QUANTUM PHYSICS Chapter 15. Chap 15 Quantum Physics Physics 2 15-1 Blackbody Radiation, Planck Hypothesis 15-4 Bohr’s Theory of Hydrogen Atom 15-3 Compton.

QUANTUM PHYSICSQUANTUM PHYSICS

Chapter 15Chapter 15

Chap 15 Quantum Physics

Physics

2

15-1 Blackbody Radiation, Planck Hypothesis

15-4 Bohr’s Theory of Hydrogen Atom

15-3 Compton Effect

15-2 Photoelectric Effect, Wave-particle Duality of Light

15-0 Basic Requirements

Chapter Index

*15-5 Franck-Hertz Experiment

15-6 de Broglie Matter Wave, Wave-particle Duality of Particles


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15-8 Introduction to Quantum Mechanics

15-9 Introduction to Quantum Mechanics of Hydrogen Atom

Chapter Index

*15-11 Laser*15-12 Semiconductor*15-13 Superconductivity

*15-10 Electron Distributions of Multi-electron Atoms

15-7 Uncertainty Principle


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1.Understand experimental laws of

thermal radiation ： Stephan-Boltzmann

law and Wein displacement law, and

difficulties of classical physics theory in

explanation of energy-frequency

distribution of the thermal radiation.

Understand Planck quantum hypothesis



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3. Understand experimental laws of

Compton effect, and its explanation by photon.

Understand wave-particle duality of light.

2. Understand difficulties of classic physics

theory in explanation of experimental

discoveries of photoelectronic effect.

Understand Einstein photon hypothesis, grasp

Einstein equation



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5. Understand de Broglie hypothesis and

electron diffraction experiment and wave-

particle duality of particles; Understand the

relation between physical quantities (wave-

length, frequency) describing wave property

and ones (energy, momentum) describing

particle property.

4. Understand experimental results of

Hydrogin atom spectra, and Bohr’s theory



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7. Understand wave function and its

statistical explanation. Understand 1-

dimension stationary Schrodinger equation,

and the quantum mechanical method deal

with 1 dimensional infinity potential well

etc.

6. Understand 1-dimension coordinate momentum uncertainty principle



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The foundations of quantum mechanics were established during the first half of the twentieth century by Niels Bohr, Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli, David Hilbert, and others.


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In the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics.

In the summer of 1925, Bohr and Heisenberg published results that closed the "Old Quantum Theory". Light quanta came to be called photons (1926).

Quantum physics emerged, its wider acceptance was at the Fifth Solvay Conference in 1927.


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The study of electromagnetic waves such as light was the other exemplar that led to quantum mechanics

M. Planck, in 1900, found that the energy of waves could be described as consisting of small packets or quanta, A. Einstein further developed this idea to show that an EM wave could be described as a particle - the photon - with a discrete quanta of energy that was dependent on its frequency


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1. Thermal Radiation(1) Fundamental concepts and basic laws

(1a) Monochromatic radiant

emittance: the power of electro-

magnetic radiation whose

frequency around (or

wavelength ) per unit area and

unit time radiated by a surface.

)(TM 3mW -

12 HzmW - -)(TM


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(2) Radiation emittance

power emitted from a surface per unit time

and unit area

0

d)()( TMTM

0

d)()( TMTM


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14

0 2 4 6 8 10 12

Hz10/ 142

12

10

4

6

8

))/(( 128 HzmW10 TMSUN))/(( 129 HzmW10 -TMTi

K 800 5T

visible

SUN

Ti

Monochroma-tic radiation emittance of Sun and Ti


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15

Incident absorption

Reflection transmission

(3) Monochromatic absorption ratio and reflection ratio

monochromatic absorption ratio (T) ：

The ratio of absorbed energy to the incident

energy between wavelength and

d


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16

For opaque object (T ) + r(T )=1

the ratio of reflected energy to the incident

energy between wavelength and

monochromatic reflection ratio r(T ):

