modern physics - 物理系

Modern Physics

Spring, 2008

Chen GuanJunPhysics Department, TaiYuan Normal University

Email:[email protected]

Preface

What is “Modern Physics”?

This class is usually called “modern” physics. People often use the word“modern” to mean “contemporary”, but that’s not the usage here. Modern(in art as well as physics) often means work that was done in the first part ofthe twentieth century. In the case of this class, most of this work was done inthe years 1890-1930. These were years of a great revolution in physics, andmost of the work done today directly descends from work done in this period.A lot of time in this class will indeed be devoted to physics discovered in thisperiod. This is still a big step forward in time from your earlier classes, wherethe physics you studied was done in the nineteenth century or earlier (yougot up to 1905 if you did special relativity). Therefore, even though a lot ofthis work is fairly old, you should view this class as being an introductionto current problems in physics.

Closed fields of physics at the end of the 19th century:

• Mechanics (Newton, Hamilton, Lagrange . . . )

• Statistical mechanics (Boltzmann, Gibbs)

• Electrodynamics (Maxwell)

Apparently “Small” discrepancies between theory and obser-vation:

• Black body radiation (ultraviolet divergence)

• Photoelectric effect (properties unexplained)

• Line spectra (no classical explanation)

• . . .

i

ii PREFACE

Figure 1: What is “Modern Physics”

From the mid-19th through the early 20th century, scientists studied aset of new and puzzling phenomena concerning the nature of matter and,indeed, of energy in all its forms. The program that brought these questionsto the point where we are today has provided some of the most remarkablesuccess stories in all of science.

This is the history of quantum mechanics, which began in mystery, yetat the end of the century has come to dominate the economies of modernnations.

Grading

• Attendance & Discussion(9%)

• Homework(9%, No late homework will be accepted.)

• Midterm(12%)

• Final(70%)

Contents

Preface i

1 Black Body Radiation 11.1 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . 11.2 Rayleigh-Jeans Formula . . . . . . . . . . . . . . . . . . . . . 41.3 Planck’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Photoelectric Effect 152.1 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Photoelectric Effect Data . . . . . . . . . . . . . . . . . . . . 172.4 Einstein’s Solution of the Photoelectric Effect . . . . . . . . . 172.5 Illustration and Example . . . . . . . . . . . . . . . . . . . . . 192.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Compton Effect 233.1 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Compton Shift Equation . . . . . . . . . . . . . . . . . . . . . 243.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Thomson Model of Atoms 294.1 Discovery of the Electron . . . . . . . . . . . . . . . . . . . . 294.2 Millikan’s Experiment(1909) . . . . . . . . . . . . . . . . . . . 304.3 Thomson Model of Atoms . . . . . . . . . . . . . . . . . . . . 314.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Rutherford Experiment 335.1 Alpha(α) Particle . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Rutherford Experiment . . . . . . . . . . . . . . . . . . . . . 345.3 Disproof of the Pudding . . . . . . . . . . . . . . . . . . . . . 345.4 Emergence of the Nucleus . . . . . . . . . . . . . . . . . . . . 365.5 Rutherford Model of Atoms(1911) . . . . . . . . . . . . . . . 365.6 Rutherford Formula . . . . . . . . . . . . . . . . . . . . . . . 37

iii

iv CONTENTS

5.7 Geiger-Marsden Data on α Scattering . . . . . . . . . . . . . 39

6 Line Spectra 436.1 The Type of Spectra . . . . . . . . . . . . . . . . . . . . . . . 436.2 Line Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Hydrogen Spectra . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Difficulties of the Rutherford Model . . . . . . . . . . . . . . 496.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 The Bohr Model 517.1 Bohr’s Postulates and Rydberg Equation . . . . . . . . . . . 517.2 The Effect of Nucleus Motion: Reduced Mass . . . . . . . . . 557.3 Some Energy Level and Orbit Diagrams . . . . . . . . . . . . 567.4 Bohr’s Corresponding Principle . . . . . . . . . . . . . . . . . 567.5 The Franck-Hertz Experiment . . . . . . . . . . . . . . . . . . 587.6 Failures of the Bohr Model . . . . . . . . . . . . . . . . . . . 607.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8 Wave-Particle Duality 638.1 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.2 de Broglie Hypothesis . . . . . . . . . . . . . . . . . . . . . . 648.3 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658.4 de Broglie’s Hypothesis Applied to Atoms . . . . . . . . . . . 668.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9 Davisson and Germer Experiment 699.1 Bragg Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.2 Bragg’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.3 Bragg Spectrometer . . . . . . . . . . . . . . . . . . . . . . . 709.4 Davisson-Germer Experiment(1927) . . . . . . . . . . . . . . 719.5 Two-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . 749.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

10 Schrodinger Equation 7710.1 Schrodinger Equation: 1926 . . . . . . . . . . . . . . . . . . . 7710.2 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . 7710.3 Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.4 Probability Interpretation: Max Born . . . . . . . . . . . . . 7910.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7910.6 Operators and Expectation Values . . . . . . . . . . . . . . . 8310.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CONTENTS v

11 Hydrogen-Quantum Mechanics 8711.1 Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . 8711.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . 8811.3 Solution to Azimuthal Angle Equation . . . . . . . . . . . . . 8911.4 Solution to Polar Angle Equation . . . . . . . . . . . . . . . . 9011.5 Solution to Radial equation . . . . . . . . . . . . . . . . . . . 9111.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9311.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9411.8 Selection Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 9511.9 Probability Distributions . . . . . . . . . . . . . . . . . . . . . 9611.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

12 Angular Momentum 10312.1 Classical Angular Momentum . . . . . . . . . . . . . . . . . . 10312.2 Angular Momentum Operator . . . . . . . . . . . . . . . . . . 10412.3 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . 10512.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.5 Vector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.6 Space Quantization . . . . . . . . . . . . . . . . . . . . . . . . 10712.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

