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LECTURE NOTES ON

Physics 18324410

QUANTUM PHYSICS

Tjipto Prastowo, Ph.D

Endah Rahmawati, M.Si

Department of PhysicsFaculty of Mathematics and Natural Sciences

The State University of SurabayaDecember 2012


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TO THE STUDENT WE LOVE

Lecture Notes on Quantum Physics contain topics given to the third year students of anInternational Class Program in Department of Physics, Faculty of Mathematics and NaturalSciences, The State University of Surabaya twice a week. These notes focus upon four majordiscussions written as separate chapters: quantum theory revisited given in Chapters 1-5; basicphysics principles of quantum mechanics discussed in Chapter 6; Schrodinger wave-mechanicsapproach and three dimensional formulation for hydrogen atom using spherical coordinates inChapters 7 and 8, respectively. Each chapter is accompanied with some exercises suitable forstudents homework assignments. To master materials covered, you need not just knowledgebut skill. This can only be obtained through continual practices. Perhaps, you may obtaina superficial knowledge by listening to lectures, but you cannot reach the skill expected bythat way. It is common to come across conversation between students like this I understand itbut I cant do the problem ! This student feels uncomfortable with some problems althoughthey look so easy to do.

The above example shows lack of practice and hence lack of skill required in this course.Our dearest students, please always study with pencil and paper at hand. You will find thatthe more able you are to choose effective methods of solving problems the easier it will be

for you to master new materials. This costs you nothing but practice, practice and againpractice. Please do remember that the best way to learn to solve problems is to solve them.We eventually welcome good comments on the content of this course from all readers forfurther improvement of these notes as the availability of the notes is important to improvethe quality of learning and teaching processes, particularly in the course of Quantum Physics.Hope these notes are useful for all users in the department.

All the best,Kampus Unesa Ketintang, 31 December 2012TjiptoPrastowo

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iii

General Guidance

PHYSICS 18324410: QUANTUM PHYSICS

Pre-requisites: Fundamental Physics (I and II)Modern PhysicsMathematical Physics (I and II)

Lecturers: Tjipto Prastowo, Ph.D and Endah Rahmawati, M.Si

References: Liboff,1980;Beiser,1988;Gasiorowicz, 1996; Serway et al., 2005;McMahon, 2006;Harris, 2007

Time and Place: Monday 7-8.30 am, C12 and Friday, 9-10.30 am, D4

Marking Scheme: NA= 20%P+ 20%UTS+ 30%T+ 30%UAS

NA=Final Mark, P=Presence, UTS=Mid-Exam, T=Homework, UAS=Final Exam

Notes:

1. Students are not allowed to join the class for being late (a maximum of 15 minutes fromthe starting time is permitted), except for reasonable arguments.

2. Each lecturer contributes an equal proportion of mark to the final mark.

3. P is possiblyreduced to a minimum.

4. UTS= 100% taken from Quiz

5. T = 100% taken from Homework

6. UASnormally contains 4 but possibly 5 problems.

7. Homework will be distributed to class members and all students are required to handthe completed assignments in within a given time. Various penalties will be given forany delay, i.e, 25% discounted mark for a one-day delay and 50% for a two-day delay.

There will be no mark given for those who submit the assignments more than two-daydelays.

8. No additional assignments or examinations after formal exam (both Mid and Final),except for specified reasons with very limited permission given or medical examinationrequired.

9. Students are allowed to work with their notes and books in both Mid and Final Exams.

10. Other important issues, if any, will be discussed in the class. Students are stronglyencouraged to be active and well-prepared. If possible, tutorial is available for a further,detailed description of each topics.


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Course Contents

1. Chapter One:

The Black-Body Radiation (Week 1) classical black-body radiation, Stefan-Boltzmanns law, Wiens displacement law,

Rayleigh-Jeans law, Plancks radiation formula

2. Chapter Two: The Nature of Electromagnetic Radiation (Week 2)

photo-electric effect, Compton effect3. Chapter Three: The Bohr Model of the Atom (Week 3)

early atomic model, Bohrs theory of an atom, Bohrs explanation to the theory,hydrogen spectral lines, Bohrs correspondence principle

4. Chapter Four: The Wave Behaviour of Sub-Atomic Particles (Week 4)

de Broglie hypothesis and its implications, Davisson-Germer experiment5. Chapter Five: The Heisenberg Uncertainty Principle (Week 5)

mathematical basis of the Heisenberg uncertainty principle, its interpretation andconsequences of the uncertainty principle

6. Quiz I : Chapters 1, 2, 3, 4, and 5 (Week 6)

7. Chapter Six: The Basic Physics Principles of Quantum Mechanics (Week 7)

observables, operators, wavefunctions, quantum measurements in microscopic world,Born interpretation, normalisation procedure, Hilbert space in quantum mechanicsand superposition principle, expectation values

8. Chapter Seven: The Wave-Mechanics Approach (Weeks 8, 9, 10, and 11)

steady state time-independent Schrodinger equation, stationary states, eigen valuesof energy, free particle, infinite square well, simple-step potential, reflection andtransmission coefficients, harmonic oscillator

9. Chapter Eight: The Schrodinger Equation in Spherical Coordinates (Weeks 12, 13,14, and 15)

solution to the time-independent Schrodinger equation in spherical coordinates,theory of hydrogen atom revisited, orbital and spin angular momenta, spin-orbitcoupling and total angular momentum

10. Quiz II: Chapters 6, 7, and 8 (Week 16)

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Contents

1 THE BLACK-BODY RADIATION 1

1.1 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Wiens Displacement Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Rayleigh-Jeans Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Plancks Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 THE NATURE OF ELECTROMAGNETIC RADIATION 7

2.1 Photo-Electric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 THE BOHR MODEL OF THE ATOM 15

3.1 The Early Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Bohrs Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Bohrs Explanation to the Model . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Hydrogen Spectral Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Bohrs Correspondence Principle. . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 THE WAVE BEHAVIOUR OF SUB-ATOMIC PARTICLES 27

4.1 De Broglie Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Implications of the De Broglie Hypothesis . . . . . . . . . . . . . . . . . . . . 304.3 The Davisson-Germer Experiment. . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 THE HEISENBERG UNCERTAINTY PRINCIPLE 37

5.1 Mathematical Basis for the Uncertainty Principle . . . . . . . . . . . . . . . . 385.2 Interpretations of the Uncertainty Principle . . . . . . . . . . . . . . . . . . . 415.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 THE BASIC PHYSICS PRINCIPLES OF QUANTUM MECHANICS 45

6.1 Observables, Operators, and Wavefunctions . . . . . . . . . . . . . . . . . . . 456.2 Born Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Normalisation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.4 Hilbert Space and Superposition Principle . . . . . . . . . . . . . . . . . . . . 496.5 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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vi CONTENTS

6.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 THE WAVE-MECHANICS APPROACH 53

7.1 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 The Free Particle and Infinite Square Well . . . . . . . . . . . . . . . . . . . . 567.3 Simple-Step Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.3.1 The Continuity Equation and Current Density . . . . . . . . . . . . . . 597.3.2 Reflection and Transmission Coefficients . . . . . . . . . . . . . . . . . 607.3.3 Cases where E is greater than V. . . . . . . . . . . . . . . . . . . . . . 637.3.4 Cases where E is less than V. . . . . . . . . . . . . . . . . . . . . . . . 64

7.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.4.1 The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.4.2 The Analytic Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8 THE SCHRODINGER EQUATION IN SPHERICAL COORDINATES 75

8.1 The Schrodinger Equation in Spherical Coordinates . . . . . . . . . . . . . . . 768.1.1 Separation of Dynamic Variables . . . . . . . . . . . . . . . . . . . . . 768.1.2 The Angular Component . . . . . . . . . . . . . . . . . . . . . . . . . . 788.1.3 The Radial Component. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 868.3.2 Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.3.3 Total Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 868.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography 89


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Chapter 1

THE BLACK-BODY RADIATION

The achievement made in theoretical and experimental physics towards the end of the-19th

century had been immense. Classical mechanics and thermodynamics were well understood.

The concept of electromagnetics was also remarkably established. It was found that light is a

form of electromagnetic waves, providing a firm theoretical framework for the wave theory of

light, which could account for most phenomena in optics. Thus, most physicists at that time

believed that the combination of the three subjects could account for all physical phenomena.

