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PHYSICAL CHEMISTRY II

CHEM 3720

Syllabus & Lecture Notes

Prepared by Dr. Titus V. Albu

Department of Chemistry

University of Tennessee at Chattanooga

Spring 2017

12

HH

e

34

56

78

910

LiB

eB

CN

OF

Ne

1112

1314

1516

1718

Na

Mg

AlSi

PS

Cl

Ar

1920

2122

2324

2526

2728

2930

3132

3334

3536

KC

aSc

TiV

Cr

Mn

FeC

oN

iC

uZn

Ga

Ge

AsSe

Br

Kr

3738

3940

4142

4344

4546

4748

4950

5152

5354

Rb

SrY

ZrN

bM

oTc

Ru

Rh

PdAg

Cd

InSn

SbTe

IXe

5556

57-7

172

7374

7576

7778

7980

8182

8384

8586

Cs

Ba

Hf

TaW

Re

Os

IrPt

AuH

gTl

PbB

iPo

AtR

n

8788

89-1

0310

410

510

610

710

810

911

011

111

211

311

411

511

611

711

8

FrR

aR

fD

bSg

Bh

Hs

Mt

Ds

Rg

Cn

Uut

FlU

upLv

Uus

Uuo

5758

5960

6162

6364

6566

6768

6970

71

Lant

hani

des

LaC

ePr

Nd

PmSm

EuG

dTb

Dy

Ho

ErTm

YbLu

8990

9192

9394

9596

9798

9910

010

110

210

3

Actin

ides

AcTh

PaU

Np

PuAm

Cm

Bk

Cf

EsFm

Md

No

Lr8A

Perio

dic

Tabl

e of

the

Elem

ents

1A

1.00

794

2A3A

4A5A

6A7A

4.00

2602

6.94

19.

0121

8210

.811

12.0

107

14.0

067

15.9

994

18.9

9840

3220

.179

7

30.9

7376

232

.065

35.4

5339

.948

39.0

983

40.0

7844

.955

912

47.8

6750

.941

551

.996

1

7B┌

──

──

─ 8

B ─

──

──

┐1B

2B26

.981

5386

28.0

855

22.9

8976

924

.305

03B

4B5B

6B

78.9

679

.904

83.7

9854

.938

045

55.8

4558

.933

195

58.6

934

63.5

4665

.38

85.4

678

87.6

288

.905

8591

.224

92.9

0638

95.9

6

69.7

2372

.64

74.9

2160

114.

818

118.

710

121.

760

127.

6012

6.90

447

131.

293

[98]

101.

0710

2.90

550

106.

4210

7.86

8211

2.41

1

[209

][2

10]

[222

]18

6.20

719

0.23

192.

217

195.

084

196.

9665

6920

0.59

[223

][2

26]

Actin

ides

[267

][2

68]

[271

]

204.

3833

207.

220

8.98

040

132.

9054

519

137.

327

Lant

hani

des

178.

4918

0.94

788

183.

84

[284

][2

89]

[288

][2

93]

[294

][2

94]

[272

][2

70]

[276

][2

81]

[280

][2

85]

[227

]23

2.03

806

231.

0358

823

8.02

891

[237

][2

44]

[243

]

151.

964

157.

2513

8.90

547

140.

116

140.

9076

514

4.24

2[1

45]

150.

36

[259

][2

62]

[247

][2

47]

[251

][2

52]

[257

][2

58]

168.

9342

117

3.05

417

4.96

6815

8.92

535

162.

500

164.

9303

216

7.25

9

http://chemistry.about.com/

The University of Tennessee at Chattanooga

i

Physical Chemistry II

Spring 2018

CHEM 3720, CRN 20987, 4 credit hours (including CHEM 3720L lab)

Instructor: Dr. Titus V. Albu

Phone and Email: 423-425-4143; [email protected]

Office Hours and Location: Monday 8:30-11:00 am & Wednesday 8:30-11:00 am or by

appointment; Grote 314

Course Meeting Days, Time, and Location: Monday/Wednesday/Friday 1:00 pm-1:50 pm;

Grote 411

Course Catalog Description: Continuation of 3710 with primary emphasis on kinetics, quantum

mechanics, and spectroscopy. Spring semester. Lecture 3 hours, laboratory 3 hours. Prerequisite:

CHEM 3710 with a minimum grade of C and 3710L with a minimum grade of C; or department

head approval. Corequisite: CHEM 3720L or department head approval. Laboratory/studio

course fee will be assessed.

Course Pre/Co Requisites: see Course Catalog Description above

Course Student Learning Outcomes: Through classroom lectures, assigned textbook reading

and homework, and the laboratory work the students are expected to advance the ability to

interpret and reason with physical chemistry concepts, laws, and theories. Having completed the

class, a student is expected to be able to: Understand and use basic physical chemistry language,

Identify, discuss and analyze factors influencing molecular properties and chemical kinetics;

Apply physical chemistry principles and laws to problems or issues of a chemical nature;

Critically interpret and reason physical chemistry data.

Course Fees: Laboratory/studio course fee will be assessed.

Course Materials/Resources:

Textbook and Topics: Physical Chemistry 10th edition by Atkins and de Paula (W. H.

Freeman and Company, New York, 2014, ISBN: 978-1-4292-9019-7) is the textbook of

record in this class. We will be covering topics that are presented in the textbook as

follows:

Quantum Chemistry (Chapters 7-11)

Spectroscopy (Chapters 12-13)

Statistical Thermodynamics (Chapter 15)

Reaction Dynamics (Chapter 21)

ii

The topics will be covered using lecture notes organized as units, which have a slightly

different arrangement than the textbook. In addition, selected topics of nanomaterials

will be presented and discussed toward the end of the semester.

Lecture Notes: The lecture notes are available on UTC Learn, should be printed and

brought to the class. I strongly recommend that you print and have bound the entire

course package containing the syllabus and the lecture notes. In addition to the lecture

notes, I may also refer you to or have you find additional information from various online

sites.

Technology Requirements for Course:

Computer: You need access to a computer with a reliable internet connection for this

course. Test your computer set up and browser for compatibility with UTC Learn at

http://www.utc.edu/learn/getting-help/system-requirements.php. Although not required,

the computer might need to have speakers or headphones. You should also have an

updated version of Adobe Acrobat Reader, available free from

https://get.adobe.com/reader/.

UTC Learn: Access this class by selecting “SP18.CHEM.3720.20987: Physical

Chemistry II” course on UTC Learn (http://www.utc.edu/learn/). Log in using your

utcID and password (the same as for your UTC email). In this class, UTC Learn will be

used for: (1) Course announcements; (2) Syllabus; (3) Course Materials: Lecture notes,

old/practice exams, and homework assignments; and (4) Individual grades.

Technology Skills Required for Course: You will need to have basic computer skills including

using the learning management system (UTC Learn), using MOCSNet email, completing online

homework, and downloading and printing pdf files.

Technology Support: If you have problems with your UTC email account or with UTC Learn,

contact IT Solutions Center at 423-425-4000 or email [email protected]

Course Assessments and Requirements: Your overall course grade will be computed based on:

In-class exams 45%

Final exam 20%

Homework assignments 10%

Laboratory 25%

Exams: There will be 3 one-hour exams during the class period. The lowest-scored

exam grade (or a missed exam grade) can be replaced by the final exam grade, if the final

exam grade is higher. A grade of 0 will be assigned for any exam that is not taken, and

one grade of 0 can be replaced by the final exam grade. The typical average exam score

is around 60-65. Exams are based on class lecture notes, textbook, and homework. You

should bring a working calculator and two pencils to exams. You may not share a

calculator during exams. No other paper, notes, books or stored information is to be used

except what will be provided to you. After the first person leaves an exam, no one else

can come late and start the exam. No cell phone use, texting, or checking phone in class

http://www.utc.edu/learn/getting-help/system-requirements.phphttps://get.adobe.com/reader/mailto:[email protected]

iii

at any time. The tentative exam dates are given below. NO MAKEUP exams will be

given.

Final Exam: The final exam is a standardized ACS exam that is scheduled for Friday,

April 27, 2018: 10:30 am-12:30 pm. Final exam contains 60 questions: 40 questions

covering Quantum Theory and Spectroscopy, 10 questions covering Statistical

Thermodynamics, and 10 questions covering Dynamics. The final exam grade will be

determined by dividing the number of questions answered correctly to 0.50 and will not

be adjusted any further. The typical average on this test (after adjusting) is expected to be

around 55-60. There is NO MAKEUP for the final exam.