13 The Stern-Gerlach Experiment 11113.1 Orbital Magnetic Moment . . . . . . . . . . . . . . . . . . . . 11113.2 Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . 11313.3 Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . 11513.4 The Meaning of Stern-Gerlach Experiment . . . . . . . . . . . 11713.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

14 The Periodic Table 11914.1 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 11914.2 Complex Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 12014.3 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . 12114.4 Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . 12214.5 Build up the Periodic Table . . . . . . . . . . . . . . . . . . . 12314.6 Justification of the Shell Model . . . . . . . . . . . . . . . . . 12614.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

15 Alkali Metals 12915.1 The Alkali Metals . . . . . . . . . . . . . . . . . . . . . . . . 12915.2 Screening Model: Effective Potentials . . . . . . . . . . . . . . 13015.3 Lithium and Sodium . . . . . . . . . . . . . . . . . . . . . . . 13115.4 � Dependence of Electron Energies . . . . . . . . . . . . . . . 13215.5 The Spectrum of Alkali Metals . . . . . . . . . . . . . . . . . 132

vi CONTENTS

16 Fine Structure 13716.1 Magnetic Field in Atoms . . . . . . . . . . . . . . . . . . . . . 13716.2 Spin Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . 13816.3 The Total Angular Momentum . . . . . . . . . . . . . . . . . 13816.4 Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.5 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14416.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

17 The Zeeman Effect 14717.1 The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . 14717.2 Total Spin of Atoms . . . . . . . . . . . . . . . . . . . . . . . 14717.3 The Normal Zeeman Effect . . . . . . . . . . . . . . . . . . . 14817.4 The Anomalous Zeeman Effect . . . . . . . . . . . . . . . . . 15217.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Chapter 1

Black Body Radiation

The difficulties of classical physics I: black body radiation

1.1 Black Body Radiation

1.1.1 Thermal Radiation

Any object whose temperature is greater than 0K will emit some radiantenergy(Fig. 1.1). The intensity of thermal radiation – Radiated Power is

Figure 1.1: Thermal radiation

spread out as a function of wavelength. Color(wavelength, frequency)andradiated power of heated materials depend on T , when the temperature isabout 500◦C, visible wavelengths are dominated. The challenge to scientistswas to show how this radiant energy is related to the temperature of theobject(Fig. 1.2).

1

2 CHAPTER 1. BLACK BODY RADIATION

Figure 1.2: Radiated power

1.1.2 Black Body Radiation

In 1859, German physicist Gustav Kirchhoff studied the emission and ab-sorption of radiation by heated materials: In thermal equilibration,

absorption coefficient = emission coefficient, that is a = e.

“Black body radiation” or “cavity radiation” (Fig. 1.3) refers to anobject or system which absorbs all radiation incident upon it, that is forblack body:

absorption coefficient : a = 1 =⇒ e = 1.

Ideal radiator-black body(cavity) radiation independents of material.

Figure 1.3: A cavity

1.1.3 Stefan-Boltzmann Law

Stefan-Boltzmann law(1874):

R(T ) =∫ ∞

0R(λ, T )dλ = eσT 4.

1.1. BLACK BODY RADIATION 3

R(T ) is the radiated power per unit area, R(λ, T ) is the radiant intensityof wavelength λ, e is the emission coefficient, σ is the Stefan-Boltzmannconstant,

σ = 5.67 × 10−8Wm−2K−4.

In 1884, Boltzmann proved the Stefan-Boltzmann law using the laws ofthermodynamics. By this time Maxwell had formulated his equations.

Radiation is heat transfer by the emission of electromagnetic waves whichcarry energy away from the emitting object. If the hot object is radiatingenergy to its cooler surroundings at temperature Tc, the net radiation lossrate takes the form

P = eσA(T 4 − T 4c ).

Where P is net radiated power, A is radiating area, T is temperature ofradiator, Tc is temperature of surroundings.

Example 1: The Sun (can be considered as a black body) with radius r =0.7 × 109m at 5800K give off radiation, its total radiated power?

4πr2σT 4.

a hot campfire at perhaps 800K, the radiation loss rate of the Sun?

4πr2σ(T 4 − T 4c ).

1.1.4 Wien’s Displacement Law

Wien’s displacement law(Fig. 1.4):

λmaxT = b,

whereb = 2.898 × 10−3mK.

is called Wien’s constant.This relationship is called Wien’s displacement law and is useful for the

determining the temperatures of hot radiant objects such as stars(Fig. 1.5).

Example 2: λmax of the Sun?

λmax =b

T= 499.7nm.

Star Temperatures: Stars approximate black body radiators and their visiblecolor depends upon the temperature of the radiator. The curves in Fig.1.5,1.6 show blue, white, and red stars.


Figure 1.4: Wien’s displacement law

Figure 1.5: Star temperatures

1.2 Rayleigh-Jeans Formula

1.2.1 Rayleigh-Jeans Formula

The Rayleigh-Jeans law was an important step in our understanding of theequilibrium radiation from a hot object,here are the steps which led to theRayleigh-Jeans law.

1. Equilibrium standing wave electromagnetic radiation in a cubical cav-ity of dimension L must meet the condition:

Electromagnetic standing waves in a cavity must satisfy the wave equa-tion in three dimensions:

∂2E

∂x2+

∂2E

∂y2+

∂2E

∂z2=

1c2

∂2E

∂t2.

The solution to the wave equation must give zero amplitude at thewalls, The boundary conditions can be met with a solution of theform:

E = E0 sin(n1πx

L) sin(

n2πy

L) sin(

n3πz

L) sin(

2πct

λ).

1.2. RAYLEIGH-JEANS FORMULA 5

Figure 1.6: Star temperatures

Substituting this solution into the wave equation above gives

(n1π

L)2 + (

n2π

L)2 + (

n3π

L)2 = (

2π

λ)2,

which simplifies to

n21 + n2

2 + n23 =

4L2

λ2.