In fact, it was a period of a turmoil state when there were surprising experimental results,

which based on classical theory were totally inexplicable. One dilemma lay in the observations

of thermal radiation. Existing classical theory was unable to explain the observed frequency(or wavelength) of radiant energy. The incredibly greatest minds of physics in the period

were about to begin. Out of the turmoil state came a new philosophy of science a new

way of thinking the so-called Quantum Physics. The way we think about natural laws

particularly in microscopic world has totally changed since then. In this new paradigm, light

is considered not only as a wave but also as a series of bundles of energy called quanta.

Unlike Newtonian mechanics where dynamic variables, such as position and momentum can

be accurately determined with high precision, quantum world makes these variables uncertain

in that all measurements made are undeterministic.This chapter is aimed to examine the birth of quantum physics that has shaped the world

of physics. We first begin with experimental observations of the spectrum of thermal radiation,

what comes later to be known as black-body radiation, that put physicists at that time

into a Pandora Box. The second issue to discuss is the theoretical work of Wien in 1893, who

provided a primitive formula for the distribution of radiant energy. In line with this, we then

discuss the contributions of Rayleigh-Jeans in 1900 to the problem in question. Finally, we

revisit the notion of the photon concept of electromagnetic radiation in the light of Plancks

ideas proposed in 1901. This great work of Planck, along with relativity theory suggested by

Einstein, serves as a connecting bridge between classical physics and modern physics.

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2 1. THE BLACK-BODY RADIATION

intensityinwatt/m2

(a) (b)

Figure 1.1: Plots of radiant energyE, showing the characteristics of thermal radiationof a black-body as a function of wavelength : (a) at various temperatures Tand (b) at agiven temperature with theoretical predictions suggested by Rayleigh-Jeans and Planck(taken from Ch.1, Quantum Physics, Stephen Gasiorowicz, 1996).

1.1 Thermal Radiation

In 1859, Kirchoff proposed a theorem regarding with the relation between the coefficients

of emission and absorption of electromagnetic radiation; how these could be related to the

spectrum of thermal equilibrium radiation, the so-called black-body radiation phenomenon.

He challenged the community to work this out. A black-body is an object that absorbs all

the radiant energy falling on it and hence the black-body reflects no light, for which it would

appear as black. A black-body is thus aperfect absorber, as well as aperfect emitter. Kirchoff

suggested a functional dependence of the radiant energy

Eon temperatureTand frequency

(or wavelength ),

E =E(T, ) =E(T, ) (1.1)It has since then been of fundamental interest among physicists during the last two decades of

the-19th century to find out what the explicit form of the function is. This has been examined

through an approach of both experimental and theoretical considerations.

In 1879, Stefan based on experimental results argued that the radiant energy emitted by a

hot bodyis proportional to the fourth power of its absolute temperature. The same conclusion

based on classical thermodynamics was derived by Boltzmann in 1884. The findings led to

the Stefan-Boltzmanns law. Here, we are not going to provide the derivation of the law.


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1.2. Wiens Displacement Law 3

Rather, we pick up the result as follows,

E T4 (1.2)

where Edenotes the energy emitted per unit time per unit area from the surface of a black-bodyat a given temperature T, and is the Stefan-Boltzmann constant (5.67108 Wm2T4).

1.2 Wiens Displacement Law

In 1893, Wien proposed a formula to answer to the question previously posed by Kirchoff.

Wiens displacement law was derived using a combination of classical electromagnetics

and thermodynamics, as well as dimensional analyses. From the two basic concepts, it can beshown that

T 1 (1.3)The above relation says that the wavelength of a set of radiant waves changes inversely with

absolute temperatureT, as seen in Figure1.1(a).

Wien then went further by combining his own law with the Stefan-Boltzmann law. If we

defineE() as the energy density radiated, then we can write

E() T5

5(1.4)

for which we can derive

E = 5 f(T) = 3 g(/T) (1.5)where both f = f(T) and g = g(/T) are constants. The resulting prediction (1.5) matches

experimental observations, particularly for a region of small wavelengths (or high frequencies),

but it was brokendown for a region of large wavelengths (or low frequencies). Equation (1.5)

is the complete form of Wiens law and sets constrains on the black-body radiation spectrum.But it is actually incomplete because f = f( T) and g = g(/T) are both undetermined.

The complete form of the energy density of spectral distribution cannot be determined from

classical physics.

1.3 Rayleigh-Jeans Law

In 1900, Rayleigh analysed experimental data of black-body radiation and its corresponding

theory. His interest was stimulated by the inadequacies of Wiens law in (1.5) in accounting

for the large-wavelength (or low-frequency) behaviour of black-body radiation as a function


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of temperature. The reason for the inclusion of Jeans name in what became known as the

Rayleigh-Jeans law is that there was an error in Rayleighs analysis, which was corrected

by Jeans in 1906. Based on theoretical considerations, Rayleigh-Jeans found that the spectral

energy density radiated is given by

E = 8 2

c3 E (1.6)

where E is something that must be in a unit of energy. Here Rayleigh put E=kT based on

the principle of the equipartition of energy from classical kinetic theory into (1.6). Hence, we

have

E = 8 2

c3 kT (1.7)

for the energy spectral distribution at a given temperature. This formula is only valid at low

frequencies (or large wavelengths). It does not hold for high frequencies (or small wavelengths)because the spectrum of black-body radiation does not increase as 2 to infinite frequency, as

shown in Figure1.1(b).

Thus, both Wiens law and Rayleigh-Jeans law are counter-part in that they complete one

to another. However, what people want is a comprehensive theory that can be used to explain

the black-body radiation for all ranges of frequency or wavelength. It seems that the existing

theory based on classical view of thermal radiation of a black-body where such radiation is

considered to propagate in space as waves is no longer relevant.

1.4 Plancks Formula

Here we do not intend to derive explicitly what Planck did in his best times. Rather, we try to

connect (1.6) previously derived by Rayleigh to the correct formula. From classical statistics,

Eis related to the mean energy of a harmonic oscillator having two degrees of freedom, which

should be 12

kT + 12

kT = kT. This is what exactly Rayleigh did in deriving his law, which

turns out to be the correct expression for the black-body radiation law at low frequencies (or

large wavelengths), as shown in Figure1.1(b).An interesting question is that why did Planck not make such substitution forE? Firstly,

the equipartition theorem of Maxwell-Boltzmann is a result of statistical thermodynamics,

which was a point of view that he had rejected. Secondly, the Maxwells kinetic theory could

not account for the problem of specific heats of diatomic gases at high temperatures. Thirdly,

Planck did not agree fully with the Boltzmanns statistical approach.

In 1901, Planck worked out the problem of fundamental interest the energy spectral

distribution of the black-body radiation. Planck has introduced the concept of quantisation

by assuming that radiation consists of a bundle of quanta each having an energy which is

proportional to the radiant frequency . The energy density distributionEof the black-body


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1.4. Plancks Formula 5

radiation can then be written as a function of radiant frequency as follows,

E() =

8 h3

c3

1

eh/kT

1 (1.8)

The above expression satisfies all regions of frequency and wavelength in the spectrum, as

seen in Figure 1.1(b). The Plancks formula in (1.8) for the black-body radiation is valid

over the whole spectrum of wavelengths, or accordingly frequencies. It covers the empirically

observed findings of the spectral distribution of electromagnetic radiation that are previously

unexplained by classical theories of both electrodynamics and thermodynamics.

It is straightforward to integrate (1.8) to find the total energy density of radiationU inthe black-body spectrum,

U =

0

E() d = 8 hc3

0

3

eh/kT 1 d (1.9)

for which the total power of radiation emitted per unit area can be calculated from classical

theory of radiation,I=cU/4, such that we can write

I = 2k4T4

h3c2

0

(h/kT)3

eh/kT 1 d

h

kT

=

2k4T4

h3c2

0

x3

ex 1 dx (1.10)

whereIdenotes the total energy radiated per unit time per unit area and xh/kT. Theintegral can be evaluated with the help of some steps below (see p.5, Gasiorowicz, 1996),

0

x3

ex 1 dx =

0

x3 dx ex

n=0

enx =

n=0

1

(n+ 1)4

0

y3 ey dy = 61

1

n4 =

4

15

Equation (1.10) then becomes

I = 25k4

15h3c2T4 = T4 (1.11)

where the quantity in the bracket is replaced with , and hence

= 25k4

15h3c2 (1.12)

representing the Stefan-Boltmann constant, as previously shown in (1.2).