Homework Assignments: Homework assignments will be assigned regularly and will

primarily contain exercises and problems from the textbook except the last assignment

which is a class presentation. All homework assignments will be posted in your UTC

Learn course and will be collected during the lecture period. Assignments containing

problems will be graded with scores of 1/0.5/0 point per problem. Each problem should

be worked out on one page (or more separate pages), and spread them out showing all

steps. Do not put more than one problem per page unless otherwise noted in the

instructions. If problems are not done in this format, points will be deducted. It is

assumed you are working problems in a timely manner. Late homework assignments will

be accepted as long I did not return the graded ones back but a 25% deduction per day

will be enforced. Some homework problems are more difficult and are for you to

struggle with and be satisfied with your answer. You must try to work all problems by

yourself with help only to guide you and not to replace working or thinking about the

problem. If I will be asked questions about homework problems before the due date, I

may be able to point you in the right direction, but all the details of the work are up to

you. In addition to reading the textbook and studying the lecture notes, working the

assigned homework problems as we discuss each chapter is a good way to prepare for the

exams. You can also use the worked examples in the textbook or old exams for practice.

Remember that learning chemistry requires thinking and doing, and not just listening and

reading.

Laboratory: Laboratory grades will be provided by the CHEM 3720L instructor (Dr.

Han Park), who has full responsibility for these grades. The typical class lab average is

around 90.

iv

Course Grading

Course Grading Policy: The unadjusted overall course score can be adjusted (up or

down) by the instructor, by up to 3%, based on (but not limited to) class attendance and

participation, homework effort, involvement in the lab experiments, general interest in

the presented material, the score/average of a particular exam, etc. Your letter grade in

the class will be determined based on the adjusted overall course score, and it is expected

to be determined according to the following scale:

F < 50% < D < 60% < C < 70% < B < 83% < A

Instructor Grading and Feedback Response Time: I will try my best to grade all

assignments before the next class period, and provide written feedback when necessary.

Course and Institutional Policies:

Late/Missing Work Policy: Late homework assignments will be accepted as long I did

not return the graded ones back but a 25% deduction per day will be enforced. There is

no makeup for in-class exams. As presented above, the lowest-scored exam grade (or a

missed exam grade) can be replaced by the final exam grade, if the final exam grade is

higher. There is no makeup for the final exam.

Student Conduct Policy: UTC’s Academic Integrity Policy is stated in the Student

Handbook. A violation of the honor code could result in appearing in honor court and

receiving a course grade of F. Instructor will not tolerate academic dishonesty.

Specifics: (1) Any attempt (successful or unsuccessful) to cheat during any of the exams

will automatically result in an “F” grade in the course; (2) Presenting a homework

assignment that resembles too closely the assignment of another student will result in a

grade of 0 for that assignment (for both students) for the first infraction, and an overall

grade of 0 for the homework (10% of the overall grade) for the second infraction.

Honor Code Pledge: I pledge that I will neither give nor receive unauthorized aid on any

test or assignment. I understand that plagiarism constitutes a serious instance of

unauthorized aid. I further pledge that I exert every effort to ensure that the Honor Code

is upheld by others and that I will actively support the establishment and continuance of a

campus-wide climate of honor and integrity.

Course Attendance Policy: Students are expected to attend every lecture, be punctual,

and be respectful of others in the class. Classroom behavior such as talking to your

neighbor during lecture, reading, dozing, or checking cell phone, might interfere with my

ability to teach effectively and others ability to learn. I might require you to meet with

me before you are allowed to take the next exam so I can explain more clearly why your

activities are a problem. I might also ask you to leave the classroom. Laptop computers

can be a big distraction in class so no laptops may be used at any time during class.

Similarly, there should be no cell phone use of any kind during class. You are

responsible for everything covered in the lecture. Information or points missed during

unexcused absences cannot be reclaimed from me so check with a fellow student who is

able to share notes and go over items you missed. The only acceptable (but not

necessarily accepted) excuses are the ones received from The Dean of Students Office.

http://www.utc.edu/dean-students/pdfs/academics16.pdfhttp://www.utc.edu/dean-students/pdfs/academics16.pdf

v

During lecture period, there might be some additional assignments, class pop-quizzes, or

attendance quizzes that might be added as bonus points to the next exam grade or

considered in the homework grade. Negative points will be assessed for missing class

during attendance quizzes. Class will include lecture and discussion with assumption that

you have read and studied textbook and lectures notes ahead of where we are in class and

you can discuss topic, ask relevant questions, and/or respond to questions. Always bring

your printed lecture notes to class.

Communication: Class announcements will be made through UTC Learn and email. UTC email

is the official means of communication between instructor and student at UTC. Please check

your UTC email and UTC Learn on a regular basis. (i.e., daily). I will try to answer emails from

students with questions/comments/concerns within 24 hours (Monday through Friday) although

occasionally it might take longer. I might not answer student emails if they require repeating

information already mentioned in a class that the student missed.

Course Participation/Contribution: The course contains several learning objectives that are

critical to building a solid foundation in physical chemistry. For this reason, several methods will

be employed, including (but not limited to): lecture, group study, pre-class reading, and post-

class work. To be successful in this course, I recommend that you engage in all methods.

Course Learning Evaluation: Course evaluations are an important part of our efforts to

continuously improve the learning experience at UTC. Toward the end of the semester, you will

receive a link to evaluations and are expected to complete them. We value your feedback and

appreciate you taking time to complete the anonymous evaluations.

Syllabus Changes: Although unlikely, some things on this syllabus are subject to change at the

discretion of the instructor. Every attempt will be made to follow this syllabus, however, if

changes are made, they will be announced in class, by email, and/or on UTC Learn, and it is the

responsibility of the student to keep up with the changes.

Course Calendar/Schedule: The tentative exam schedule below is based on the assumption that

no classes will be cancelled (due to weather or other emergencies).

Exam 1: February 7, 2018 (covering Units 2-5)

Exam 2: March 7, 2018 (covering Units 6-9)

Exam 3: April 16, 2018 (covering Units 11-13)

Final Exam: April 27, 2018 10:30 am – 12:30 pm

CHEM 3720

1

Unit I

Introduction

A. Introduction to Physical Chemistry

1. Physical Chemistry is the part of chemistry dealing with application of

physical methods to investigate chemistry.

2. Physical Chemistry main subdivisions are:

a. Quantum Mechanics

□ deals with structure and properties of molecules

b. Spectroscopy

□ deals with the interaction between light and matter

c. Computational Chemistry

□ deals with modeling chemical properties of reactions using

computers

d. Statistical Mechanics

□ deals with how knowledge about molecular energy levels (or

microscopic world) transforms into properties of the bulk (or

macroscopic world)

e. Thermodynamics

□ deals with properties of systems and their temperature dependence

and with energetics of chemical reactions

f. Electrochemistry

□ deals with processes in which electrons are either a reactant or a

product of a reaction

g. Chemical Kinetics

□ deals with the rates of chemical reactions or physical processes

CHEM 3720

2

B. Classical Physics Review

1. Classical Physics was introduced in the 17th century by Isaac Newton.

2. At the end of 19th century, classical physics (mechanics, thermodynamics,

kinetic theory, electromagnetic theory) was fully developed and was

divided into:

a. the corpuscular side or particle domain (the matter)

b. the undulatory side or wave domain (the light)

3. Some useful classical physics equations:

a. Total energy E:

VKE

○ K is the kinetic energy (or energy arising from motion)

○ V is the potential energy (or energy arising from position)

b. Kinetic energy K:

m

pmK

2v

2

1 22

○ m is the mass

○ v is the velocity (or speed)

○ p is the momentum

c. Frequency (Greek letter nu):

2

~ cc

○ is the wavelength (Greek letter lambda)

○ c is the speed of light

○ ~ is the wavenumber (read “nu tilde”)

○ is the angular frequency (Greek letter omega)

4. Classical mechanics was successful in explaining the motion of everyday

objects but fails when applied to very small particles. These failures led to

the development of Quantum Mechanics.

CHEM 3720

3

C. The Classical Wave Equation

1. It is a prelude to Quantum Mechanics because it introduces (or reminds

you) concepts that are similar to the ones in Quantum Mechanics.