2. The number of modes in the cavity

Figure 1.7: n-space

We need to evaluate the number of modes which can meet this condi-tion, which amounts to counting all the possible combinations of theinteger “n” values. An approximation can be made by treating thenumber of combinations as the volume of a three-dimensional grid ofthe values of “n”, an “n-space”(Fig. 1.7).


Using the relationship for the volume of a sphere, with the “n” valuesspecifying the coordinates along three “n” axes, gives

volume =4π

3(n2

1 + n22 + n2

3)32 ,

we have used both positive and negative values of “n”, whereas thewave equation solution uses only positive definite values

volume × 18,

waves can polarize in two perpendicular planes

volume × 2,

when the size of the cavity is much greater than the wavelength, thisis a very good approximation.

Number of modes N =π

3(n2

1 + n22 + n2

3)32 =

8πL3

3λ3.

3. The number of modes per unit wavelength

This may be obtained by taking the derivative of the number of modeswith respect to wavelength.

dN

dλ=

d

dλ(8πL3

3λ3) = −8πL3

λ4.

The negative sign here reveals that the number of modes decreaseswith increasing wavelength. Now to get the number of modes per unitvolume per unit wavelength:

1L3

dN

dλ=

8π

λ4.

For a large cavity, this result obtained is independent of cavity geom-etry.

4. The energy per unit volume per unit wavelength

Assigning energy to the electromagnetic standing waves in a cavitydraws on the principle of equipartition of energy. Each standing wavemode will have average energy kT

ε =

∫ ∞0 εe−βεdε∫ ∞0 e−βεdε

= − d

dβ[ln

∫ ∞

0e−βεdε]

= − d

dβln

1β

=1β

= kT,

1.2. RAYLEIGH-JEANS FORMULA 7

where β = 1kT , k is Boltzmann’s constant

k = 1.380 × 10−23J · K−1,

and T is the temperature in Kelvins. Letting u represent the energydensity:

du

dλ=

1L3

dE

dλ= kT

1L3

dN

dλ=

8πkT

λ4.

It can also be expressed in terms of the frequency:

du

dν=

du

dλ

dλ

dν=

du

dλ

d cν

dν=

du

dλ

−c

ν2.

The minus sign here just reminds us that a decrease with wavelengthimplies an increase with increasing frequency. The magnitude of theenergy density dependence on frequency is given by:

du

dν=

du

dλ

c

ν2=

8πkTν4

c4

c

ν2=

8πν2

c3kT.

this is the classical result: the Rayleigh-Jeans law.

1.2.2 Ultraviolet Catastrophe

Ultraviolet catastrophe(Fig. 1.8):

u =∫ ∞

0

du

dλdλ = ∞

Figure 1.8: Ultraviolet catastrophe


Figure 1.9: Radiated power

1.2.3 Some Different Expressions of Rayleigh-Jeans Law

1. Radiated power as a function of wavelength.For perpendicular radiated energy(Fig. 1.9)

it must be noted that half of the energy density in the waves is goingtoward the walls and half is coming out if the system is in thermalequilibrium. Radiated power per unit wavelength

dR

dλ=

12

du

dλ

Ax

t=

12

du

dλ

Act

t=

du

dλ

Ac

2,

but at an angle θ, the effective area will be A cos θ and the effectivespeed will be c cos θ, so the radiated energy will be reduced to:

dR

dλ=

du

dλ

Ac

2cos2 θ,

For a given observation point near a radiating surface, the power willbe the average power from all directions. Having averaged over allangles, the calculated radiated power per unit wavelength per unitarea is finally:

dR

dλ=

dudλ

Ac2 < cos2 θ >aver

A

=du

dλ

c

4=

8πkT

λ4

c

4=

2πc

λ4kT.

This is also the Rayleigh-Jeans formula.

2. Radiated power as a function of frequency.To express this in terms of frequency, yields a radiated power per unitfrequency per unit area:

dR

dν=

dR

dλ

c

ν2=

2πν2

c2kT.

1.3 Planck’s Formula

1.3.1 The Planck’s Hypothesis

In order to explain the frequency distribution of radiation from black body(cavity), In 1900, Planck proposed the ad hoc assumption that the radiant

1.3. PLANCK’S FORMULA 9

energy could exist only in discrete quanta which were proportional to thefrequency, a quantum has energy

ε0 = hν.

where h is Plank’s constant, with

h = 6.626 × 10−34J · s.This would imply that higher modes would be less populated and avoid

the ultraviolet catastrophe of the Rayleigh-Jeans law.

1.3.2 Planck’s Formula

The radiant energy of each mode in the cavity must be:

ε = nε0, n = 0, 1, 2, ...

this quantum idea(Analogy to “e”) was soon seized to explain the photo-electric effect(see chapter 2), became part of the Bohr theory (see chapter7) of discrete atomic spectra, and quickly became part of the foundation ofquantum theory.

The average energy per “mode” is:

ε =∑∞

n=0 nε0e−βnε0∑∞

n=0 e−βnε0= − d

dβ[ln

∞∑n=0

e−βnε0 ]

= − d

dβ[ln(

11 − e−βε0

)] =ε0

eβε0 − 1.

Sodu

dλ=

8π

λ4

ε0

eβε0 − 1=

8πhc

λ5

1

ehc

λkT − 1.

ordu

dν=

du

dλ

c

ν2=

8πhν3

c3

1

ehνkT − 1

.

This is the Planck’s black body radiation formula. The Rayleigh-Jeans curveagrees with the Planck’s radiation formula for long wavelengths, low fre-quencies.

Radiated power per unit area

dR

dλ=

du

dλ

c

4=

2πhc2

λ5

1

ehc

λkT − 1.

dR

dν=

dR

dλ

c

ν2=

2πhν3

c2

1

ehνkT − 1

.

For low frequency radiation(Fig. 1.10)


• Rayleigh-Jeans Formula:

du

dν=

8πν2

c3kT.

• Planck’s Formula:When hν � kT , e

hνkT � 1 + hν

kT ,

du

dν=

8πhν3

c3

1

ehνkT − 1

� 8πν2

c3kT.

