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PHYSICAL CHEMISTRY II CHEM 3720 Syllabus & Lecture Notes€¦ · PHYSICAL CHEMISTRY II CHEM 3720...

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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

  • CHEM 3720

    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

  • CHEM 3720

    15

    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

  • CHEM 3720

    16

    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)

  • CHEM 3720

    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.

  • CHEM 3720

    18

    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.

  • CHEM 3720

    19

    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

  • CHEM 3720

    20

    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.

  • CHEM 3720

    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

    23

    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

    24

    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

    27

    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.

  • CHEM 3720

    28

    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

    29

    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  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  , 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).

  • CHEM 3720

    35

    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  is linear if:

    xfAcxfAcxfcxfcA 22112211 ˆ)(ˆ)()(ˆ

  • CHEM 3720

    37

    d. An operator  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

  • CHEM 3720

    38

    ○ 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  and B̂ ) as:

    ABBABA ˆˆˆˆ]ˆ,ˆ[

    □ The commutator of commuting operators is the zero operator:

    0̂ˆˆˆˆ ABBA

    where 0̂ is the zero operator.

  • CHEM 3720

    39

    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  :

    )(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  , 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  ,

    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.

  • CHEM 3720

    40

    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.

  • CHEM 3720

    41

    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

  • CHEM 3720

    42

    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  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.

  • CHEM 3720

    43

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

    can be measured simultaneously to any precision.

    □ If  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

  • CHEM 3720

    44

    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

    46

    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

    *

  • CHEM 3720

    47

    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

  • CHEM 3720

    48

    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:

  • CHEM 3720

    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

    51

    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

  • CHEM 3720

    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

    xmH x

    , 2

    22

    ymH y

    , 2

    22

    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.

  • CHEM 3720

    55

    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)(

  • CHEM 3720

    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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