2. The classical wave equation describes various wave phenomena:

a. a vibrating string

b. a vibrating drum head

c. ocean waves

d. acoustic waves

3. The classical (nondispersive) wave equation for a 1-dimensional wave:

2

2

22

2 ),(

v

1),(

t

txu

x

txu

□ ),( txu is the displacement of the string from the horizontal

position

□ v is the velocity or the speed that the disturbance moves

□ t is the time

a. The classical wave equation is a partial differential equation (a linear

partial differential equation) because ),( txu and its derivatives appear

only to the first power, and there are no cross terms.

b. The x and t are independent variables.

c. The ),( txu is a dependent variable.

4. Example: A 1-dimensional wave describing the motion of a vibrating string

a. The displacement ),( txu must satisfy certain physical conditions: the

amplitude should be zero at the end of the string.

□ 0),0( tu

□ 0),( tlu

CHEM 3720

4

b. These conditions are called boundary conditions because they satisfy

the behavior at the boundaries.

c. To solve the differential equation, we assume that ),( txu factors into a

function of x times a function of t:

)()(),( tTxXtxu

d. This technique (or method) is called the separation of variables.

e. Solving further the equation: – Substituting ),( txu in the equation above:

2

2

22

2 )()(

v

1)()(

dt

tTdxX

dx

xXdtT

– Dividing by )()(),( tTxXtxu :

Kdt

tTd

tTdx

xXd

xX

2

2

22

2 )(

)(

1

v

1)(

)(

1

– In order for this equation to be true for every x and t, each side should be equal to a

constant K called the separation constant.

– The problem of finding ),( txu transformed into two problems of finding X(x) and T(t) by

solving the following linear differential equations with constant coefficient (they are

ordinary differential equations):

0)()(

2

2

xKXdx

xXd

0)(v)( 2

2

2

tTKdt

tTd

f. Solving for X(x): l

xnBxX

sin)(

– Trivial solution is obtained (that is X(x) = 0) if 0K .

– If K 0, set 2K ( is real):

0)()( 2

2

2

xXdx

xXd

– The general solution for this equation is: xixi ececxX 21)(

– Considering Euler equation ( xixe ix sincos ):

xBxAxX sincos)(

– This solution of X(x) should verify the boundary conditions:

0)0( X A = 0

0)( lX 0sin lB nl where n = 1, 2, ...

CHEM 3720

5

g. Look more closely to the solutions:

Number of

wavelength

that fits in 2l:

n = 1

n = 2

n = 3

n = 4

Number of

wavelength

that fits in 2l:

n = 1

n = 2

n = 3

n = 4

□ By generalization: n

ln

2

○ This is called the eigenvalue condition.

□ The solutions are a set a functions called eigenfunctions or

characteristic functions.

xBxl

nBxX

nnnn

2sinsin)(

□ Also, angular frequencies 0vv2

2

n

l

n

nnn (where

l

v0

) are called eigenvalues or characteristic values.

1

2

3

4

CHEM 3720

6

h. Solving for T(t) but keeping in mind that l

n

0)(v)( 22

2

2

tTdt

tTd

– Similar to above, the solution is:

tEtDtT nn sincos)( where: l

nn

vv

i. Coming back to ),( txu :

)()(),( tTxXtxu

l

xntGtFtxu nn

sin)sincos(),( ; n = 1, 2,…

– There is a ),( txu function for each n so a better notation would be:

l

xntGtFtxu nnnnn

sin)sincos(),( ; n = 1, 2,…

– The sum of all ),( txun solutions is also a solution of the equation (This is called the

principle of superposition.) The general solution is:

1

sin)sincos(),(

n

nnnnl

xntGtFtxu

– Make the transformation: )cos(sincos tAtGtF where (Greek letter phi)

is the phase angle and A is the amplitude of the wave.

– Rewrite the general equation as:

11

),(sin)cos(),(n

nn

nnn txul

xntAtxu

□ Each ),( txun is called:

○ a normal mode

○ a standing wave

○ a stationary wave

○ an eigenfunction of this problem

j. The time dependence of each mode represents a harmonic motion of

frequency: n

nn

l

n

v

2

v

2 where the angular frequency is:

nnn

l

nv

v2v2v .

CHEM 3720

7

k. Solutions:

□ First term is l

xt

lA

sin)

vcos( 11

○ First term is called fundamental mode or first harmonic.

○ The frequency is: l2/v1

□ Second term is l

xt

lA

2sin)

v2cos( 22

○ Second term is called first overtone or second harmonic.

○ The frequency is: l/v2

○ The midpoint has a zero displacement at all times, and it is

called a node.

□ Third term is l

xt

lA

3sin)

v3cos( 33

○ Third term is called second overtone or third harmonic.

○ The frequency is: l2/v33

○ This term has two nodes.

□ Fourth term is l

xt

lA

4sin)

v4cos( 44

CHEM 3720

8

l. Let’s consider now the case of:

l

xt

l

xttxu

2sin)

2cos(

2

1sin)cos(),( 21

4

t1

2

4

30

□ This is an example of a sum of standing waves yielding a traveling

wave.

m. Thinking backwards, any general wave function can be decomposed

into a sum or superposition of normal modes.

n. The number of allowed standing waves on a string of length l:

□ increases as the wavelength decreases the possible high-

frequency oscillations outnumber the low-frequency ones.

n

ln

2

– Consider that l so we can approximate the set of integers n by a continuous

function )(n .

dl

dnl

nn

2

22

– The negative sign indicates that the number of standing waves decreases as

increases.

o. The number of standing waves in an enclosure of volume V (use c not

v for the speed):

d

Vdn

4

4 but

v

c and

cv ;

d

cdv

2 dv

cd

2

dvvc

Vdv

c

Vdn 2

3

2

4

4))(

4(

CHEM 3720

9

5. Example: A 2-dimensional wave equation = the equation of a vibrating

membrane:

2

2

22

2

2

2

v

1

t

u

y

u

x

u

where ),,( tyxu

a. Solving this equation:

– Similar to the one-dimensional problem, use separation of variables:

)(),(),,( tTyxFtyxu

22

2

2

2

2

2

2 ),(

1

)(v

1

y

F

x

F

yxFdt

Td

tT

– Use separation of variables for ),( yxF :

)()(),( yYxXyxF

– Divide by ),( yxF : 0)(

)(

1)(

)(

1 22

2

2

2

dy

yYd

yYdx

xXd

xX

– Solve two equations:

2

2

2 )(

)(

1p

dx

xXd

xX and 2

2

2 )(

)(

1q

dy

yYd

yY

where 222 qp

– Solutions for )(xX and )(yY are:

a

xnBxX

sin)( (n = 1,2,…) and

b

xmDyY

sin)( (m = 1,2,…)

where 2

2

2

2

b

m

a

nnm

– Solution for )(tT :

)cos(sincos)( nmnmnmnmnmnmnmnm tGtFtEtT

where

2

2

2

2

vvb

m

a

nnmnm

b. The general solution for ),,( tyxu :

1 1

1 1

sinsin)cos(

),,(),,(

n mnmnmnm

n mnm

b

ym

a

xntA

tyxutyxu

x

y

a

b

x

y

a

b

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10

c. Again, the general function is a superposition of normal modes

),,( tyxunm but in this case one obtains nodal lines (lines where the

amplitude is 0) instead of nodes.

d. Examples:

m =

n =

m =

n =

m =

n =

m =

n =

1

1

2

1

1

2

2

2

e. The case of a square membrane ( ba ), the frequencies of the normal

modes are given by:

22v mna

nm

□ For the cases of n = 1, m = 2 and n = 2, m = 1 one can see that:

a

v52112

although ),,(),,( 2112 tyxutyxu

f. This is an example of a degeneracy.

□ The frequency 2112 is double degenerate or two-fold

degenerate.

□ This phenomenon appears because of the symmetry ( ba ).

CHEM 3720

11

D. Unit Review

1. Important Terminology

frequency

wavelength

wavenumber

angular frequency

independent variables

boundary conditions

separation of variables

eigenfunctions

eigenvalues

stationary wave

traveling wave

node

degeneracy

CHEM 3720

12

2. Important Formulas

VKE

m

pmK

2v

2

1 22

2

~ cc

CHEM 3720

13

Unit II

The Development of Quantum Mechanics

A. Introduction

1. Failures of classical physics

a. The classical physics predicts the precise trajectory of a particle and

allows the translational, rotational, and vibrational modes of motion to

be excited to any energy by controlling the applied force.

b. These observations are found in everyday life in macroscopic world

but do not extend to individual atoms.

c. Classical mechanics fails when applies to transfers of very small

quantities of energy and to objects of very small mass.