Most people at the time Planck proposed his ideas in verifying the experimental data of

black-body radiation got an impression that the Plancks concept of energy quantization was

only for the case of thermal radiation. It remained until Einstein who used the work of Planck

to show that light is quantized in the case of photo-electric effect.


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1.5 Exercises

1. Show that the Rayleigh-Jeans spectral distribution of black-body radiation given in (1.7)

is of the form required by Wiens law given in (1.5).

(taken from Ch.2, Introductory Quantum Mechanics, Richard Liboff, 1980).

2. Obtain the correct form Wiens undetermined function f( T) from Plancks formula

given in (1.8).


3. Use (1.8) to prove Wiens displacement law in the form of

max T= constant


4. (a) Use the Stefans law to calculate the total power radiated per unit area by a tungsten

filament at a temperature of 3000 K (assume that the filament is an ideal radiator).

(b) If the tungsten filament of a lightbulb is rated at 75 W, what is the surface area of

the filament ? (assume that the main energy is lost due to radiation).

(taken from Ch.3, Modern Physics, 3rd edition, Serway et al., 2005).

5. At what wavelength does the human body emit the maximum electromagnetic radiation?

(assume that the skins temperature is about 70F).

(taken from Ch.3, Modern Physics, 2nd edition, Randy Harris, 2007).


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Chapter 2

THE NATURE OF

ELECTROMAGNETIC RADIATION

As discussed in Chapter1, Planck proposed a remarkable formula for the spectral distribution

of black-body radiation, which shed light on the photon concept of quantisation. There soon

followed an understanding of the quantum nature of electromagnetic radiation. However, it

was not until Einstein who suggested that the concept was applied to real cases when metals

were irradiated with light. Another support for the concept came from experimental evidence

of theparticle nature of photonsobserved by Compton. These two fascinating phenomena

contribute significantly to the development of modern physics. One of the corner stones ofquantum physics is wave-particle duality, meaning that things may behave as waves or

particles depending upon a physical situation. In a classical situation, say the propagation

of sunlight in space for example, such electromagnetic radiation is considered as waves with a

continuous ranges of energy spread out. In this chapter, however, we study the complementary

topic electromagnetic radiation behaves as a collection of discrete particles.

2.1 Photo-Electric Effect

An important contribution to the energy quantisation of electromagnetic radiation came from

the work of Einstein who in 1905 applied the concept of the quantum nature of light to explain

unresolved, experimental findings in the photo-electric effectoriginally discovered by Hertz

in 1887. Hertz found peculiar properties of metals while observing his famous experiments on

electromagnetic waves to validate the Maxwells hypothesis of the speed of light.

The experimental set up of the photo-electric effect experiments is depicted in Figure2.1,

where a metal plate of a certain material placed as a cathode in the apparatus is irradiated with

light of a given frequency . There were a number of surprising results from the experiments.

The most remarkable feature was that the kinetic energies of the photo-electrons emitted from

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8 2. THE NATURE OF ELECTROMAGNETIC RADIATION

Figure 2.1: Sketch of the experimental set up of photo-electric effect experiments (takenfrom Ch.2, Introductory Quantum Mechanics, Richard Liboff, 1980).

the metal surface are independent of the intensity of the incident radiation, but are linearly

dependent upon the incident frequency (or wavelength). This feature could not be explained

by classical theory of electromagnetic radiation at that time.

Einstein proposed a brilliant idea to solve the problem in question by assuming that the

radiation consists of a collecting bundle of energy calledquantaof the same energyh. When

the bundle of energy is absorbed by the metal, electrons may receive sufficient energy to escapefrom the metal surface, against the energy that binds them. When the intensity of the incident

light is increased more electrons will be ejected from the surface, with their kinetic energies

remaining unaltered although the associated photo-electric current measured by Ammeter is

larger, as shown in Figure2.2(a). The result for an increase in the photo-current as the light

intensity increases is actually predicted from classical view of electromagnetic radiation. But,

the result for the independence of kinetic energy on the light intensity is totally unexpected.

This is a point where the glorious story of the quantum nature of light begins.

To cope with the unexpected results above, Einstein further suggested that the maximumkinetic energy Kmax of the photo-electrons linearly depends on both the incident frequency

(or wavelength) and the work function Wof a given material, as shown in Figure2.2(b).

The work function of a metal here is defined as the minimum amount of work necessary to

remove the electrons from the metal surface. Thus, this kinetic energy is expressed as

Kmax = h W=

hc

hc

o

(2.1)

where the term max is used to show thatW is needed to free theleaststrongly bond electrons.


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2.2. Compton Effect 9

Figure 2.2: Some of the surprising results for experiments of photo-electric effect,showing (a) photo-current is only affected by the intensity of incident light with a fixedstopping potential for a given material; and (b) the dependence of the maximum kineticenergy of photo-electrons on both the incoming frequency and the material used for acathode-plate (taken from Ch.3,Modern Physics, Serway et al., 2005).

Determination of the magnitude of the work function for a given material involves placing

a photo-cathode in an opposing potential so that, when the potential reaches a certain value,

the ejected electrons can no longer reach a collecting plate that serves as an anode, causing

photo-electric currents to fall to zero. This situation occurs at a value of an applied voltage

called the stopping potential Vs. In this situation, Kmax=eVs such that (2.1) becomes

eVs = h ( o) (2.2)

where we have usedW =ho witho being the threshold frequency, that is, the minimum

frequency needed to eject the electrons from the metal surface. Note that based on (2.2) eVs

must be a linear function of the incident frequency, with the slope of the straight line being h

known as the Plancks constant experimentally found to be 6.63 1034 Js, independent ofthe nature of the material chosen.

The central point in Einsteins contribution to the photo-electric effect experiments is that

electromagnetic radiation behaves as a collection of particles, each having a discretized energy

as opposed to the classical view of continuous wave energy for the radiation. With all respects,

Einsteins explanations agree well with experimental evidence, for which the work earned him

the Nobel Prize in physics in 1921.

2.2 Compton Effect

In 1922, Compton provided direct experimental evidence for the correctness of the photon

concept of electromagnetic radiation. In the experiment known as theCompton scattering,


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Figure 2.3: Sketch of the Compton effect, showing (a) initial and (b) final states ofthe X-ray scattering experiment (taken from Ch.2, Introductory Quantum Mechanics,Richard Liboff, 1980).

he used a beam of X-ray of initial energy hand momentump to bombard a targeted electron

initially at rest, as shown in Figure2.3(a). It was found that the X-ray beam was scattered in

a manner which is not consistent with classical theory of electromagnetic radiation. According

to the classical wave theory, the electron would scatter electromagnetic energy in all directions

at the same frequency as the incoming radiation frequency. But in fact, the scattered X-ray

beam has a different, outgoing frequency. To this end, Compton argued that the incoming

radiation should be treated as a beam of photons with individual photons scattering elastically

off individual electrons.

Based on the sketch of the Compton effect experiment (see Figure 2.3) where both energy

and momentum must be conserved, the conservation of energy is then given by

h + mo c2 = h + mo c

2 (2.3)

where mo is the rest mass of electron, and are the frequencies of incoming and scattered

X-ray beam, respectively, and is defined to be (1 2)1/2, where = v/c. Note thathere we use the relativistic expression for energy, and later momentum, as the recoil electron

is frequently observed to move very fast after the collision. We do not need to worry about

the expression for photons as the non-relativistic expression for objects that always move at

speed c does not exist.

Unlike energy, conservation of linear momentum is separately given for each direction, one


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2.2. Compton Effect 11

component in the direction of the incident radiation and the other component perpendicular

to the former. The equations denoting the momentum conservation are

h = mo c cos + h cos (2.4)

and

0 =mo c sin + h

sin (2.5)

where is an angle through which the photon is scattered and is an angle of the recoil

electron, kicked off from its original position as seen in Figure 2.3(b).

Compton pointed out that the scattered beam has a larger wavelength than the incident,

associated with less amount of energy for the scattered beam. The difference in wavelength

between the two beams can therefore be directly derived from the energy and momentum

conservation, and is measured as an increase in wavelength of the incoming radiation in

a billiard-ball type of collision with a stationary electron. We can then write

= = hmo c

(1 cos ) (2.6)

where and denote the wavelengths of the incoming radiation and scattered photon,

respectively, andh/moc= 0.024Ais called the Compton wavelength. Thus, the difference

in the wavelength between the incident and scattered photons depends only on the angle

of the scatter.

