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.



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



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



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.



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 /



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



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



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



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.



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,



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




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,



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




We have

kwv

wtkxAv

w

t

wtkxA

t

tx

vx

tx

/ gives results two theseEquating

sinsin

,1,

22

2

2

2

2

2



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.



1 2( , ) ( ) ( ) sin )cos(x t x t A kx t

Standing (stationary) waveStanding (stationary) wave




a. Express the complex number 4+4i in the form .b. Express the complex number in the form a+ib.

2/33 ie

ire



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



SolutionSolution

b. Using the relation can be written

iii 3032

3sin

2

3cos3

2/33 , sincosexp ii eiie



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



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




• 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




• 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 ,



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



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.




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



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



SolutionSolution

This equation shows that is an eigenfunction of the operator with the eigenvalue k2.

22 / dxd

x

xkBeAekBikAeikdx

BeAed ikxikxikxikxikxikx

2222

2

2



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



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



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



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



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:




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



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.



SolutionSolution



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



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




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



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



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



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



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



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



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



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



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.

Documents

Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 13 The Schrödinger Equation