2. Historic prospective on Quantum Mechanics (QM)

a. 1887 Hertz The discovery of photoelectric effect

b. 1895 Roentgen The discovery of x-rays

c. 1896 Becquerel The discovery of radioactivity

d. 1897 J. J. Thomson The discovery of the electron

e. 1900 Plank The quantum hypothesis of blackbody radiation

f. 1905 Einstein The quantum hypothesis of photoelectric effect

g. 1907 Thomson Model of atom

h. 1909 Rutherford Scattering experiment with particles

i. 1911 Rutherford The nuclear model of atom

j. 1913 Bohr The quantum hypothesis applied to the atom

k. 1924 de Broglie The prediction of the wave nature of the matter

l. 1925 Heisenberg QM in a form of matrix mechanics

m. 1926 Heisenberg Uncertainty principle

n. 1926 Schrödinger QM in a form of wave mechanics

CHEM 3720

14

B. Blackbody Radiation

1. Background

a. All objects are absorbing and emitting radiation and their properties as

absorbers or emitters may be extremely diverse.

b. It is possible to conceive the existence of objects that are perfect

absorbers of radiation, and they are called blackbodies.

c. A blackbody is an ideal body, which absorbs and emits all frequencies.

d. Blackbody radiation is the radiation emitted by the blackbody.

e. Ideal blackbodies do not exist. Most of substances absorb (or emit) all

frequencies only in a limited range of frequencies.

f. The best lab blackbody is not a body but a cavity

that is constructed with insulating walls, and in

one of which a small orifice is made.

g. When the cavity is heated, the radiation from the

orifice is a good sample of the equilibrium

radiation within the heated enclosure, which is

practically ideal blackbody radiation.

2. Experimental observations

a. Materials at the same temperature T have the

same blackbody radiation spectrum.

□ “Materials look the same”.

b. The brightness increases as T increases.

c. As the temperature increases, the maximum

shifts toward higher frequencies (or toward

lower wavelength).

d. There is a simple relationship between the wavelength at the maximum

intensity and the temperature, relationship known as the Wien

displacement law: Km108979.2const 3max T

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3. Analogy to classical systems

a. There is a similarity between the behavior of radiation within such a

cavity and that of gas molecules in a box.

b. Both the molecules and the radiation are characterized by a density,

and both exert pressure on the confining walls.

c. One difference is that the gas density is a function of V and T, whereas

radiation density is a function of temperature alone.

4. The classical explanation of the blackbody radiation

a. Rayleigh and Jeans try to explain the observed blackbody radiation

based on the laws of classical physics.

b. Assumptions:

□ Blackbody radiation is coming from standing electromagnetic

waves in the cavity that are at equilibrium with the vibrating atoms

(or electrons) in the walls.

□ The waves that are leaked out are observed.

□ The atoms in the blackbody are assumed to vibrate like harmonic

oscillators (these harmonic oscillators may be seen as

electromagnetic oscillators), and to be in thermodynamic

equilibrium with the radiation in the cavity.

□ According to the principle of equipartition energy, an oscillator in

thermal equilibrium with its environment should have an average

energy equal to TkB (that is TkB21 for kinetic energy and TkB2

1

for the potential energy).

□ We already found out that 32 /4 cdVdn for electromagnetic

waves from classical physics but we should multiply this result by

2 because for the electromagnetic radiation, both electric and

magnetic fields are oscillating

dc

Vdn 2

3

8

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c. According to Rayleigh-Jeans law, the radiant energy density is the

product of the density of states with the average energy of the state:

dc

Tk

V

dnTkdTTd 2

3B

B8

)(),(

where dT is the radiant energy density between the

frequencies and + d.

d. Rayleigh-Jeans law reproduces the experiment at low frequencies but

diverges at high frequencies (at low wavelength) as the radiation enters

the ultraviolet region.

e. Because of that, this divergence was called the ultraviolet catastrophe.

f. This was the first failure of classical physics in explaining theoretically

naturally occurring phenomena that could be explained by quantum

ideas.

5. The quantum explanation of the blackbody radiation

a. Proposed by Planck in 1900.

b. Assumptions:

□ The vibrating atoms in the walls have quantized energies or there

is a collection of N oscillations with fundamental frequency:

nhEn

where En is the energy of an oscillation, is the frequency, h is

a constant, and n = 0,1,2,…

○ Another way to say it is that oscillators take up energy in

increments h.

○ These increments (discrete units) are called quanta.

□ All frequencies are present.

c. Planck distribution law for blackbody radiation in terms of frequency:

– The number of oscillators having energy nh is given by the Boltzmann distribution: Tknh

n eNNB/

0

where 0N is the number of oscillations in the lowest energy state (n = 0)

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17

– Total number of oscillators:

0

/0

/20

/00

BBB ......

n

TknhTkhTkheNeNeNNN

– Total energy of oscillators is the product of the number of oscillators and their energy:

0

/0

/20

/00

BBB .....20

n

TknhTkhTkhneNheNheNhNE

– The average energy of an oscillator becomes:

11 BB

B

/

0

0

0

/

0

/

Tkhx

n

nx

n

nx

n

Tknh

n

Tknh

e

h

e

h

e

ne

h

e

neh

N

E

where Tk

hx

B

1

8

1)(),(

BB /

3

3/

TkhTkh

e

d

c

h

V

dn

e

hdTTd

where dT )( is the radiant energy density between the

frequencies and + d.

d. Planck distribution law for blackbody radiation in terms of wavelength:

1

8

1)(),(

BB /5/

TkhcTkhe

dhc

V

dn

e

hdTTd

where dT )( is the radiant energy density between the

wavelength and + d.

e. Successes of Planck’s distribution law

□ It reproduces experimental data for all frequencies and

temperatures within the experimental error if Js10626.6 34h .

○ Units of energy·time = action.

○ The constant h is now known as the Planck constant.

□ It explains the constant in the Wien distribution law:

Bmax

965.4 k

hcT

□ It introduces the idea of energy quantization: an oscillator acquires

energy only in discrete units called quanta.

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C. Photoelectric Effect

1. Experimental observations:

a. Hertz (1887) a spark would jump a gap more readily when the gap

electrodes were illuminated by light from another spark gap than when

they were kept in the dark.

b. The phenomenon was due to the emission of electrons from the surface

of solids upon incidence of light having suitable wavelengths. These

emitted electrons were called photoelectrons.

c. Whether or not electrons are emitted from the surface (plate of

electrodes) depends only on the frequency of the light and not at all on

the intensity of the beam.

d. The number of electrons emitted is proportional

to the intensity of the light.

e. There is no time delay between the light beam

striking the surface and the emission of the

electrons.

f. Lenard (1902) determined that the maximum kinetic energy of the

emitted electrons depends on the frequency of the incident light, and

below a certain frequency called threshold frequency 0 no electrons

were ejected. Above 0, the kinetic energy of the electrons varies

linearly with the frequency .

2. The classical interpretation of the photoelectric effect

a. Electromagnetic radiation is an electric field oscillating perpendicular

to its direction of propagation, and the intensity of radiation is

proportional to the square of the amplitude of the electric field.

b. Increasing intensity of the light, the electrons oscillate more violently

and break away from the surface.

c. KE will depend on the amplitude (intensity) of the field.

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d. Photoelectric effect should occur for any frequency of light as long as

the intensity is sufficiently high.

e. For weak intensities and reasonable values of the frequency a long time

should intervene before any electron would soak up enough energy to

be emitted from the metal.

f. None of these predictions were verified experimentally. Classical

physics failed badly.

3. The quantum interpretation of the photoelectric effect

a. Einstein extends Plank’s idea of quantized oscillators comprising the

blackbody radiation by suggesting that radiation itself is quantized.

b. He suggests that the radiation itself exists as small packets (quanta) of

energy known as photons:

hE

c. Entire quantum of energy is accepted by a single electron and cannot

be divided among all the electrons present.

d. Kinetic energy of the ejected electrons is the difference between the

energy of the incident photons h and the minimum energy required to

remove an electron from the surface of a particular metal called the

work function of the metal and denoted .

hm 2v2

1KE

e. One can write as: = h0

where 0 is called the threshold frequency.

)(KE 00 hhh

f. The work function is usually

expressed in eV:

J10602.1eV1 19

where V1C1J1

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D. Atomic Spectrum of Hydrogen Atom; The Bohr Model

1. Experimental facts known at the beginning of 20th century

a. Structure of the atom

□ The nuclear model for the atom had been proposed and accepted.

□ The model has a positively charged nucleus but the model was

unstable according to classical electromagnetic theory.

b. Existence and properties of atomic spectra

□ It was known that the emission spectra of atoms consist of certain

discrete frequencies called line spectra. The simplest such spectra

was the spectrum of hydrogen atom.