Figure 1.10: Planck vs. Rayleigh-Jeans

1.3.3 Applications of the Planck’s Radiation Formula

1. Integrate over wavelength to get the Stefan-Boltzmann law

dR

dλ=

2πhc2

λ5

1

ehc

λkT − 1.

R = 2πhc2

∫ ∞

0

dλ

λ5(ehc

λkT − 1).

make the substitution

x =hc

λkT, dx =

−hc

λ2kTdλ,

gives λ = 0 → ∞, x = ∞ → 0

R =2π(kT )4

h3c2

∫ ∞

0

x3

ex − 1dx.


Making use of the standard form integral∫ ∞

0

x3

ex − 1dx =

π4

15,

gives the final form of the Stefan-Boltzmann law

R =2π5k4

15h3c2T 4 = σT 4.

σ =2π5k4

15h3c2= 5.67 × 10−8Wm−2K−4.

2. Take derivative to get the Wien’s displacement lawWhen the temperature of a black body radiator increases, the overallradiated energy increases and the peak of the radiation curve movesto shorter wavelengths. When the maximum is evaluated from thePlanck radiation formula, the product of the peak wavelength and thetemperature is found to be a constant.

du

dλ=

8πhc

λ5

1

ehc

λkT − 1.

To find the peak of the black body radiation curve, we take the deriva-tive: Let

a =hc

k,

d

dλ(

1λ5

1e

aλT − 1

) = 0,

(−5λ6

1e

aλT − 1

+1λ5

−ea

λTaT

−1λ2

(ea

λT − 1)2= 0.

Simplifying1 gives the maximum condition:

λT =ae

aλT

5(ea

λT − 1)=

a

5(1 − e−aλT )

.

which must be solved numerically to give:

λmaxT =hc

4.965k= b = 2.898 × 10−3mK.

Using the constants c, b and σ, Plank calculated related constants hand k. With h and k, Planck radiation formula agrees with experi-mental curves well.

1× 15λ7T (e

aλT − 1)


3. 3K background radiationA uniform background radiation in the microwave region of the spec-trum is observed in all directions in the sky. It shows the wavelengthdependence of a “black body” radiator at about 3 Kelvins tempera-ture(Fig. 1.11).

The discovery of the 3K microwave background radiation was one ofthe crucial steps leading to the calculation of the standard “Big Bang”model of cosmology.

Figure 1.11: 3K background radiation

Example 3: Temperature of stars.Suppose a star radiates maximally at a wavelength of 300nm. Assume thatthe radius of the star is the same as that of the sun, rsun = 6.96×108m. Whatis the power per unit area reaching the surface of the earth if the star is 1000light years(1ly = 9.46 × 1015m) away from the earth? How many Planck’squanta are collected by the 10m2 mirror of the telescope each second? (Hint:Star can be considered as a black body.)Solution:

• Consider the temperature of the star first: The maximal intensity atthe maximum of Planck’s formula:

T =hc

4.965kλmax=

b

λmax= 9600K,


• The radiated power(unit area), radiated by the star, is given by theStefan-Boltzman law:

Rsun(T ) = σT 4 = 5.67 × 10−8 × 96004

= 4.82 × 108Wm−2,

• Now the radiated power(unit area)reaching the surface of the earth is

Rearth(T ) =4πr2

sun × Rsun(T )4πd2

s−e

= 2.6 × 10−12Wm−2,

• The energy of 300nm quanta (photons) is

E =hc

λ= 6.626 × 10−19J,

• Then the number of Planck quanta per second on the 10m2 mirror is

N =Area × Rearth(T)

E= 3.9 × 107s−1.

Exercise 1: Find the energy density of black body radiation at T = 6000Kin the range from 450 to 460nm. Assume that this range is so narrow thatthe energy density function du

dλ does not vary much over it.Solution: The fact that du

dλ does not vary much over a range of a wavelengthsδλ means that the energy density is given simply by du

dλδλ rather than thecorresponding integral over a range of wavelengths.

For dudλ , we can simply evaluate it at any wavelength in the range, and

we do so at the midpoint λ = 455nm = 455 × 10−9m. We have

hc

λkT=

(6.63 × 10−34) × (3 × 108)(455 × 10−9) × (1.38 × 10−23) × 6000

= 5.28,

du

dλ=

8πhc

λ5

1

ehc

λkT − 1= 1.312 × 106,

du

dλδλ = 1.312 × 10−2Jm−3.

Exercise 2: Find the average energy contained in quanta of a given fre-quency at temperature T in the two limits hν � kT and hν � kT .Solution: Solve this problem is a matter of taking limits of the more generalresult for the average energy for a given frequency. For the limit hν � kT ,


the exponential factor in the denominator greater greater than 1, and theaverage energy is

hνe−hνkT .

For the classical limit hν � kT , the exponential factor is approximately

ehνkT ≈ 1 +

hν

kT

andε ≈ hν

1 + hνkT − 1

= kT.

We see that in this limit we recover the average energy kT that was usedin the development of the Rayleigh-Jeans expression for the low-frequencypart of the spectrum.

1.4 Problems

1. Is a black body black?

2. The cosmic background radiation is that of a black body at 2.7K.What is the value λmax at which the distribution has a maximum?What is the energy of the corresponding quantum?

3. Solar radiation falls on Earth’s surface at a rate of 1400W/m2. As-suming that the radiation has an average wavelength of 550nm, Howmany photons per square meter per second fall on the surface?

Chapter 2

Photoelectric Effect

The difficulties of classical physics II: photoelectric effect

2.1 Photoelectric Effect

In the latter part of the 19th century, experiments showed that light incidenton certain metallic surfaces cause electrons to be emitted from surface(Fig.2.1), this phenomenon, first discovered by Hertz, is known as the Photoelec-tric Effect.

Figure 2.1: Photoelectric effect

Electrons ejected from a sodium metal surface were measured as an elec-tric current. Finding the stopping voltage(V0) it took to stop all the electronsgave a measure of the maximum kinetic energy of the electrons in electronvolts.