d

Incident absorption

Reflection transmission


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17

(4) Black body

Blackbody is an

idealized model

An idealized physics object whose absorption

ratio equals 1, i.e., it absorbs all incident EM

radiation, regardless

of its frequency


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18

For a body of any arbitrary material, emitting

and absorbing thermal EM radiation in

thermodynamic equilibrium, the ratio of its

emissive power to its dimensionless coefficient

of absorption is equal to a universal function

only of radiative wavelength and temperature,

the perfect black-body emissive power

(5) Kirchhoff’s Law


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In other words, for a body of any arbitrary

material, emitting and absorbing thermal EM

radiation in thermodynamic equilibrium, the

ratio of M(T ) to (T) equals to MB( ,T)

under the same temperature T

),()(

)(B TM

T

TM


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21

0 1 000 2 000

0.5

)mW10/()( 314 TM

nm/

Visible Region

3 000 K

6 000 K

m

Exp. Curve

1.0

2. Experimental Observations of Blackbody Radiation


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1. Stephan-Boltzmann Law

4

0d)()( TTMTM

428 Km W10670.5 Stephan-Boltzmann const.

Total Radiation Emittance

where

0 1 000 2 000

1.0

)mW10/()( 314 TM

nm/

Visible region

3 000 K

6 000 K

m

0.5


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2. Wien’s Displacement Law

bT m

Km 10898.2 3 bConst.

Peak wave length

0 1 000 2 000

1.0

)mW10/()( 314 TM

nm/

Visible

region

3 000 K

6 000 K

m


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nm 890 9nm293

10898.2 3

1m

T

b

Solu: (1) From Wien’s displacement law

E.g-1 (1) Suppose a blackbody with

temperature T= , what is the wave-

length of its monochromatic peak ？ (2) the

monochromatic emittance peak wave

length , estimate the surface

temperature of the sun; (3) 上 what is the

ratio of above two ？

C20

nm 483m


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541212 1076.1)()()( TTTMTM

（ 2 ）

（ 3 ） From Stephan-Boltzmann law

K 000 6K10483

10898.29

3

m2

b

T

From Wien displacement law


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3. Rayleigh-Jeans formula Failures of classical physics

)HzmW10/()( -1-29 TM

0 1 2 3

6

Hz10/ 14

2

4

Rayleigh-Jeans

kTc

TM2

2π2)(

Rayleigh-Jeans

Violet Catastrophy

Exp. Curve

***

*

** *

****

**

* * *

T = 2 000 K


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27

M. Planck (1858 - 1947)

German theoretical physicist and the founder of quantum mechanics and one of the most important physicists of the 20th century.His talk under the title “On the Law of Distribution of Energy in the Normal Spectrum” *in 1900, was regarded as the “birthday of quantum theory” (by M. Laue)

* M. Planck, On the Law of Distribution of Energy in the Normal Spectrum, Ann. der Physik, Vol. 4, 1901, p. 553 ff.


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4. Planck’s hypothesis and blackbody radiation formula

1e

dπ2d)(

/

3

2

kThν

c

hTM

(1) Planck’s blackbody radiation formula

sJ1063.6 34 h

Planck’s Constant


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0 1 2 3

6

Hz 10/ 14

)HzmW10/()( 1-29 TM

Reighley - Jeans

2

4

Planck’s formula

Exp. Data

***

*

** *

****

**

* * *

Exp. Data vs. Planck theoretical curve

T = 2 000 K


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The vibration modes of molecules and atoms in

blackbody can be viewed as harmonic oscillators

(HO). The energy states of these HOs are

discrete, their energies are integer of a minimum

energy, i.e., , 2 , 3, … n, is called energy

quanta, n is quantum number

2. Planck’s quantum hypothesis

),3,2,1( nnhε

Planck quantum hypothesis is the milestone

of quantum mechanics


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(2) when quantum number increases from

to ， how much does the amplitude

change？

n

1n

E.g-2 Suppose a tuning fork mass m = 0.05 kg ，frequency , amplitude

.

mm 0.1AHz 480(1) quantum number of vibration；

Solu: (1)

J 227.0)π2(2

1

2

1 2222 AmAmE


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nhE 291013.7 h

En

energy J1018.3 31h

（ 2）

m

nh

m

EA

2222

π2π2

nhE

nm

hAA d

π2d2

2


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2

A

n

nA

1n

m1001.7 34A

Macroscopically, the effect of energy

quantization is not obvious, namely, the

energy of macroscopic object is continuous


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1. Photoelectric Effect and Phenomenon

VA

(1) Experimental Setup and Phenomenon


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35

(2) Discoveries

(2a) Current linearly proportional to the

intensity.