Let us take a closer look at (2.6). If the targeted electron is replaced with an atom, for

example, thenmowould be the mass of the atom, which is much more larger than the electron

mass. Consequently, the shift in the wavelength is sufficiently small to observe so that it

is effectively zero. This situation is similar to the case of an elastic collision between a small

projectile and a relatively massive body as a target. After a head-on collision, the projectile

is reflected back from the body with the same values of both kinetic energy and momentum,

but is in the opposite direction. For this case, the increase in wavelength is maximum for

which this type of scattering is called backward scatterof the incoming photon with = 180

because the collision imparts the maximum possible energy to the recoil electron.

The measurements of the shift in the wavelength and the kinetic energy of the recoil

electron are in good agreement with the Comptons theoretical predictions. These provide

convincing justification of the correctness of the classical, billiard-ball collision interpretation,

demonstrating the particle behaviour of the photon. The results for the Compton scattering

experiments opened up the problems of classical view of electromagnetic radiation theory, for

which Compton won the Nobel Prize in 1927.


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2.3 Exercises

1. The photo-electric threshold of tungsten is 2300 A. Determine the kinetic energy of the

electrons ejected from the metal surface by ultraviolet light of wavelength 1900 A.


2. The work function of zink is known to be 3.6 eV. What is the energy of the most energetic

photo-electron emitted by ultraviolet light of wavelength 2500 A?


3. Photo-electrons emitted from a cesium plate that are illuminated with ultraviolet light

of wavelength 2000 A are stopped by a potential of 4.21 V. What is the work function

of cesium ?


4. Ultraviolet light of wavelength 3500Afalls on a potassium plate. The maximum kinetic

energy of the photo-electrons is 1.6 eV. What is the work function of potassium ?

(taken from Ch.1, Quantum Physics, Stephen Gasiorowicz, 1996).

5. Consider the following metals lithium, beryllium, and mercury, having work functions

of 2.3 eV, 3.9 eV, and 4.5 eV, respectively. If light of wavelength 300 nm is incident oneach of these metals, then determine

(a) which metal exhibit the photo-electric effect;

(b) the maximum kinetic energy for the photo-electrons in each case

(taken from Ch.3, Modern Physics, Serway et al., 2005).

6. A photon of energy hcollides with a stationary electron of rest mass mo. Show that it

is not possible for the photon to impart all its energy to the electron.


7. A 100 MeV photon collides with a proton at rest. What is the maximum possible energy

loss for the photon ?


8. A 100 keV photon collides with an electron at rest. It is scattered at 90. What is its

energy after the collision ? What is the kinetic energy in eV of the electron after the

collision, what is the direction of the recoil electron ?



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2.3. Exercises 13

9. A beam of X rays is scattered by electrons at rest. What is the energy of the Xray if

the wavelength of the X-ray scattered at 60 to the beam axis is 0.035 A?


10. Gamma rays (high-energy photons) of energy 1.02 MeV are scattered from electrons

that are initially at rest. If the scattering is symmetric, that is, if = then determine

(a) the scattering angle ;

(b) the energy of the scattered photons



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Chapter 3

THE BOHR MODEL OF THE ATOM

3.1 The Early Atomic Model

The first half of the 20th century was generally considered as a period of remarkable advances

in modern physics, pioneered by the triumphs of relativity and quantum theories, as well as

similar discoveries in the fields of atomic and nuclear physics. As with relativity theory and

quantum theory for electromagnetic radiation previously discussed in the first four chapters,

this chapter discusses a new way of looking at an atom, in particular the introduction of

quantization concept to atomic scales yet in its primitive form as only a single quantum

number involved in describing the dynamics of the atom. To start with, we here give a brief

introduction to the development of atomic model.

The early modern, if it were worth to say, atomic model was the Thomson model in 1898,

in which an atom was viewed as a homogeneous solid sphere of uniformly distributed mass

having positive charge and negatively charged electrons distributed over the sphere surface to

produce a totally, electrically neutral atom. But, this model failed to explain the various lines

of atomic spectra found from direct observations. The mysteriouslines and the corresponding

revised model were then suggested by Rutherford in 1912 after a series of experiments using

a beam of-particles scattered by a thin metal foil (see Figure3.1). The important result ofthese experiments was that most of atomic mass and its corresponding positive charge lie in

a central region of the atom called the nucleus, while the electrons surrounding it. However,

the idea of such a nuclear atom raised some fundamental questions as follows: (1) If there are

only Z protons within the nucleus comprising roughly a half of the total mass of the nucleus,

what composes the other half ? (2) What provides the cohesive force to keep protons within

a confined region of 1014 m, the size of the nucleus ? (3) What drives the electrons to move

around the nucleus ? (4) How does this motion account for the observed spectral lines ?

Among these, Rutherford gave his best speculation to answer to the first two questions. But,

he had certainly no answer to the other two until the work of Bohr came into play in 1913.

15


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16 3. THE BOHR MODEL OF THE ATOM

Figure 3.1: A beam of -particles scattered by a dense positively charged nucleus(taken from Ch.4, Modern Physics, Serway et al., 2005).

3.2 Bohrs Atomic Model

The Rutherford atomic model is based on the fact that the electrons must revolve the nucleus.

Meanwhile, classical electromagnetic theory of radiation claims that accelerated electrons

revolving the nucleus will radiate some energy that makes orbital radius of the electronsbecome less and less. In this sense, the radius decreases steadily followed by an ever-increasing

radiant frequency corresponding to the energy released. The atom may therefore be unstable

and collapsing as the electrons plunge into the nucleus, as shown in Figure 3.2.

The wrong deductions above that lead to a continuous emission spectrum were well tackled

by Niels Bohr, a young Danish physicist. He argued that classical electromagnetic radiation,

theoretically proposed by Maxwell and experimentally confirmed by Hertz long before Bohrs

new model of the atom shook the world of physics to its foundation, is false when applied to

systems at a microscopic scale. Based on the great work of Planck, he applies the concept ofquantized levels of energy and angular momentum to orbital electrons that occupy stable

stationary states in which no energy is lost. At the same time, based on the corresponding

work of Einstein, he applies the concept of photon emitted when there is a quantum jump

of an electron from a particular stationary state associated with certain values of energy and

angular momentum to another. In this way, Bohr developed a new theory of an atom in 1913

by combining basic physic principles of classical mechanics with a simple quantum theory of

light emission. The Bohr model is then able to explain the existence of atomic spectral lines

of the hydrogen atom, in good agreement with experimental evidence and thereby resolving

the shortcomings of the previous, classical atomic models.


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3.2. Bohrs Atomic Model 17

Figure 3.2: Classical radiation theory of an orbital electron having less and less energyas it approaches the nucleus (taken from Ch.4, Modern Physics, Serway et al., 2005).

In this context, let us now examine in detail the semi-classical, Bohr theory for an atom.

As it originally applies to a hydrogen atom with a single proton and a single electron only,

the basic idea of the Bohrs theory is that the dynamics of the hydrogen is determined by

the Coulomb attraction of a nucleus (the proton) and the electron that drives the electron to

orbit the nucleus. The orbiting electron has an energy derived from a classical expression for

the total energy. This concept is then combined with two quantum postulates as follows :

The hydrogens electron is assumed to orbit the nucleus in concentric circular pathsassociated withstationary orbits, each having its own angular momentum L equal to

an integer multiplied by h/2, where h is the Plancks constant. These orbits are then

referred to atomic shells. This can be mathematically expressed as

L d = nh

L = n

(3.1)

wheren= 1, 2, 3 .... is an integer, the so-called principal quantum number associated

with thenth stationary orbit at which the electron may be located. The stationary orbits

are also related to discrete values of energy representing quantum states, each having its

own quantum number n, for the orbiting electron. In such stable orbits, this electron

radiates no energy.