2.2 Classical Theory

The minimum energy required to eject an electron(photoelectron) from thesurface is called the photoelectric work function φ. So we have

Kmax = E − φ,

15

16 CHAPTER 2. PHOTOELECTRIC EFFECT

where Kmax is the maximum kinetic energy of the photoelectrons, E isenergy that the electron absorbed from incident light.

The details of the photoelectric effect were in direct contradiction to theexpectations of very well developed classical physics. Because The remark-able aspects of the photoelectric effect when it was first observed were:

1. The electrons were emitted immediately, no time lag !

Example 1: Estimate time lag.Size of the atoms is a few times 10−10m. Area is about 10−19m2.Suppose we consider sodium, φ = 2.3eV, with light of intensity I =10−8W/m2. How long should one wait to produce a photoelectron ofenergy 1eV?Solution: Power over atomic area is

W = 10−8W/m2 × 10−19m2 = 10−27J/s

=10−27J/s

1.6 × 10−19J/eV= 6.3 × 10−9eV/s,

t =2.3eV + 1eV

6.3 × 10−9eV/s= 5.2 × 108s = 12year.

2. Increasing the intensity of the light increased the number of photoelectrons(I),but not their maximum kinetic energy Kmax(Fig. 2.2)!

Figure 2.2: Photocurrent vs. light intensity

3. Red light will not cause the ejection of electrons, no matter what theintensity(Fig. 2.3:left)!

4. A weak violet light will eject only a few electrons, but their maxi-mum kinetic energies are greater than those for intense light of longerwavelengths(Fig. 2.3:right)!

2.3. PHOTOELECTRIC EFFECT DATA 17

Figure 2.3: Frequency vs. Kmax

2.3 Photoelectric Effect Data

See Fig. 2.4,2.5

Figure 2.4: Photoelectric effect data

2.4 Einstein’s Solution of the Photoelectric Effect

2.4.1 The Planck’s Hypothesis

The remarkable fact showed that the interaction between light and metalsurface must be like that of a particle which gave all of its energy to theelectron! This fit in well with Planck’s hypothesis that light could exist onlyin discrete bundles with energy. Each quantum has energy

ε0 = hν,

called a photon-the quantum of light, where h is Plank’s constant, with

h = 6.626 × 10−34J · s = 4.136 × 10−15eV · s.


Figure 2.5: Photoelectric effect data

Figure 2.6: Electromagnetic Spectrum

2.4.2 Einstein’s Photoelectric Equation

All the problems are solved if we assume that the incoming light of frequencyν is quantized. It consists of photons each having energy E = hν(sameconstant found by Planck!).

Maximum (negative) eV0 (called Stopping Potential) with nonzero Iequals to maximum kinetic energy

eV0 = Kmax = hν − φ.

1. Instantaneous emission is due to the arrival of at least one photon assoon as light is turned on.

2. Maximum kinetic energy depends on frequency only but not on lightintensity.

2.5. ILLUSTRATION AND EXAMPLE 19

Table 2.1: Work functions for photoelectric effect

Element Work Function(eV)

Aluminum 4.08Calcium 2.9Carbon 4.81Copper 4.7Gold 5.1Iron 4.5Magnesium 3.68Mercury 4.5Nickel 5.01Potassium 2.3Silver 4.73Sodium 2.46

3. Threshold Frequency(νth) is required by work function φ = hνth(Table:2.1).

Most commonly observed phenomena with light can be explained bywaves, but the photoelectric effect suggested a particle nature for light. Theexplanation marked one of the major steps toward quantum theory.

2.5 Illustration and Example

See Fig. 2.7

Figure 2.7: Illustration

Example 2: find h and φ from data.


In a photoelectric experiment it is found that a stopping potential 1.0V isneeded to stop all the electrons when incident light of wavelength 260nm isused and 2.3V is needed for light of wavelength is 205nm. From these datadetermine Planck’s constant and the work function of the metal in eV.Solution: Use

eV0 = Kmax = hν − φ = hc/λ − φ,

for each of the two measurements

h(c

λ1− c

λ2) = e(V1 − V2),

h =e(V1 − V2)

c(1λ1

− 1λ2

)=

1.6 × 10−19 × (2.3 − 1)

3 × 108(1

2.05 × 10−7− 1

2.6 × 10−7)

= 6.7 × 10−34Js.

φ = hc/λ − eV0 = e(6.7 × 10−34 × 3 × 108

1.6 × 10−19 × 2.6 × 10−7− 1) = 4.83eV.

The emitter is made of carbon.Exercise 1: The kinetic energies of photoelectrons range from zero to 4.0×10−19J when light of wavelength 3000A falls on a surface. What is thestopping voltage for this light?Solution:

Kmax = 4.0 × 10−19J × 1eV1.6 × 10−19J

= 2.5eV.

Hence, from Kmax = eV0 we get stopping voltage

V0 = 2.5V.

Exercise 2: What is the threshold wavelength for the material in Exercise1?Solution:

hc = (4.136 × 10−15eV · s) × (3 × 1018A/s)

= 12.4 × 103eV · A,

eV0 = hν − φ =hc

λ− hc

λth,

2.5eV =12400eV · A

3000A− 12400eV · A

λth,

Solving,λth = 7607.4A.

2.6. PROBLEMS 21

2.6 Problems

1. Are photoelectrons different from other electrons?

2. Can you find the momentum of photons? Ans. p = hλ

3. The emitter in a photoelectric tube has a threshold wavelength of6000A. Determine the wavelength of the light incident on the tube ifthe stopping voltage for this light is 2.5V. Ans. λ = 2713A

4. Find the work function for Potassium if the largest wavelength forthe electron emission in a photoelectric experiment is 5350A. Ans.φ = 2.3eV

5. A sodium surface is illuminated with light of wavelength 300nm. Thework function for sodium metal is 2.46eV. Find:(1) the maximum kinetic energy of the ejected photoelectrons.(2) the maximum speed of the photoelectrons (me = 9.1 × 10−31kg).(3) the threshold wavelength for sodium. Ans. Kmax = 1.67eV,vmax = 7.66 × 105ms−1, λth = 504.1nm.