□ For the hydrogen atom the line spectra is composed from a number

of series.

□ One of these series is in the visible range of the radiation and is

called Balmer series in honor of Balmer who showed (1885) that

the emission lines could be described by the equation:

Hz4

1102202.82

14

n where n = 3,4,5,…

○ Using wavenumbers (c

~ ) instead of frequency:

122

cm1

2

1109680~

n where n = 3,4,5,…

○ This is the Balmer formula and it describes the lines in the

hydrogen spectrum occurring in VIS and near UV regions.

○ As n increases the lines bunch up toward the series limit.

□ Other series have been discovered in UV and IR regions and

Balmer’s formula had been generalized by Rydberg and Ritz:

22

21

111~

nnRH

where n1 < n2

CHEM 3720

21

○ This is called Rydberg formula.

○ 1-H cm109677.57R is the Rydberg constant.

□ These series were named as:

Lyman 11 n ,...3,22 n UV

Balmer 21 n ,...4,32 n VIS

Pashen 31 n ,...5,42 n Near IR

Bracket 41 n ,...6,52 n IR

○ Series limits are obtained for 2n .

□ Ritz (1908) showed that:

21~ TT

where 21

1n

RT H and

22

2n

RT H are called terms.

○ This is called the Ritz combination rule.

□ Conway (1907) proposed that a single atom produces a single

spectral line at a time, and the emission is due to a single electron

in an “abnormal state”.

2. The Bohr atomic model (1911)

a. Assumptions:

□ The spectral lines are produced by atoms one at a time.

□ A single electron is responsible for each line.

□ The Rutherford nuclear atom is the correct model.

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22

□ The quantum laws apply to jumps between different states

characterized by discrete values of angular momentum and, Bohr

added, energy.

□ The angular momentum L of the electron is given by the

Ehrenfest’s rule, i.e., the angular momentum L is given by:

nh

nrmL 2

ve where n is an integer.

□ Two different states of the electron in the atom are involved. They

are called allowed stationary states, and the spectral terms of Ritz

correspond to these states.

□ Planck-Einstein equation E = h holds for the emission and

absorption. If the electron makes a transition between two states

with energy Em and En, the frequency of the spectral line is given

by h = Em – En.

□ Bohr said: “We must renounce all attempts to visualize or to

explain classically the behavior of the active electron during a

transition of the atom from one stationary state to another.”

b. The picture of the atom: a massive nucleus (proton) considered fixed

with the electron revolving around it.

c. The forces between the proton and the electron:

□ Attraction given by the Coulomb’s law:

20

2

4 r

ef

□ Repulsion given by the centrifugal force:

r

mf

2ev

CHEM 3720

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d. Bohr assumed the existence of stationary states so the electron is not

accelerated toward nucleus.

□ The radius of the stationary orbit is given by:

r

m

r

e2

e2

0

2 v

4

2e0

2

v4 m

er

For hydrogenic atoms (He+, Li2+, etc): 2

e0

2

v4 m

Zer

e. The angular momentum of the electrons is quantized:

nh

nrm 2

ve with n = 1,2,… rm

nh

e2v

f. The allowed radius must satisfy the condition:

nrem

n

em

hnr

2e

220

2e

220 4

g. The smallest radius, obtained for n = 1, is called the first Bohr radius.

bohr1A5292.0m10292.5 110 a

h. The energies of the stationary states (or allowed orbits):

□ The total energy is given by:

r

e

r

emVKE

0

2

0

22

e84

v2

1

□ By introducing the expression of the allowed orbit radius:

2220

4e 1

8 nh

emEn

with n = 1,2,…

□ When n = 1, one gets the ground state and the ground state energy: 1

1 cm109690kJ/mol1312eV6.13E

□ When n = 2,3,…, one gets excited states and excited state energies: 1

2 cm74202kJ/mol328eV4.3E

13 cm12187kJ/mol134eV5.1

E

CHEM 3720

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i. The lines in the spectrum were a result of two allowed stationary states

(corresponding to the spectral terms of Ritz) and Plank-Einstein

equation E = h holds with the energy being equal to energy difference

between two states n1 and n2:

EEEh 12

□ This is the Bohr frequency condition.

j. The observed spectrum is due to transitions from one allowed energy

state (also called stationary state or orbit) to another one.

□ The energy difference is being given to a photon:

hnnh

emE

22

21

220

4e 11

8

□ In terms of wavenumbers:

22

21

320

4e 11

8

~

nnch

em

c

where 32

0

4

8 ch

eme

is the Rydberg constant

1cm32.109737 R

○ R∞ is only 0.005% off the RH.

Energy Level

Diagram:

CHEM 3720

25

E. The wavelike properties of the matter

1. Wave-particle duality

a. In the beginning of the 1920’s it was well established that the light

behaves as a wave in some experiments and as a stream of photons in

others.

b. This is known as the wave-particle duality of the light.

c. For the light:

chhE and 2mcE

2mcc

h

p

h

mc

h

where p is the photon momentum

d. In 1923-24 Louis de Broglie proposed the idea that the matter might

also display wavelike properties under certain conditions. Under those

conditions, similar equations should hold for the matter.

e. A particle of mass m moving with the speed v will exhibit a de Broglie

wavelength given by:

vm

h

p

h

f. For particles with big mass the de Broglie wavelength is so small that it

is completely undetectable and of no practical consequences.

g. Examples:

Baseball: m102.1m/s40mph90v

kg0.14oz5.0 34

m

Electron: m1043.2m/s102.998v

kg109.109 106

31

m

h. The wave property of the electron (as well as other particles like the

neutron or the hydrogen atom) has been observed experimentally.

i. The electron diffraction is actually used in the electron microscopy.

CHEM 3720

26

2. The de Broglie interpretation of Bohr radius

a. The Bohr condition that says that angular momentum of the electron

should be a multiple of :

nh

nrm 2

ve

b. This is equivalent with saying that an integral number of complete

wavelength must fit around the circumference of the orbit:

nr 2

c. Substituting vem

h

p

h one obtains:

v2

em

hnr n

hnrm

2ve

d. This is equivalent of saying that the de Broglie waves of the orbiting

electron must “match” or be in phase as the electron makes one

complete revolution.

e. Without the matching the amplitude of the wave gets cancelled during

each revolution and the wave will disappear.

CHEM 3720

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F. Heisenberg’s Uncertainty Principle (Principle of indeterminacy)

1. Heisenberg’s Uncertainty Principle in terms of position and momentum

a. In classical mechanics it was possible to be able to determine both the

position and the momentum of a particle. What happens if the particle

has wave properties?

b. Look at the example of an electron: during the measurement of the

position of an electron, the radiation used to do that changes its

momentum.

c. If we wish to locate an electron within a region x there will be an

uncertainty in the momentum of the electron (denoted p).

d. Heisenberg showed that:

4

hpx

□ This is the Heisenberg Uncertainty Principle.

2. Heisenberg’s Uncertainty Principle in terms of energy and time

a. A similar expression can be deduced for the uncertainty in the energy

E and the uncertainty in the time t:

m

pE x

2

2

; xxxxx p

t

xpp

m

pE

v

xptE x

4

htE

b. The uncertainty in terms of E and t is used in spectroscopy:

□ stable states sharp lines (large t small E)

□ unstable states diffuse lines (small t large E)

c. To measure the energy of a system with accuracy E the measurement

must be extended over a period of time of order of magnitude h/t.

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G. Unit Review

1. Important Terminology

blackbody

blackbody radiation

Wien displacement law

ultraviolet catastrophe

quanta

Planck constant

photoelectrons

photon

threshold frequency

work function

line spectra

Lyman, Balmer, Pashen, Bracket series

series limit

CHEM 3720

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Balmer formula

Rydberg formula

Rydberg constant

term

angular momentum

stationary state or orbit

first Bohr radius

ground state

excited states

Bohr frequency condition

wave-particle duality

de Broglie wavelength

Heisenberg Uncertainty Principle

CHEM 3720

30

2. Important Formulas

nhEn or hE

1

8)(

B/

3

3

Tkh

e

d

c

hdT

1

8)(

B/5

Tkhce

dhcdT

)(v2

1KE 00

2 hhhhm

22

21

111~

nnRH

nh

nrm 2

ve

2e

2204

em

nrn

2220

4e 1

8 nh

emEn

EEEh 12

2

221

11~

nnR

vm

h

p

h

4

hpx

4

htE

CHEM 3720

31

Unit III

Postulates and Principles of Quantum Mechanics

A. The Schrödinger Equation

1. Introduction

a. It is the fundamental equation of quantum mechanics.

b. The solutions of the time-independent Schrödinger equation are called

stationary-state wave functions.