When an electron from a quantum state at a particular level of energy jumps into anotherquantum state at a lower level of energy, then photon with a certain value of frequency

is emitted (see Figure3.3). In this case, the radiant frequency is then proportional to

the difference in the level of energy Ebetween the two states, and is written as

= Eh

= Ei Ef

h (3.2)


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Figure 3.3: A photon of frequency is emitted as a result of quantum transition froman initial state with a particular level of energy Ei to a final state with a lower level ofenergy Ef (taken from http://www.kottan-labs.bgsu.edu).

whereEiand Efare levels of energy for the initial and final quantum states, respectively.

Note that the other way around of atomic transition may occur when an electron absorbs

some amount of energy to move from an initial state of one energy level to a final state

of a higher energy level. For this case, (3.2) still holds with is the absorbed frequency.

3.3 Bohrs Explanation to the ModelThis section describes the uniqueness of the Bohrs model in that it combines well classical and

quantum concepts. The combined ideas then lead to discretized orbit radii and levels of the

electrons energy, and are at the same time prove the Bohrs postulates to be inter-connected.

Let us start with the hydrogen having a single electron only. From the classical point of view,

the total energy Eof the hydrogens electron is given by

E = 1

2

mv2

k

e2

r

= L2

2mr2

k

e2

r

(3.3)

where m= 9.11 1031 kg denotes the mass of the hydrogens electron, e= 1.6 1019 C isthe electron charge, and k= 9 109 Nm2C2 is the Coulomb constant for vacuum.

Next, the dynamics of such an electron can be written in the form of Newtons second law

of motion as follows,mv2

r =

L2

mr3 = k

e2

r2 (3.4)

Inserting (3.1) to (3.4) yields the hydrogens electron orbit radii rn as follows,

rn = n22

mke2 = n2 ao for n = 1, 2, 3,... (3.5)


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3.3. Bohrs Explanation to the Model 19

Figure 3.4: The first three Bohr atomic orbits for the hydrogen atom (taken from Ch.4,Modern Physics, Serway et al., 2005).

where = h/2 andao = 2/mke2 = 0.529Ais here called the Bohr radius. It is then clear

from (3.5) that only orbits illustrated in Figure3.4 with certain values of quantum number n

are allowed to occupy. These discrete orbits are due to the non-classical requirement for the

electrons angular momentum to be an integral multiple of. Such orbits form atomic shells

surrounding the nucleus, where the smallest orbit is calledKshell having radius r1=ao andis associated with n = 1. For larger orbits withn > 1, the shells are calledL,M,N........Later on, according to quantum mechanics these orbits are divided into sub-shells.

The above quantization of the electrons orbit radii for the hydrogen atom immediately

leads to thediscretized levels of the electrons total energy, which will be derived below.

Substituting (3.4) into (3.3) results in

E = L2

2mr2

L2

mr2

=

L2

2mr2

(3.6)

With the help of (3.1) and (3.5) to be inserted into (3.6), we can write the allowed levels En

of the total energy for the hydrogens electron as

En = mk2e4

22n2 = R

n2 for n = 1, 2, 3,... (3.7)

whereR =mk2e4/22 = 13.6 eV is called the Rydberg constant for the electrons energy.In other literatures, the Rydberg constant is differently valued as it is meant for a different

quantity (see further

3.4). The negative sign for the electrons total energy in ( 3.7) suggests

that the electron is bound to the nucleus and thereby requiring some amount of energy to


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Figure 3.5: A diagram showing the energy-discretized states of the hydrogen atomand some spectral lines following atomic transitions (taken from Ch.4, Modern Physics,Serway et al., 2005).

liberate it from the Coulomb attraction. The lowest possible stationary state corresponding

to quantum numbern = 1 is generally called the ground statehaving the smallest energy of

E1= 13.6 eV. The next state is the first excited state, corresponding to quantum numbern= 2 and hence having an energy ofE2= 3.4 eV.

An energy-level diagram for the hydrogen atom showing the energy-discretized states with

the corresponding quantum numbers is depicted in Figure3.5. The uppermost level of energy

corresponds to n = and hence En = 0, representing a state in which an electron is freefrom Coulomb nuclear field. When this electron experiences an external force it has a kinetic

energy only for which it is usually termed as a free electron. The minimum energy required to

remove the bound electron from its ground state to a large distance from the nuclear influence

is called the ionization energy. Thus for the hydrogen, the ionization energy is 13.6 eV.

3.4 Hydrogen Spectral Lines

The great success of the Bohr atomic model for the hydrogen is achieved when it is used

to describe the existence of the hydrogen spectral lines. Instead of a continuous emission

spectrum argued by Rutherfords model, the Bohrs theory predicts that atomic transitions

between levels of energy produce a spectral series of separated lines (see Figure 3.5), each

having its own characteristics; all the observed lines in Lymann series end at a state ofn = 1,

lines of Balmer series are down to n = 2, and Paschen lines towards n = 3.


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3.4. Hydrogen Spectral Lines 21

Here we derive a mathematical expression for the observed hydrogen spectral lines using

(3.2) and (3.7). Let us suppose a quantum transition from an outer orbit at an initial state of

quantum numberni having energyEi=

R/n2i to an inner orbit at a final state of quantum

number nf having energy Ef =R/n2f. Such a transition emits a photon of frequencygiven by

=R

h

1

n2f 1

n2i

(3.8)

The above equation is equivalent to

1

=

Rhc

1

n2f 1

n2i

= R

1

n2f 1

n2i

(3.9)

where represents the photon wavelength corresponding to the emitted frequency in (3.8)

and R is exactly the same as R/hc= 1.097 107 m1, defined to be theRydberg constantfor the emitted wavelengths of the hydrogen spectral series depicted in Figure3.5.

When Bohr proposed his theory in 1913, the spectral lines of Balmer series (nf= 2 and

ni > 2) and Paschen series (nf = 3 and ni > 3) had already been found two years before.

The Rydberg constant R in (3.9) theoretically derived from the Bohrs quantum theory for

the hydrogen is found to be in good agreement with an empirical value for a constant used to

describe Balmer and Paschen lines. This was followed by the Lymann series (1916) for which

nf= 1 and ni >1. In the years to come since the first three spectral series, Brackett (1922)

and Pfund (1924) observed the spectral lines for nf= 4 and nf= 5, respectively. Among all

these series, only Balmer lines constitute a range of wavelengths in a visible light spectrum.

Impressed by his successful explanation to the experimental observations of the hydrogen

spectral lines, Bohr then immediately extended his theory to hydrogen-like atoms, such

as He+ and Li2+, in which all but one electron had been removed from a nucleus of positive

charge Ze, where Z denotes the number of proton. For a single electron of negative chargee

orbiting the nucleus of a hydrogen-like atom, the quantized orbit radii in (3.5) becomes

rn = n22

mkZe2 = n2

aoZ

for n = 1, 2, 3,... (3.10)

and the corresponding discrete energies in (3.7) becomes

En = mk2Z2e4

22n2 =Z2 R

n2 for n = 1, 2, 3,... (3.11)

The extended theory was used to explain mysterious lines observed in hot stellar atmospheres.

In the first paper of his great trilogy published in late 1913, Bohr noted that a formula similar

to (3.9) could account for the Pickering series(1896), a series of lines describing the spectra


28/95


of stars. Bohr argued that singly ionized helium atoms would have exactly the same spectrum

as that of the hydrogen, but the corresponding wavelengths would be four times shorter.

In his effort to explain such spectral lines, Bohr even went further to take into account the

contributions of the motion of both the electron and the nucleus about their centre of mass

to derive the so-called reduced mass. Using the reduced mass of the hydrogen and He+, Bohr

found that the ratio of the Rydberg constants for ionized helium and hydrogen is 4.00160,

compared to the value of 4.00163 obtained from laboratory experiments of Fowler in 1912. He

then predicted the lines to be due to the presence of singly ionized helium atoms, instead of

the hydrogen.

Although Bohrs remarkable achievement in explaining both emission and absorption lines

of the hydrogen and hydrogen-like atoms and in describing the shell structure of an atom

is of paramount importance in the development of atomic model using an approach of anuneasy mixture of classical and quantum ideas, the Bohr model of the atom is fundamentally

incomplete. The model has, however, some difficulties regarding with discrete circular orbits.

These orbits give a classical, deterministic propertyto the position of an orbiting electron

in that the electrons position at any time can thus be determined with a high accuracy. This

property is one in which it is not acceptable in the context of quantum mechanical model of

an atom, where the electrons exact position cannot be precisely determined but only with

some confidence by a probabilistic value.