Chapter 3

Compton Effect

The difficulties of classical physics III: Compton effect

3.1 Compton Effect

The scattering of photons from charged particles is called Compton scat-tering after Arthur Compton who was the first to measure photon-electronscattering(Fig. 3.1) in 1922.

Figure 3.1: Compton scattering

Compton’s original experiment made use of molybdenum K-alpha x-rays,which have a wavelength of 0.0709nm. These were scattered from a blockof carbon and observed at different angles with a Bragg spectrometer.

The spectrometer consists of a rotating framework with a calcite crystalto diffract the x-rays and an ionization chamber for detection of the x-rays.Since the spacing of the crystal planes in calcite is known, the angle ofdiffraction gives an accurate measure of the wavelength.

23

24 CHAPTER 3. COMPTON EFFECT

A. H. Compton observed the scattering of x-rays and found them witha longer wavelength than those incident upon the target. The shift of thewavelength increased with scattering angle(Fig. 3.2).

Figure 3.2: Experimental results

3.2 Compton Shift Equation

In the explanation of the Compton scattering experiment, Arthur Comp-ton treated the x-ray photons as particles, modeled scattering as collisionbetween a photon and an electron(Fig. 3.3).

When the incoming photon gives part of its energy to the electron, thenthe scattered photon has lower energy and according to the Planck’s rela-tionship has longer wavelength. Using the Planck’s relationship and therelativistic energy expression, he worked out the Compton shift equation.

1. Conservation of energy takes the form:

hνi + mec2 = hνf +

√p2

ec2 + m2

ec4.

3.2. COMPTON SHIFT EQUATION 25

Figure 3.3: Compton scattering model

2. Conservation of momentum requires:

pi = pf + pe,

where p = E/c is used for the photon momentum.

3. Squaring this equation using the scalar product gives

p2e = p2

i + p2f − 2pipf cos θ.

4. Again using the Planck’s relationship and the relativistic energy expression(×c2):

(pec)2 = (hνi)2 + (hνf )2 − 2h2νiνf cos θ.

5. The energy conservation expression above can be squared to give

(pec)2 = (hνi)2 + (hνf )2 − 2h2νiνf + 2mec2(hνi − hνf ).

6. These two forms can be equated to give

−2h2νiνf cos θ = −2h2νiνf + 2mec2(hνi − hνf ),

7. which can be rearranged to

1hνf

− 1hνi

=1

mec2(1 − cos θ),


8. and finally to the standard Compton formula:

λf − λi = ∆λ =h

mec(1 − cos θ)

= λC(1 − cos θ),

where λC =h

mec= 0.00243nm is called the Compton wavelength for

the electron. This is the Compton Shift Equation 1. The wavelengthchange in such scattering depends only upon the angle of scatteringfor a given target particle.

For scattering from free electrons, the formula gives a wavelength of0.0733nm for scattering at 90 degrees. That is consistent with the right-hand peak in Fig. 3.1 above.

The peak which is near the original x-ray wavelength is considered tobe scattering off inner electrons in the carbon atoms which are more tightlybound to the carbon nucleus, this causes the entire atom to recoil from thex-ray photon. Putting the entire carbon nuclear mass into the scatteringequation yields a wavelength shift almost 22, 000 times smaller than that foran unbound electron, so those scattered photons are not seen to be shifted.

At 1920’s when the particle nature of light suggested by the photo-electric effect was still being debated, the Compton experiment gave clearand independent evidence of particle-like behavior. Compton was awardedthe Nobel Prize in 1927 for the “discovery of the effect named after him”.

Example 1: X-rays of wavelength 0.14nm are scattered from a block of car-bon. What will be the wavelengths of x-rays scattered at(1)0◦; (2)90◦; (3)180◦?Solution:(1) For θ = 0◦, cos θ = 1,

λf = λi = 0.14nm.

The photon goes straight through without interacting.(2) For θ = 90◦, cos θ = 0

λf = λi +h

mec= 0.14nm + 0.00243nm = 0.14243nm.

(3) For θ = 180◦, cos θ = −1

λf = λi +2h

mec= 0.14nm + 2 × 0.00243nm = 0.14486nm.

1A. H. Compton, Phys. Rev. 21, 483; 22, 409 (1923)

3.3. PROBLEMS 27

Example 2: A photon (λ = 0.4nm) strikes an electron at rest and reboundsat an angle of 150◦. Find the speed and wavelength of the photon after thecollision.Solution:(1) The speed of a photon is always c.(2)

λf = λi +h

mec(1 − cos θ)

= 0.4nm + 0.00243 × (1 + 0.866)nm = 0.405nm.

3.3 Problems

1. Calculate the fractional change in the wavelength of an x-ray of wave-length 0.4A that undergoes a 90◦ Compton scattering from an electron.Ans. ∆λ

λi= 0.0608.

2. An x-ray of wavelength 0.3A undergoes a 60◦ Compton scattering.Find the wavelength of the scattered photon and the kinetic energy ofthe electron after scattering. Ans. λf = 0.312A, K = 1.59keV

List of Figures

1 What is “Modern Physics” . . . . . . . . . . . . . . . . . . . ii

1.1 Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Radiated power . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 A cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Wien’s displacement law . . . . . . . . . . . . . . . . . . . . . 41.5 Star temperatures . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Star temperatures . . . . . . . . . . . . . . . . . . . . . . . . 51.7 n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Ultraviolet catastrophe . . . . . . . . . . . . . . . . . . . . . . 71.9 Radiated power . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 Planck vs. Rayleigh-Jeans . . . . . . . . . . . . . . . . . . . . 101.11 3K background radiation . . . . . . . . . . . . . . . . . . . . . 12