2. Time-independent Schrödinger equation

a. Schrödinger equation is the equation for finding the wave function of a

particle and come up based on idea that if the matter possesses

wavelike properties there must be a wave equation that governs them.

b. Schrödinger equation cannot be demonstrated (it can be seen as a

fundamental postulate) but it can be understood starting from classical

mechanics wave equation:

– Classical wave equation:

2

2

22

2

v

1

t

u

x

u

– The solution is: txtxu cos)(),(

– The t dependence appear as cost or T(t) or exp(2ivt).

– The spatial amplitude of the wave, , is obtained from the equation:

0)(v

)(

2

2

2

2

xdx

xd

where

v22 v

0)(4)(

2

2

2

2

xdx

xd

– Rearrange the equation considering:

Vm

pVKE

2

2

)(2 VEmp

)(2 VEm

h

p

h

0)()]([2)(

22

2

xxVEm

dx

xd

CHEM 3720

32

)()()()(

2 2

22

xExxVdx

xd

m

c. This is the one-dimensional time-independent Schrödinger equation.

d. The solutions (wave functions) of this equation are called stationary-

state wave functions.

e. Schrödinger equation for three dimensions:

□ Rewrite the equation and generalize to three dimensions:

),,(),,(),,(

),,(2 2

2

2

2

2

22

zyxEzyxzyxV

zyxzyxm

□ Rewrite including the notation for the Laplacian operator:

○ 22

2

2

2

2

2

zyx is the Laplacian operator.

○ An operator is a symbol that tells you to do something (a

mathematical operation) to whatever (function, number, etc)

follows the symbol.

EzyxVm

),,(

2

22

□ Rewrite including the notation for the Hamiltonian operator:

○ HzyxVm

ˆ),,(2

22

is the Hamiltonian operator.

EH ˆ

This is the simple form of the Schrödinger equation.

CHEM 3720

33

3. Eigenvalue-eigenfunction relation in quantum mechanics

a. The eigenvalue-eigenfunction relation in quantum mechanics is written

in the form:

)()(ˆ xxA

where Â is an operator

(x) is an eigenfunction or characteristic function

is an eigenvalue or characteristic value

b. The wave functions in Quantum Mechanics are eigenfunctions of the

Hamiltonian operator and the total energy is the eigenvalue.

4. Interpretation of the wave function :

a. The *(x)(x)dx is the probability that the particle to be located

between x and x + dx (the one-dimensional case).

b. The function * is the complex conjugate of the wavefunction .

□ The complex conjugate * is obtained by replacing i with –i in the

expression of the wavefunction .

□ The * product becomes real.

CHEM 3720

34

B. Postulates of Quantum Mechanics

1. Enunciations

a. Postulate 1: The state of a quantum-mechanical system is completely

specified by a function (x) that depends upon the coordinate of the

particle. All possible information about the system can be derived

from (x). This function, called the wave function or the state

function, has the important property that *(x)(x)dx is the probability

that the particle lies in the interval dx, located at the position x.

Note: This principle can be stated using the time-dependent wave

function (x,t) instead of the time-independent wave function (x).

b. Postulate 2: To every observable in classical physics there corresponds

a linear, Hermitian operator in quantum mechanics.

c. Postulate 3: In any measurements of the observable associated with the

operator Â , the only values that will ever be observed are the

eigenvalues an, which satisfy the eigenvalue equation: nnn aA ˆ

d. Postulate 4: If a system is in a state described by a normalized wave

function , then the average value of the observable corresponding to

Â is given by:

space all

ˆ dxAa

e. Postulate 5: The wave function, or state function, of a system evolves

in time according to the time-dependent Schrödinger equation:

t

txitxH

),(),(ˆ

f. Postulate 6: All electronic wave functions must be antisymmetric under

the exchange of any two electrons.

g. Note: The principles above are stated for a one-dimensional space. To

generalize to three dimensions one can replace (x) with (x,y,z) and

(x,t) with (x,y,z,t).

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2. First postulate of quantum mechanics

a. Classical mechanics deals with quantities called dynamical variables:

position, momentum, angular momentum, and energy. A measurable

dynamical variable is called an observable.

b. The movement of a particle can be described completely in classical

mechanics. This is not possible in quantum mechanics because the

uncertainty principle tells one cannot specify or determine the position

and the momentum of a particle simultaneously to any desirable

precision. This leads to the first postulate of quantum mechanics.

c. Postulate 1: The state of a quantum-mechanical system is completely

specified by a function (x) that depends upon the coordinate of the

particle. All possible information about the system can be derived

from (x). This function, called the wave function or the state

function, has the important property that *(x)(x)dx is the probability

that the particle lies in the interval dx, located at the position x.

d. This principle can also be stated using the time-dependent wave

function (x,t) instead of the time-independent wave function (x).

e. The one-dimensional probability will be replaced by the three-

dimensional probability:

dxdydzzyxzyx ),,(),,(*

f. The case of two particles:

212121* ),(),( dxdxxxxx

g. The total probability of finding a particle somewhere must be unity:

1)()(

spaceall

* dxxx

□ The wave functions that satisfy this condition are said to be

normalized.

CHEM 3720

36

□ The functions are called normalizable if:

1)()(

spaceall

* Adxxx

because they can be normalized once they are divided by A .

h. Other conditions for the wave function(x),

its first derivative dx

xd )(, and its second

derivative 2

2 )(

dx

xd should be:

○ single-valued

○ continuous

○ finite

□ If these conditions are met, the wave

functions is said to be “well behaved”.

3. Second postulate of quantum mechanics

a. Postulate 2: To every observable in classical physics there corresponds

a linear, Hermitian operator in quantum mechanics.

b. An operator is a symbol that tells to do a mathematical operation to

whatever follows the symbol.

□ The operators are usually denoted by a capital letter with a little hat

over it called a carat (like in Ĥ ).

□ What follows the operator is called operand.

c. An operator Â is linear if:

xfAcxfAcxfcxfcA 22112211 ˆ)(ˆ)()(ˆ

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37

d. An operator Â is Hermitian if it has the property of being linear and if:

dxAdxA

spaceall

12spaceall

2*1 )

ˆ(ˆ

or dxxfAxgdxxgAxf

spaceallspaceall

)](ˆ)[()(ˆ)(

for any pair of functions 1 and 2 (or f and g) representing a physical

state of a particle.

□ These expression are true in derivative form as well.

e. All the quantum operators can be written starting from the operators in

the table below using classical physics formulas:

Classical

Variable

QM

Operator

Expression for operator

x X̂ x

xP xP̂ x

ixi

t T̂ (or t̂ ) t

E Ê t

iti

or zyxVm

,,2

22

□ Examples:

○ Kinetic energy: m

pK

2

2

1-dimensional: 2

222

22

ˆˆ

xmm

pK

3-dimensional: 22

2

2

2

2

2

22

22ˆ

mzyxmK

○ Potential energy: VV ˆ

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○ Total energy: VKE

The operator for the total energy is the Hamiltonian Ĥ :

Vm

Vzyxm

VKH

2

2

2

2

2

2

2

22

22ˆˆˆ

○ The angular momentum along the x axis:

yzx pzpyL

)(ˆy

zz

yiLx

f. Commutation

□ It is a property of operators in which )(ˆˆ xfBA is compared to

)(ˆˆ xfAB .

○ )](ˆ[ˆ)(ˆˆ xfBAxfBA

○ )](ˆ[ˆ)(ˆˆ xfABxfAB

□ Operators usually do not commute: )(ˆˆ)(ˆˆ xfABxfBA

□ When )(ˆˆ)(ˆˆ xfABxfBA for every compatible f(x) the operators Â

and B̂ are said to commute.

□ Example: dx

dA ˆ and 2ˆ xB

dx

xdfxxxfxfBA 22ˆˆ

dx

xdfxxfAB 2ˆˆ

□ Rewrite and drop f(x).

□ Define the commutator (i.e., the commutator of Â and B̂ ) as:

ABBABA ˆˆˆˆ]ˆ,ˆ[

□ The commutator of commuting operators is the zero operator:

0̂ˆˆˆˆ ABBA

where 0̂ is the zero operator.