The above argument is in line with Heisenbergs uncertainty principle (which will bediscussed in Chapter5), where it is argued that the electrons of a multi-electron atom occupy

a region like a cloud around the nucleus with the cloud density being the probability to find

an electron in a particular region. As more electrons are introduced to a stable multi-electron

atom, the paths through which the electrons move around the nucleus are complicated, and are

likely to be overlapping inelliptical orbits, making the circular orbits are no longer applicable.

This implies that additional quantum numbers other than the principal quantum number n

are needed to describe the complex structure of a multi-electron atom. Detailed calculations

of quantized energy levels of such an atom reveal the presence ofatomic sub-shells, associated

with orbital quantum numbers. The complete description of all quantum numbers required to

describe the dynamics of a system at atomic scales is given in the course of Quantum Physics.

3.5 Bohrs Correspondence Principle

Bohrs correspondence principleis a simple principle that provides a smooth and gradual

change of the new but in some sense yet primitive quantum theory into classical theory

in the limit of a very large quantum number previously defined. This principle naturally comes

from the basic idea that a new theory should be able to capture all the essence of the old law


29/95

3.5. Bohrs Correspondence Principle 23

Figure 3.6: A sketch showing the classical limit of the Bohr theory as the principal

quantum number approaches infinity, associated with very large orbit radii (taken fromCh.4, Modern Physics, Serway et al., 2005).

with no exception, and at the same time to demonstrate when the new theory approaches the

old one. If this requirement is satisfied, then the new theory is said to be firmly established

with no doubt. A similar situation can also be found in the context of special relativity theory.

This theory shows its strength not only by combining spatial and temporal components of

the four-dimensional space-time coordinates with the concept of the four-vector thoroughly

discussed in Lecture Notes on Modern Physics, but also by demonstrating that calculations of

the physical quantities based on the special relativity gradually change into classical results

of Newtonian mechanics when the speed is relatively small compared to the speed of light.

In the hands of Bohr, the correspondence principle becomes a master tool to demonstrate

where the quantization of the electrons orbital angular momentum comes from. Once derived,

the validity of the quantization concept depends upon the agreement with measurements of

lines of atomic spectra. Here, we simply use Figure 3.6 to derive the first Bohr postulate

earlier mentioned in

3.2 from which the quantized electrons orbit radii shown in (3.5) and

the discretized electrons total energy in (3.7) are derived. Let us assume thatr1 and r2 are

the radii of the two adjacent quantum orbits, each having its own orbital angular frequencies

1and2, respectively. We go further by assuming thatr1 r2 rand hereby1 2 .We also assume that is the frequency of a photon emitted as a result of a quantum transition

from the initial state of radius r1 to the final state of radius r2.

Using the fact that the orbiting electron is under the influence of the Coulomb force, we

can show that the total energy for the electron is given by

E = mk2e42L2

(3.12)


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Note that we obtain the close relationship of the electrons energy and the angular momentum

for the Bohr model. Taking the derivative of (3.12), we have

dEdL

= mk2

e4

L3 = mk

2

e4

(mk2e4)/ = (3.13)

or it can be simply written as

dE = dL (3.14)

where dE is the energy of a photon emitted when a quantum transition from r1 to r2 occurs

and dL is the corresponding change in the electrons angular momentum. Thus based on

Figure3.6and the definitions of dEand , we have dE= and so (3.14) becomes

= dL (3.15)

As earlier noted, the correspondence principle must be able to predict the same frequency

occurs for the emitted photon following the transition, as suggested by the quantum theory of

light emission, and for the electrically charged electron moving around the nucleus, based on

classical electromagnetic theory of radiation. In this way, it is true that =, and therefore

we can write

dL = (3.16)

It is understood from (3.16) that the electrons orbital angular momentum in a specific orbit

is an integral multiple of , from which the first postulate of Bohr L = ngiven in (3.1) is

proposed. What an amazing result ! Bohr realized that although (3.16) originally was derived

for the special case of orbits with large radii, it was a universal quantum principle that it was

relevant to a wider range of applicabilities than the classical Maxwells theory of radiation.

3.6 Exercises

1. If an electron moves from an inner orbit to an outer orbit, does its total energy increase

or decrease ? Does its kinetic energy increase or decrease ?

(taken from Ch.37, Physics for scientists and engineers, Tipler, 1999).

2. Show that the speed of an electron in the nthBohr orbit of the hydrogen atom is given

byvn=e2/2onh.


3. The binding energy of an electron is the minimum energy required to remove the electron

from its ground state to a large distance from the nucleus.

PR, UTS


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3.6. Exercises 25

(a) What is the binding energy for the hydrogen atom ?

(b) What is the binding energy for He+ ?

(c) What is the binding energy for Li2+

?Note that a singly ionized helium He+ and a doubly ionized lithium Li2+ behave as

hydrogen-like atoms in that such ionized elements consist of a positively charged nucleus

and a single bound electron.


4. What is the radius of the first Bohr orbit in He+ and Li2+ ?


5. A hydrogen atom is in its ground state. Using the Bohr theory of the atom, calculate

(a) the radius of the orbit

(b) the linear momentum

(c) the angular momentum

(d) the kinetic energy

(e) the potential energy

(f) the total energy of the electron


6. A photon is emitted when a hydrogen atom undergoes a jump of electronic transition

from the initial quantum state ofn = 3 to the final quantum state ofn = 2. Determine

(a) the energy

(b) the wavelength

(c) the frequency of the emitted photon


7. A hydrogen atom initially at rest in then = 3 quantum state decays to the ground state

with the emission of a photon.

(a) Calculate the wavelength of the emitted photon

(b) Estimate the recoil momentum of the atom

(c) Estimate the kinetic energy of the recoiling atom

(d) Where does this energy come from ?



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8. If one assumes that in a stationary state of the hydrogen atom the electron fits into a

circular orbit with an integral number of wavelengths, one can reproduce the results of

the Bohr theory. Work this out.


9. Use the Bohr quantisation rules to calculate the energy levels for a harmonic oscillator,

for which the energy is p2/2m + m2r2/2 directly given by the driving force m2r.

Restrict yourself to circular orbits. What is the analog of the Rydberg formula ? Show

that the corresponding principle is satisfied for all values of the principal quantum

numbern used in quantizing the angular momentum.


10. A muon is a particle with a charge equal to that of electron and a mass equal to 207

times the mass of an electron. Muonic lead is formed when Pb208 captures a muon to

replace an electron. Assume that the muon moves in such a small orbit that it sees a

nuclear charge of Z = 82. According to the Bohr theory, what are the radius and energy

of the ground state of muonic lead ? Use the concept of reduced mass here.



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Chapter 4

THE WAVE BEHAVIOUR OF

SUB-ATOMIC PARTICLES

As widely known, classical wave theory suggests that the propagation of light is considered

as a natural wave phenomenon. Examples of this can be found in many physical situations.

In particular, the laws of geometric optics are empirically proved to be held when light is

incident on a surface (reflection) or arrives at a boundary between two media (refraction).

In other optical phenomena, such as interferenceand diffractionof light, the laws of physical

optics are observed to occur when a beam of light rays passes through a very small aperture.

In Chapter 2, however, it was shown that electromagnetic radiation in the form of photonsmay behave as particles when interacting with matter, such as those in the photo-electric and

Compton effects. A somewhat bizarre, fundamental question based on the reverse mechanism

is raised: can a classical particle, say an electron, with its own character behave as a wave

with a totally different character ? If this is the case, then the revolutionary idea about

matter waves that leads to the new concept of the way we are looking at the dynamics of

moving particles at microscopic scales has to be proven by both theoretical considerations and

experimental measurements. These topics are the primary issues we discuss in this chapter.

4.1 De Broglie Hypothesis

In classical physics, the concept of particle distinguishes both qualitatively and quantitatively

from that of wave. In the context of a classical particle, it is common to characterize particle

with its inertial mass; a particle is spatially localized at a certain time. Whereas a wave is

characterized by its spatial periodicity, i.e., wavelength, or temporal periodicityi.e., frequency

for which it is also defined as an energy propagated from one point to another in space-time

coordinates; the wave spreads and hence its energy distributes over space. It is clear from this

that none can demonstrate itself to be a particle and at the same time behaves as a wave.