2.1 Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Photocurrent vs. light intensity . . . . . . . . . . . . . . . . . 162.3 Frequency vs. Kmax . . . . . . . . . . . . . . . . . . . . . . . 172.4 Photoelectric effect data . . . . . . . . . . . . . . . . . . . . . 172.5 Photoelectric effect data . . . . . . . . . . . . . . . . . . . . . 182.6 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . 182.7 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . 233.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 243.3 Compton scattering model . . . . . . . . . . . . . . . . . . . . 25

4.1 Thomson’s experiment . . . . . . . . . . . . . . . . . . . . . . 294.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Millikan’s experiment . . . . . . . . . . . . . . . . . . . . . . 314.4 Thomson model of atoms . . . . . . . . . . . . . . . . . . . . 324.5 Some Thomson atoms . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Alpha(α) particle . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Rutherford experiment . . . . . . . . . . . . . . . . . . . . . . 34

159

160 LIST OF FIGURES

5.3 Geiger-Marsden Data . . . . . . . . . . . . . . . . . . . . . . . 345.4 Scattering by Thomson Atoms . . . . . . . . . . . . . . . . . 355.5 Rutherford scattering experiment . . . . . . . . . . . . . . . . 375.6 The trajectory of the alpha particle . . . . . . . . . . . . . . . 375.7 Differential cross section . . . . . . . . . . . . . . . . . . . . . 385.8 Spatial Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.9 dn vs. θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.10 dn vs. t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.11 Estimate nuclear size . . . . . . . . . . . . . . . . . . . . . . . 415.12 An atom vs. the solar system . . . . . . . . . . . . . . . . . . 42

6.1 Continuous spectra . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Line spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 Absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . 436.4 Spectra photography . . . . . . . . . . . . . . . . . . . . . . . 446.5 Gas-discharge tube . . . . . . . . . . . . . . . . . . . . . . . . 446.6 Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.7 Diffraction grating . . . . . . . . . . . . . . . . . . . . . . . . 456.8 Hydrogen(left) and Argon(right) . . . . . . . . . . . . . . . . 456.9 Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.10 Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.11 Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.12 Sodium, Helium . . . . . . . . . . . . . . . . . . . . . . . . . . 476.13 Solar absorption spectrum . . . . . . . . . . . . . . . . . . . . 486.14 And more. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.15 Hydrogen:Balmer Series . . . . . . . . . . . . . . . . . . . . . 486.16 Hydrogen:Lyman series . . . . . . . . . . . . . . . . . . . . . 496.17 Hydrogen:Paschen series . . . . . . . . . . . . . . . . . . . . . 49

7.1 Classical electron orbit . . . . . . . . . . . . . . . . . . . . . . 517.2 Orbits in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . 537.3 Orbits and energy levels . . . . . . . . . . . . . . . . . . . . . 547.4 Atomic transition . . . . . . . . . . . . . . . . . . . . . . . . . 557.5 Energy level diagram 1 . . . . . . . . . . . . . . . . . . . . . . 567.6 Energy level diagram 2 . . . . . . . . . . . . . . . . . . . . . . 577.7 Energy level diagram 3 . . . . . . . . . . . . . . . . . . . . . . 577.8 Orbit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.9 Franck-Hertz apparatus . . . . . . . . . . . . . . . . . . . . . 587.10 Franck-Hertz data 1 . . . . . . . . . . . . . . . . . . . . . . . 597.11 Franck-Hertz data 2 . . . . . . . . . . . . . . . . . . . . . . . 59

8.1 Interference, diffraction and polarization . . . . . . . . . . . . 638.2 Wave and particles nature . . . . . . . . . . . . . . . . . . . . 648.3 Wavelength expression for a particle . . . . . . . . . . . . . . 65

LIST OF FIGURES 161

8.4 Standing wave . . . . . . . . . . . . . . . . . . . . . . . . . . 668.5 Electron standing wave . . . . . . . . . . . . . . . . . . . . . . 678.6 Electron standing wave 2 . . . . . . . . . . . . . . . . . . . . 678.7 Resonant electron “standing” waves . . . . . . . . . . . . . . 68

9.1 Cubic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.2 X-Rays scattering by Bragg planes . . . . . . . . . . . . . . . 709.3 Bragg spectrometer . . . . . . . . . . . . . . . . . . . . . . . . 709.4 Davisson-Germer apparatus . . . . . . . . . . . . . . . . . . . 719.5 Davisson-Germer data . . . . . . . . . . . . . . . . . . . . . . 729.6 Bragg plane d3 . . . . . . . . . . . . . . . . . . . . . . . . . . 739.7 Two-slit experiment . . . . . . . . . . . . . . . . . . . . . . . 749.8 Neutron-interference data . . . . . . . . . . . . . . . . . . . . 75

10.1 The infinite well . . . . . . . . . . . . . . . . . . . . . . . . . 8010.2 The infinite well: wavefunction . . . . . . . . . . . . . . . . . 8110.3 The infinite well: energy levels . . . . . . . . . . . . . . . . . 82

11.1 Rectangular and spherical coordinates . . . . . . . . . . . . . 8711.2 Azimuthal function . . . . . . . . . . . . . . . . . . . . . . . . 8911.3 Associated Legendre Polynomials . . . . . . . . . . . . . . . . 9011.4 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 9111.5 Radial wavefunctions . . . . . . . . . . . . . . . . . . . . . . . 9211.6 Radial wavefunctions . . . . . . . . . . . . . . . . . . . . . . . 9211.7 Hydrogen wavefunctions . . . . . . . . . . . . . . . . . . . . . 9311.8 Spectroscopic notation: electron states . . . . . . . . . . . . . 9411.9 Electron states with n = 5 . . . . . . . . . . . . . . . . . . . . 9511.10Degenerate energy levels . . . . . . . . . . . . . . . . . . . . . 9511.11Volume element . . . . . . . . . . . . . . . . . . . . . . . . . . 9611.12Radial probability distribution . . . . . . . . . . . . . . . . . 9711.13Radial probability distribution 2 . . . . . . . . . . . . . . . . 9811.14Radial probability distribution: 1s . . . . . . . . . . . . . . . 9811.15Radial probability distribution: 2s . . . . . . . . . . . . . . . 9811.16Radial probability distribution: 2p . . . . . . . . . . . . . . . 9911.17Radial probability distribution: 3s . . . . . . . . . . . . . . . 9911.18Radial probability distribution: 3p . . . . . . . . . . . . . . . 9911.19Radial probability distribution: 3d . . . . . . . . . . . . . . . 10011.20Angular probability distribution . . . . . . . . . . . . . . . . 100