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g. A special property of linear operators (for example, the operator Â) is

that a linear combination of two eigenfunctions of the operator with the

same eigenvalue is also an eigenfunction of the operator:

□ Consider that two eigenfunctions have the same eigenvalue:

11ˆ aA and 22

ˆ aA

(This is a two-fold degeneracy.)

□ Then any linear combination of 1 and 2 is also an eigenfunction

of Â :

)(ˆˆ)(ˆ 221122112211 ccaAcAcccA

□ Example:

– Solving )()( 2

2

2

xfmdx

xfd gives the eigenfunctions imxexf )(1 and

imxexf )(2 .

– A linear combination of 1f and 2f , )()()( 21112 xfcxfcxf x , is also an

eigenfunction:

)()(

)()()(

122

212

22

212

122

xfmececm

emcemcdx

xfd

imximx

imximx

4. Third postulate of quantum mechanics

a. Postulate 3: In any measurements of the observable associated with the

operator Â , the only values that will ever be observed are the

eigenvalues an, which satisfy the eigenvalue equation:

nnn aA ˆ

b. For an experiment designed to measure the observable associated to Â ,

we will find only the values naaa ,...,, 21 corresponding to the states

n ,...,, 21 and no other values will be observed.

c. Example: If HA ˆˆ nnn EH ˆ (Schrödinger equation) and only

the nE energies (eigenvalues) will be experimentally observed.

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5. Fourth postulate of quantum mechanics

a. Postulate 4: If a system is in a state described by a normalized wave

function , then the average value of the observable corresponding to

Â is given by:

space all

dxÂa

□ a is the symbol that represents the average value.

b. Determine the variance, 2, of experiments:

□ Variance is a statistical mechanics quantity.

□ Assume we have: )()(ˆ xaxA nnn

nnn adxxAxa

)(ˆ)(*

□ Also: )()(ˆˆ)(ˆ 22 xaxAAxA nnnn

22*2 )( nnnn adxxaxa

□ Variance of the experiments become:

022222 nna aaaa

□ The standard deviation a is zero so the only values observed are

the an values.

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6. Fifth postulate of quantum mechanics

a. Postulate 5: The wave function, or state function, of a system evolves

in time according to the time-dependent Schrödinger equation:

t

txitxH

),(),(ˆ

b. Time dependence of ),( tx :

– Assume Ĥ does not depend on time.

– Separation of variable: )()(),( tfxtx

Edt

tf

tf

ixH

x

)(2

)()(ˆ

)(

1

(where E is a constant)

)()(ˆ xExH (time-independent Schrödinger equation)

)()(

tEfi

dt

tdf

– The solution is: tiiEt eetf /)( where hvE

/

)(),(tiEnextx

(assuming a set of solutions nE )

c. The probability density and the average values are independent of time

(the function of time cancels out):

dxxxdxexexdxtxtxtiEtiE nn )()()()(),(),( *//**

)(xn are called stationary-state wave functions

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C. More Properties

1. Properties of the Eigenfunctions of Quantum Mechanical Operators

a. The eigenfunctions of QM operators are orthogonal.

b. The eigenfunctions of QM operators are functions satisfying the

equation: nnn aA ˆ where na are real (although Â and n can be

complex) when the operator is Hermitian.

c. If the eigenvalues na are real then eigenfunctions have the property:

0)()(*

dxxx nm

□ If a set of wave functions satisfies this condition then the set is said

to be orthogonal or that the wave functions are orthogonal to each

other.

□ If they are also normalized, 1)()(*

dxxx nn , than the set is

said to be orthonormal and wave functions are orthonormal.

d. Generalizing:

ijji dx

*

where

ji

jiij

if0

if1 is the Kroenecker delta symbol.

e. Example: sin and cos functions are

orthogonal in 0–2 (or 0–) interval.

f. An even function is always orthogonal to an

odd function over an interval centered at 0.

□ An even function is a function for which f(x) = f(–x).

□ An odd function is a function for which f(x) = –f(–x).

g. The eigenfunctions of QM operators form a complete set.

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2. Uncertainty Principle written based on operators

a. The uncertainty in the measurements of a and b ( a and b ) are

related by:

dxxBAxba )(]ˆ,ˆ[)(

2

1 *

where ABBABA ˆˆˆˆ]ˆ,ˆ[ is the commutator.

□ If Â and B̂ commute then 0̂]ˆ,ˆ[ BA then 0ba a and b

can be measured simultaneously to any precision.

□ If Â and B̂ do not commute then a and b cannot be simultaneously

determined to arbitrary precision.

b. Example:

□ For x the operator is xX ˆ

□ For xp the operator is dx

diPx ˆ

□ The commutator of X̂ and xP̂ is IiPXXPXP xxxˆˆˆˆˆ]ˆ,ˆ[

where Î is the identity operator (multiplication to 1).

dxxIix xxp )()ˆ)((2

1 *

22

1 ixp

where 22 xxx and 22 ppp

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D. Unit Review

1. Important Terminology

Schrödinger equation

wavefunctions

operator

operand

Laplacian operator

Hamiltonian operator

eigenvalue-eigenfunction relation

complex conjugate

QM postulates

wavefunction interpretation

Hermitian operator

observable

average value

CHEM 3720

45

normalized/normalizable

commutation

commutator

linear operators

variance/standard deviation

orthogonal

orthonormal

even function

odd function

complete set

CHEM 3720

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2. Important Formulas

)()()()(

2 2

22

xExxVdx

xd

m

2

2

2

2

2

2

2

zyx

EzyxVm

),,(

2

22

HzyxVm

ˆ),,(2

22

)()(ˆ xxA

space all

ˆ dxAa

xi

xiPx

ˆ

1)()(

spaceall

* dxxx

0)()(*

dxxx nm

ijji dx

*

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Unit IV

Applications of Quantum Theory

A. The Particle in a Box

1. Introduction

a. The particle-in-a-box refers to the quantum mechanical treatment of

translational motion.

b. One tries to solve the Schrödinger equation to obtain the wavefunctions

and the allowed energies for the particle.

c. The time-independent three-dimensional Schrödinger equation:

),,(),,(),,(2 2

2

2

2

2

22

zyxEzyxzyxVzyxm

2. Particle in a one-dimensional box (or one-dimensional motion)

a. Consider only a one-dimensional motion

(along x coordinate) drop y and z in the

equation above.

b. Consider that the particle experiences no

potential energy between the position 0

and a V(x) = 0 (for 0 x a)

c. The Schrödinger equation for this problem:

Edx

d

m

2

22

2

d. The mathematical solution of this equation is:

kxBkxAx sincos)(

where h

mEmEk

22)2( 2/1

m

khE

2

22

8

aa

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e. Apply the boundary conditions and the normalization condition to

determine A and B:

0)0( 0A

0)( a 0sin kaB nka where n = 1,2,…

f. The allowed energy levels are:

2

22

8ma

nhEn where n = 1,2,…

where n is called quantum number.

□ The energy is quantized.

□ The energy levels increase as the quantum

number increases.

□ The energy separation between the allowed

energy levels increases as the quantum

number increases.

□ The energy levels and the energy separation

between the energy levels increases as the

size of the box or the mass decreases.

g. Quantum numbers appear naturally when the boundary conditions are

put in the Schrödinger equation (like in the string problem) and are not

introduced ad hoc like in Plank model for blackbody radiation or Bohr

model of hydrogen atom.

h. Determining the wavefunctions:

□ Find B by setting the condition that the function (x) is normalized

1)()(

0

* dxxxa

1sin

0

22 a

a

xnB

1

2

2 a

B a

B2

○ B is called the normalization constant.

a2)

a2)

CHEM 3720

49

□ The normalized eigenfunctions n(x) are given by:

a

xn

axn

sin

22/1

where 0 x a and n = 1,2,…

i. Solving the Schrödinger equation for the particle in a box problem

gives a set of allowed energies (or eigenvalues) and a set of wave

functions (or eigenfunctions).

j. Representations including the energies and the wave functions (a) and

the probability densities (b,c) for the first few levels for the particle in a

box:

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50

k. The particle-in-a-box model can be applied to electrons moving freely

in a molecule (also called free-electron model).

l. Example:

□ Butadiene has an adsorption band at 4.61104 cm–1. As a simple

approximation, consider butadiene as being a one-dimensional box

of length 5.78 A = 578 pm and consider the four electrons to

occupy the levels calculated using the particle in a box model.

○ The electronic excitation is given by )23(8

22

2e

2

am

hE

○ The calculated excitation energy

14 cm1054.4~

hc

E

compares very well with the

experimental value.