27


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28 4. THE WAVE BEHAVIOUR OF SUB-ATOMIC PARTICLES

Figure 4.1: The equivalence between (a) a moving particle of mass m and speed vowith (b) a wave packet of wavelength and speed vg (taken from Ch.5, Modern Physics,Serway et al., 2005).

In 1925, de Broglie postulated a radical idea with no experimental supports at the time

it was proposed. His postulate was finally known as the de Broglie hypothesis derived

from theoretical considerations of classical wave, relativity theory, and quantum concept for

a photon with zero rest mass. The mathematical expression of his postulate is as follows,

= h

p (4.1)

where h is the Plancks constant, and pare the wavelength and momentum of the photon,

respectively. At this stage, de Broglie argued that the above expression holds also for any

moving object of mass m. The following paragraphs show logical reasons for this.

De Broglie generalized (4.1) to introduce the concept of a matter waveby assuming that

a moving particle can be in principle viewed as a wave packetor a wave group. A wave

of this type must reflect the fact that such a particle has a large probability of being foundwithin a small, confined region of space at a limited time (see Figure4.1). It follows that a

single traveling sinusoidal wave with constant amplitude and infinite extent is not relevant to

model the particle. Instead, a group of waves of limited spatial extent consisting of individual

waves with different wavelengths can then represent the particle. In this case, the resulting

wave group travels at a speed vg defined to be the group velocity of the wave, which is

identical to the observed speed vo of the corresponding particle.

Let us then apply the de Broglie hypothesis given in (4.1) to a photon of energy E=

and momentum p = k. These quantities are attributed to a wave character through two

related parameters, namely the angular frequency = 2and the wave number k = 2/.


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4.1. De Broglie Hypothesis 29

From these quantities, or parameters, we have = ck for the photon. If the group velocity of

de Broglie wavecorresponding to this photon is defined as

vg = ddk = dEdp (4.2)

then we have vg = c for the speed of the wave group, as it should be for photons with zero

rest mass, mo= 0.

As noted, we have already made (4.1) true for photons. An interesting question is that

whether such an equation also holds for a classical particle of mass m moving at speed v.

Here, we provide a somewhat crudeargument for this. During its motion, the particle has

momentum p = mv and kinetic energy E = p2/2m. If the motion is considered as a wave

group, then the group velocity can be calculated by assuming d/dk to be equivalent todE/dpand by inserting this equivalence into the definition ofvg to get the result for vg =v,

consistent with the particles speed. The result also implies that, for photons, the wave group

travels at speed c. Based on this simple calculation, it is shown that (4.1), originally derived

for photons, is relevant to any moving particle with non zero rest mass.

A more detailed use of (4.1) for any moving particle with non zero rest mass is given here,

as we are now in a position to apply the de Broglie hypothesis to sub-atomic particles, such

as a proton, a neutron, and an electron. In doing so, we need to define what is called the

phase speed, that is, the speed of a point of constant phase on a wave. The phase speed of

the wave is given by

vp =

k =

E

p (4.3)

whereEdanprepresent the relativistic expressions for the total energy and momentum of the

particle, respectively. We can also write the phase speed in (4.3) as a function ofk only with

the help ofE2 =E2o + (pc)2, where Eo is the particles rest energy. Inserting this relativistic

relation into (4.3) results in

vp = c1 + mck

2

(4.4)

where m is the particles rest mass. Ask varies with wavelength, or equivalently frequency,

then (4.4) is meant for individual waves of the wave packet. Note that the expression in (4.4)

is also called adispersion relationfor the phase speed of each component of the wave group.

The group velocity of the wave group can then be calculated from

vg = vp + kdvp

dk (4.5)

to be evaluated at ko, the central wave number of a continuous distribution of wavelengths

constituting the wave group.


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Substituting (4.4) to (4.5) and after some simple algebra, we can write the group velocity

of the de Broglie wave as

vg = c

1 + (mc/k)2=

c2

vp(4.6)

Let us go back for a moment to the phase speed given in ( 4.3). Solving for this phase speed

yields

vp = E

p =

mc2

mv =

c2

v (4.7)

where is defined as

= 11 v2/c2 (4.8)

and v is the particles speed. We eventually get the right expression for the group velocity

by inserting (4.7) into (4.6) to have vg = v . The final result obtained forvg is convincing in

that the group velocity of the wave packet is the same as the corresponding particles speed,

as expected.

4.2 Implications of the De Broglie Hypothesis

Although it looks so simple, the de Broglie hypothesis defined in (4.1) is actually powerful.

The hypothesis combines well fundamental aspects of both classical wave of wavelength

and classical particle of momentum p. The Plancks constant h serves as a connecting bridge

between the two properties, implying that the application of the hypothesis is limited only

to the case of moving particles at a microscopic scale. In this context, the hypothesis is then

used to validate the particle property of a photon and the quantization of the orbital angular

momentum of the electron for the Bohr model of the hydrogen atom. These two implications

will be discussed in the following paragraphs.

It has been known that photons have zero mass, will always be moving at speedc, and may

behave as particles. This particle property, however, would have made photons to have mass.

This seems to be contradictory with the fact that photons are massless. To examine this, letus define what is called the effective inertial mass of a photon, a quantity describing how

a photon responds to an applied force acting on it. The photons effective inertial massmeff

may reasonably be taken to be proportional to its total relativistic energy Eas follows,

meff = E

c2 =

h

c2 =

h

c (4.9)

for which (4.9) can be rearranged to become

= hmeffc

= hpeff

(4.10)


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4.2. Implications of the De Broglie Hypothesis 31

Figure 4.2: De Broglie standing waves in Bohrs stationary orbits (taken from Ch.4,

Konsep Fisika Modern, The Houw Liong, 1987, adapted from the work of Arthur Besier).

which is identical to (4.1) in the sense that the associated wavelength of a moving particle

is inversely proportional to its momentum, where meffc is the photons effective momentum.

By (4.9) and (4.10) the de Broglie hypothesis demonstrates its-self to be self consistent with

both relativity and quantum theories.

Another important aspect of the hypothesis is that it provides a physical feature of the

Bohrs atom theory. Although the Bohrs model is useful to describe the dynamics of theatoms, it has also some shortcomings regarding with, say for example, only certain values of

electronic energy are allowed to occupy in the model. De Broglie recognized this problem and

made it clear by visualizing the orbiting electrons around the nucleus as standing wavesbent

into circles of discrete radii (see Figure4.2). This point of view actually comes from the simple

but brilliant idea that an atom confines its orbiting electrons to a very small atomic dimension,

in which case the wave nature of the electrons should predominate over their particle property.

From classical wave theory we know that when a wave is restricted to a small area, then only

a discrete set of standing waves are possible to occur within that area. The Bohrs stationary

orbits are thus viewed as a result of constructive interference of these waves for a range of


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wavelengths. This constructive interference fits into the circumference of circular orbits and

corresponds to an integral number of the de Broglie waves, each having wavelength . We can

then express this de Broglies argument by rewriting the first Bohrs postulate given in ( 3.1)

for the quantized orbital angular momentum L = nof the hydrogens electron to start from.

After a simple algebra we have

2rn = n for n = 1, 2, 3,... (4.11)

where rn is the Bohrs orbit radii defined in (3.5). The fact that the de Broglie hypothesis

originating from the wave nature of the electrons revolving the nucleus matches with the Bohrs

theory is a key to understanding the nature of microscopic world. Following this, scientists

have started realizing that electrons and hence other sub-stomic particles have a dual property,

the so calledparticle-wave duality. It can then be inferred from many cases considered that

it is not possible to observe both the particle and wave properties simultaneously. Rather, one

property completes the other for which Bohr called this as a complementary principle.

4.3 The Davisson-Germer Experiment

Following his successful doctoral degree, de Broglie as a consequence of his postulate suggested

that a stream of electrons passing through a very small aperture would produce diffraction

pattern, as it would for a beam of light. In this context, a series of laboratory experiments

using Davisson-Germer experimental apparatus (see Figure 4.3) were completed in 1927 to

test the de Broglie hypothesis of a matter wave. The primary result of these experiments

was that electrons experience diffraction, providing convincing support for the wave nature

of an electron. The apparatus allows for the variations of three experimental parameters,

namely the energy of an electron, the orientation of a nickel target, and the angle of an

elastic scattering. Initially, the progress of these experiments seemed to give nothing. After

an unexpected but fortunate experimental accident occurred, however, Davisson and Germer

realized that further analyses on the experimental procedure give different results that exhibit

a diffraction pattern. Thus, they took this peculiar chance to calculate the wavelength of an

electron using both the de Broglie hypothesis given in (4.1) and a simple diffraction formula,

corresponding to various values of the three parameters. In their theoretical calculation on

the basis of conservation of energy, Davisson and Germer used the non-relativistic expression

for the speed v of the electron as follows,

1

2mv2 = P (4.12)

where m= 9.11 1031 kg is the electrons rest mass and Pis the potential energy supply.


