) cosu y f x xydx
1( ) ( )siny f x xydx
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 13: The Schrödinger Equation
For students who are not familiar with orthogonal For students who are not familiar with orthogonal expansion of functions, you may find the expansion of functions, you may find the following ppt tutorial helpful: following ppt tutorial helpful:
• http://140.117.34.2/faculty/phy/sw_ding/teaching/chem-math1/cm07.ppt
You should also read a chemistry math book and do some exercisesfor better understanding.