○ This simple free-electron model can be quite successful.

m. The probability of finding the particle between x1 and x2 is given by:

2

1

)()(*x

xnn dxxx

□ If x1 = 0 and x2 = a/2 then 2/1)2/0(Prob ax for all n.

□ For n = 1: )2/)4/(Prob)4/0(Prob axaax

□ As n increases (for example n = 20) these 2 probabilities become

equal.

n. Generalizing, the probability density becomes uniform as n increases.

o. This is an illustration of the Correspondence Principle that says that

quantum mechanics results and classical mechanics results tend to

agree in the limit of large quantum numbers.

□ The large-quantum-number limit is called the classical limit.

CHEM 3720

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p. The eigenfunctions are orthogonal to each other:

0)()(

0

* a

nm dxxx (where m n)

□ Example: look at 1(x) and 3(x):

03

sinsin2

)()(

003

*1

aa

dxa

x

a

x

adxxx

q. Average values and variances for the position and the momentum of

the particle:

– Average value of position: 2

sin2

)()(

0

2

0

* adxa

xnx

adxxxxx

aa

nn

– Average value of position square: 22

22

0

2*2

23)()(

n

aadxxxxx

a

nn

– Variance in position:

2

32

222222 n

n

axxx

– Standard deviation in position:

2/122

22 232

n

n

axxx

– Average value of momentum: a

dxa

xn

a

xn

a

nip

02

0cossin2

– Also: 2

222

2

2222

222

a

n

ma

nmEmp

;

2

2222

a

np

– Finally: 2

232

2/122

npx (Heisenberg Uncertainty Principle)

3. Particle in a two-dimensional box (or two-dimensional motion)

a. The Schrödinger equation for this problem:

yxEyxm

,2 2

2

2

22

b. Allowed energies (or eigenvalues)

2

2

2

22

8 b

n

a

n

m

hE

yxnn yx

b

a

b

a

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52

b. The wavefunctions (or eigenfunctions)

b

yn

a

xn

bayYxXyx

yx

sinsin22

)()(),(

4. Particle in a three-dimensional box (or three-dimensional motion)

a. The Schrödinger equation:

zyxEzyxm

,,2 2

2

2

2

2

22

b. Solutions:

c

zn

b

yn

a

xn

cbazyx z

yxnnn zyx

sinsinsin

222),,(

2

2

2

2

2

22

8 c

n

b

n

a

n

m

hE z

yxnnn zyx

c. Average values of the position and the momentum of the particle:

kjiRr222

),,(ˆ),,(*

0 0 0

cbazyxzyxdzdydx

a b c

where kjiR ZYX ˆˆˆˆ

0),,(ˆ),,(*

0 0 0

zyxzyxdzdydxa b c

Pp where

zyxi kjiP ˆ

d. The case of a cubic box (a = b = c):

□ Three sets of quantum numbers

give same energy:

2

2

1121212118

6

ma

hEEE

CHEM 3720

53

□ The energy level 2

2

8

6

ma

hE is degenerate.

□ The energy level 2

2

1118

3

ma

hE is nondegenerate.

□ Degeneracy is equal to the number of states with same energy.

○ Once the symmetry is destroyed then degeneracy is lifted.

5. Separation of variables

a. The Hamiltonian for the particle in a 3-dimensional box:

zyx HHHzyxm

H ˆˆˆ2

ˆ2

2

2

2

2

22

□ It can be written as a sum of terms where:

2

22

2ˆ

xmH x

, 2

22

2ˆ

ymH y

, 2

22

2ˆ

zmH z

○ The operator is said to be separable.

b. The eigenfunctions zyx nnn

are written as a product of eigenfunctions

of each operator xĤ , yĤ , and zĤ , and the eigenvalues zyx nnnE are

written as a sum of the eigenvalues of each of the operator xĤ , yĤ ,

and zĤ .

c. This is a general property in quantum mechanics: If the Hamiltonian

(or an operator in general) can be written as a sum of terms involving

different coordinates (i.e., the Hamiltonian is separable) then the

eigenfunctions of Ĥ is a product of the eigenfunctions of each

operator constituting the sum and the eigenvalues of Ĥ is a sum of

eigenvalues of each operator constituting the sum.

d. Example: )(ˆ)(ˆˆ 21 wHsHH )()(),( wswsnm and mnnm EEE

where )()()(ˆ1 sEssH nnn and )()()(ˆ

2 wEwwH mmm

CHEM 3720

54

B. The Harmonic Oscillator

1. Introduction

a. The harmonic oscillator refers to the quantum mechanical treatment of

vibrational motion.

b. The harmonic oscillator in classical mechanics: – Force: kxkf )( 0 (Hooke’s Law) where k is the force constant.

– Hooke’s Law combined with Newton’s equation: 02

2

kxdt

xdm

– The solution of this equation: tAtx cos)( where m

k

– The potential energy of the oscillator: tkAxk

xV 222 cos2

1

2)(

– The kinetic energy of the oscillator: tkAdt

dxmK 22

2

sin2

1

2

1

– The total energy of the oscillator: 2

2

1kAKVE

The total energy is conserved; it is transferred between K and V.

c. The harmonic motion in a diatomic molecule:

– Consider the movement of atoms of masses m1 and m2.

)( 01221

2

1 xxkdt

xdm

)( 01222

2

2 xxkdt

xdm

– By summing the two equation above:

02

2

dt

XdM where 21 mmM and

M

xmxmX 2211

where X is the center the mass coordinate.

– Subtracting equation 2 divided by m2 from equation 1 divided m1:

02

2

kxdt

xd where 012 xxx and

21

21

mm

mm

where is the reduced mass of the system and x is the relative coordinate.

□ The movement of a two-body system can be reduced to the

movement of a one-body system with a mass equal to the reduced

mass of the two-body system.

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2. The quantum-mechanical harmonic oscillator

a. The Schrödinger equation for quantum-mechanical harmonic

oscillator:

)()()(2 2

22

xExxVdx

d

where 2

2

1)( kxxV

0)(2

12)( 222

2

xkxE

dx

xd

b. The eigenvalues are:

2

1

2

1

2

1nhnn

kEn

where ,...2,1,0n ;

k

;

k

2

1

c. The wave functions (eigenfunctions) are:

2/2/1 2)()( xnnn exHAx where

2

k

□ The normalization constant is given by

4/1

!2

1

n

An

n

□ The wave functions form an orthonormal set.

□ The )( 2/1 xHn are polynomial functions called Hermite

polynomials where )(nH is a nth degree polynomial in .

○ Here are the first few Hermite polynomials: 1)(0 ξH 2)(1 ξH

24)( 22 ξH 128)(3

3 ξH

124816)( 244 ξH 12016032)(35

5 ξH

even polynomials: f(x) = f(–x) odd polynomials: f(x) = –f(–x)

□ Continuous odd functions properties: 0)0( f and

A

A

dxxf 0)(

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56

d. The normalized wave functions and the probability density:

e. The existence of the zero-point energy

□ The minimum energy (the ground-state energy) is not zero even for

n = 0.

□ This energy is called the zero point energy (ZPE):

h2

1ZPE

□ It is a result in concordance with the Uncertainty Principle that

says one cannot determine exactly both the position (for example x

= 0) and the momentum (for example p = 0) at the same time.

f. The quantum-mechanical harmonic oscillator model accounts for the

IR spectrum of a diatomic molecule.

□ It is a model for vibrations in diatomic molecules.

□ The transitions between various levels in harmonic oscillator

model follow the selection rule:

1n

□ The quantum-mechanical harmonic oscillator predicts the

existence of only one frequency in the spectrum of a diatomic, the

frequency called fundamental vibrational frequency:

nn EEhE 1obs

CHEM 3720

57

khnn

kE

22

1

2

1)1(

k

2

1obs and

k

c2

1~obs

○ The quantity x from above is, in this case, the difference

between the interatomic distance during the vibration and the

“equilibrium” distance.

0llx

○ If the fundamental vibrational frequency is known, one can

determine the force constant as:

2obs2

obs 2~2 ck

g. Typical values of obs~ are in the 100 – 4000 cm–1 range.

h. The average value of position and momentum for the harmonic

oscillator:

dxxxxx )()(* 0x

dxxdx

dixp

)()(* 0 p

i. Note that vibrational quantum number n is usually labeled as v in other

textbooks.

j. The probability density is different than 0 (bigger than 0) even in

regions where E < V.

□ This is equivalent to a negative kinetic energy and is an example of

quantum mechanical tunneling, a property of quantum mechanical

particles that is nonexistent in classical mechanics.

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