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4.4. Exercises 33

Figure 4.3: The Davisson-Germer experimental apparatus, used for examining thewave nature of an electron by elastically scattering a beam of low-speed electrons froma polycrystalline nickel target (taken from Ch.5, Modern Physics, Serway et al., 2005).

The remaining step is straight forward, solving for v from (4.12) and inserting the result

into (4.1) yields

= h

2meV(4.13)

where e = 1.6 1019 C is the electron charge and Vis the experimental voltage supply. Forthe special case of corresponding to the diffraction maximum where V = 54 volt, we have

= 1.67 1010 m, in good agreement with = 1.65 1010 m obtained from formula forconstructive interference,

d sin = n (4.14)

where d = 2.15 1010 m obtained from previous measurements of X-ray diffraction, = 50and n= 1 corresponds to the diffraction maximum pattern.

4.4 Exercises

1. Is light a wave or a particle ? Is an electron a particle or a wave ? Support your answer

by citing specific experimental evidence.



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2. If matter has a wave nature, why is this wave-like character not observable in our daily

experiences ?


3. Show that the de Broglie wavelength of an electron of kinetic energyE, measured in eV,

is given by

e = 12.3 1010

E1/2 m

and that of a proton is given by

p = 0.29 1010

E1/2 m


4. Show that in order to associate a de Broglie wavelength with the propagation of photons

(electromagnetic radiation), photons must travel with the speed of light c and their rest

mass must be zero. Do this relativistically.


5. Calculate the de Broglie wavelength for

(a) an electron of kinetic energy 250 eV (the rest energy of the electron is 0.511 MeV)(b) a neutron of kinetic energy 0.02 eV (the rest energy of the neutron is 940 MeV)

(c) a proton of kinetic energy 2 MeV (the rest energy of the proton is 938 MeV)


6. To observe small objects, one measures the diffraction of particles whose de Broglie

wavelength is approximately equal to the objects size. Find the kinetic energy, measured

in eV, required for electrons to resolve

(a) a large organic molecule of size 10 nm

(b) atomic features of size 0.10 nm

(c) a nucleus of size 10 fm

Repeat these calculations using alpha particles in place of electrons.


7. Find the de Broglie wavelength of a ball of mass 0.2 kg just before it strikes the Earth

after being dropped from a building 50 m tall.



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4.4. Exercises 35

8. An electron has a de Broglie wavelength equal to the diameter of the hydrogen atom.

What is the kinetic energy of the electron ? How does this kinetic energy compare with

the ground-state energy of the hydrogen atom ?


9. For an electron to be confined to a nucleus, its de Broglie wavelength would have to be

less than 1014 m.

(a) What would be the kinetic energy of an electron confined to this region ?

(b) On the basis of this result, would you expect to find an electron in a nucleus ?


10. The dispersion relation for free electron waves is

(k) =

c2k2 + (mc2/)2

From the above equation, obtain expressions for the phase velocity vp and the group

velocity vg of these waves and show that their product is constant, independent ofk.

From your results, what can you learn about vg when vp > c ?



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Chapter 5

THE HEISENBERG UNCERTAINTY

PRINCIPLE

One of the most basic and far-reaching concepts of modern physics is Uncertainty Principle,

proposed by Werner Heisenberg in 1927. This beautiful principle is fundamentally concerned

with the limit of our ability in simultaneously determining with high accuracy and precision

two independent butconjugate variablesin physics, which can be in the form of a pair of two

quantities, such as position and momentum, energy and time, angle and angular momentum.

The limit itself has nothing to do with imperfections in practical measuring instruments, or

equivalently with careless measurements. Rather, it arises automatically from the nature of

microscopic systems. Although this principle in its simple form is later known to be one of

cornerstones in modern physics, Heisenberg is also famous for his contributions to develop a

complete theory of quantum mechanics. Indeed,modern quantum theory was pioneered

by the works of Schrodinger and Heisenberg, and some other physicists in the late of 1920s.

Apparently, there were two different quantum theories, namely wave mechanics proposed

by Schrodinger andmatrix mechanicssuggested by Heisenberg. Although the latter is quite

elegant, it only paid little attention to physicists at the time it was launched for some reasons.The matrix mechanics involved an unfamiliar way of complicated mathematics to describe the

dynamics of sub-atomic particles, and was arguably based on rather vague physical concepts.

However, these two formulations were later shown to be completely equivalent, in the sense

that both theoretical approaches confirm that the principalquantum number is not adequate

to describe the dynamics of microscopic particles. Instead, additional parameters associated

with orbital, magnetic, and intrinsic quantum numbers are needed to complete an atomic

description. A detailed derivation of these quantum numbers will be given in the course of

Quantum Physics. Instead, this chapter focuses on the Heisenberg uncertainty principle,

providing a mathematical basis for the principle and exploring its consequences.

37


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38 5. THE HEISENBERG UNCERTAINTY PRINCIPLE

x

f(x)

-L 0

Eo

L

A(k)

0

(a)

(b)

kp-kpk

kp+/Lkp-/L

EoL

Figure 5.1: The idealised profile of (a) a cosine pulse of width 2L and amplitude Eo,and (b) its Fourier transform with a carrier spatial frequency kp in the space domain.

5.1 Mathematical Basis for the Uncertainty Principle

In the previous chapter, it was stated that what was meant by a wave packet or a wave group

associated with a moving particle was a group of waves of limited spatial extent consisting of

individual waves of different frequencies. Wave of this type is commonly found in the form

of a pulse propagated in space. The term space here is used in a general sense, for which it

could be eitherx-axis ort-axis depending upon an explicit mathematical function of the pulse.

To examine how a pulse is transformed from one form to another in a particular domain, here

we consider an idealised harmonic pulse in the form of a cosine periodical function, that is,

the cosine wavetrain in the space domain(see Figure5.1a) given by

f(x) = Eocos kpx for L x L

= 0 otherwise

(5.1)

The choice to work in the space domain is optional in that the time-dependent disturbance

is equivalently applicable (see also Figure5.2). In the time domain, the corresponding cosine

wavetrain is written as

g(t) = Eocos pt for T x T= 0 otherwise

(5.2)

Here we solve the problem using spatial variables in (5.1) than using temporal variables

in (5.2), as it is easier to examine the wave profile at t= 0 rather than the profile at x= 0.


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5.1. Mathematical Basis for the Uncertainty Principle 39

t

g(t)

-T 0

Eo

T

a()

0

(a)

(b)

p-p

p+/Tp-/T

EoL

Figure 5.2: The idealised profile of (a) a cosine pulse of width 2Tand amplitude Eo,and (b) its Fourier transform with a carrier temporal frequency p in the time domain.

Asf(x) given in (5.1) is an even function, the corresponding cosine Fourier transform

A(k) in the space domain is calculated from

A(k) = f(x)cos kx dx (5.3)

The above integral tells us about how dynamic variablesk and xare interchangeable, meaning

that a function of a particular variable in a given domain can be transformed into another

function of its counter-part variable in the same domain. Thus based on Figure5.1and by

substituting (5.1) into (5.3), we have

A(k) =

LL

Eocos kpxcos kx dx

=

L

L

Eo2

cos(kp+k)x + cos(kp k)x

dx

= EoL

sin(kp+k)L

(kp+k)L +

sin(kp k)L(kp k)L

= EoL

sinc (kp+k)L + sinc(kp k)L

(5.4)

Several interesting possibilities are here discussed regarding with (5.4). When there are

many waves in the train, that is, p L, then the left and right peaks are both narrowand widely spaced. When k =k

p, the second sinc function reaches a maximum value of one,

meaning that A(k) has a maximum value ofEoL (Figure5.1b). In particular, the first sinc


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40 5.

fisika_kuantum

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