12.1 Vector model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712.2 Vector model: � = 2 . . . . . . . . . . . . . . . . . . . . . . . 10812.3 Vector model: � = 3 . . . . . . . . . . . . . . . . . . . . . . . 108

13.1 Electron orbital motion . . . . . . . . . . . . . . . . . . . . . 11113.2 Orbital magnetic moment . . . . . . . . . . . . . . . . . . . . 112

162 LIST OF FIGURES

13.3 Stern-Gerlach apparatus . . . . . . . . . . . . . . . . . . . . . 11313.4 Classical expectation . . . . . . . . . . . . . . . . . . . . . . . 11413.5 Quantum expectation . . . . . . . . . . . . . . . . . . . . . . 11413.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 11513.7 Spin vector model . . . . . . . . . . . . . . . . . . . . . . . . 11613.8 Calculating deflection distance . . . . . . . . . . . . . . . . . 117

14.1 Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12114.2 Shell letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12214.3 Subshell filling . . . . . . . . . . . . . . . . . . . . . . . . . . 12214.4 Shell filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12314.5 Sodium shell model . . . . . . . . . . . . . . . . . . . . . . . . 12314.6 Order of filling . . . . . . . . . . . . . . . . . . . . . . . . . . 12414.7 Number of electrons in subshells . . . . . . . . . . . . . . . . 12514.8 Ground state configurations . . . . . . . . . . . . . . . . . . . 12514.9 The Noble Gases . . . . . . . . . . . . . . . . . . . . . . . . . 12514.10The periodic table . . . . . . . . . . . . . . . . . . . . . . . . 12614.11First ionization potentials(Z=1-20) . . . . . . . . . . . . . . . 128

15.1 Ground states electron configurations . . . . . . . . . . . . . . 12915.2 Shell models of alkali metals . . . . . . . . . . . . . . . . . . . 13015.3 Effective potential Veff (r) for the valence electrons of an atom

with atomic number Z. . . . . . . . . . . . . . . . . . . . . . 13015.4 Energy levels: Lithium . . . . . . . . . . . . . . . . . . . . . . 13215.5 Quantum defect: sodium . . . . . . . . . . . . . . . . . . . . . 13315.6 Energy levels: Sodium . . . . . . . . . . . . . . . . . . . . . . 13315.7 The radial probability densities P (r) = |Rn(r)|2r2 for the 2s

and 2p orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . 13415.8 The radial probability densities P (r) = |Rn(r)|2r2 for the 3s

and 3p orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . 13415.9 Hydrogen, Lithium and Sodium energy levels . . . . . . . . . 13515.10The yellow D-lines of sodium . . . . . . . . . . . . . . . . . . 135

16.1 The nucleus coordinate system . . . . . . . . . . . . . . . . . 13716.2 The electron coordinate system . . . . . . . . . . . . . . . . . 13716.3 The total angular momentum . . . . . . . . . . . . . . . . . . 13816.4 j = 3

2(left) and j = 12(right) . . . . . . . . . . . . . . . . . . . 140

16.5 Total angular momentum vector model . . . . . . . . . . . . . 14016.6 Fine structure of P states . . . . . . . . . . . . . . . . . . . . 14316.7 The yellow sodium D-lines . . . . . . . . . . . . . . . . . . . . 14316.8 Sodium fine structure . . . . . . . . . . . . . . . . . . . . . . 14416.9 Fine structure of Hydrogen(n = 2) . . . . . . . . . . . . . . . 14516.10Fine structure of Hydrogen spectra . . . . . . . . . . . . . . . 145

17.1 The normal Zeeman effect . . . . . . . . . . . . . . . . . . . . 147

LIST OF FIGURES 163

17.2 The anomalous Zeeman effect . . . . . . . . . . . . . . . . . . 14817.3 The Zeeman effect of 1P1 state . . . . . . . . . . . . . . . . . 14917.4 The Zeeman effect of 1D2 state . . . . . . . . . . . . . . . . . 15017.5 Zeeman Transition from 1D2 to 1P1 State . . . . . . . . . . . 15017.6 Polarization of the Zeeman lines . . . . . . . . . . . . . . . . 15117.7 Polarization of the Zeeman lines . . . . . . . . . . . . . . . . 15117.8 Polarization of the Zeeman lines . . . . . . . . . . . . . . . . 15217.9 Larmor precession . . . . . . . . . . . . . . . . . . . . . . . . 15317.10Sodium Zeeman effect: spectrum . . . . . . . . . . . . . . . . 15517.11Sodium Zeeman effect: spectra . . . . . . . . . . . . . . . . . 156

164 LIST OF FIGURES

List of Tables

2.1 Work functions for photoelectric effect . . . . . . . . . . . . . 19

12.1 Possible states for n = 3 . . . . . . . . . . . . . . . . . . . . . 109

14.1 Possible states for n = 3 . . . . . . . . . . . . . . . . . . . . . 12114.2 Shell radius and electron energy . . . . . . . . . . . . . . . . . 127

15.1 Quantum defect: Lithium . . . . . . . . . . . . . . . . . . . . 132

17.1 The normal Zeeman effect: frequencies . . . . . . . . . . . . . 15017.2 Sodium Zeeman effect: energy levels . . . . . . . . . . . . . . 15517.3 Hydrogen Zeeman effect . . . . . . . . . . . . . . . . . . . . . 156

165

modern physics - 物理系

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