1I2I

i

m1im2i

o0U U

12 II


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(2b) threshold frequency

0

Threshold frequency depends on type of metal, but not on intensity

For a given metal, electrons only emitted if frequency of incident light exceeds a threshold0. 0 is called threshold frequency


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Stopping voltage linearly related incident light frequency

(2d) Current appears with no delay

Applied reverse voltage that

makes zero current is so-

called stopping voltage ,

different metal has different

0U

0U

(2c) Stopping Voltage 0U

0U

0

sC nZ tP

O


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For very low intensities, expect a time lag

between light exposure and emission, while

electrons absorb enough energy to escape

from material

(3) Failures of Classical Theory

Threshold frequency

No time delay

Electrons should be emitted whatever the frequency ν of the light, so long as electric field E is sufficiently large


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2. “Photon”, Einstein Equation

(1) “light quanta” hypothesis

hε

Light comes in chunks (composed of

particle-like “photon”), each light quanta

has energy

(2) Einstein photoelectric equation

Wmh 2

2

1v Escape work

depends on material


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Theoretical Explanation:

Approximate escape work value of different metals (in eV)

Na Al Zn Cu Ag Pt

2.46 4.08 4.31 4.70 4.73 6.35

the greater the intensity, the more photons,

the more photo-electrons, and hence the larger

current ( )0


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Stopping voltage

VA

Applied reverse

stopping voltage

stops electrons

0U

20 2

1vmeU


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42

Einstein’s theory successfully explained the

photoelectric effect and won 1921 Nobel prize

of physics (not for relativity)!

0hW hW0thrshold frequency

Threshold frequency: 0

No lag: photon energy ( ) is

absorbed by a electron and the electron

then emits without time delay

h 0


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43

WeUh 0

e

W

e

hU 0

(3) Measurement of Planck const.

Wmh 2

2

1v

ehU 0 eU

h

0

0U

0

Stopping voltage vs. frequency

O


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44

E.g.-1. Consider a thin circular plane with

radius , 1.0 m far from an 1W

power light source. The light source emits

monochromatic light with wave length 589

nm. Suppose the energy goes off all directions

equally. Calculate the number of photons on

the plate per unit time.

m100.1 3


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111 s 104.7 hc

E

h

EN

172

sJ 105.2π4

r

SPE

Solution:

2623 m10π)m100.1(π S


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46

Photomultiplier

Amplifier

Controller

light

Demo. of photo-relay

3. Applications in Modern Technology

Photo-relay circuit, Automatic

counter, measuring device etc.


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48

4. Wave-particle Duality of Light

20

222 EcpE Relativistic energy-momentum relation

hE (2) particle: , photo-electric effect etc.

(1) wave:diffraction and interference

pcEE ,00 photon


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49

hE

h

p Particle

characterWave

character

h

c

h

c

Ep

pcEE ,00 photon


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50

Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles. He found that peak in scattered radiation () shifts to longer wavelength than source (0), i.e., > 0. Amount depends on θ, but not on the target material. A.H. Compton, Phys. Rev. 22 (1923) 409


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1. Experimental Asparatus


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52

2. Experimental Results 0

45

90

135

Relative Intensity

I

0

0

(1) shift in wave

length

depends on

0

(2) is indep.

of targets


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3. Difficulties of Classical Theory

According to the classical picture: oscillating electromagnetic field causes oscillations in positions of charged particles, which re-radiate in all directions at same frequency and wavelength as incident radiation.

Change in wavelength of scattered light is completely unexpected classically!


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54

(1) Physical model

Incident photon (X-ray or -ray) with higher energy

4. Quantum Explanations

eV10~10 54hE

0

00 vx

yPhoton

Electron x

y

Electron

Photon


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55

electron with large bouncing velocity, use

relativistic mechanics

electron energy of thermal motion , so we can

treat electron as at-rest approximately

h

electrons near the surface of solids with weak binding, quasi-free

0

00 vx

yphton

electron x

y

electron

photon


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56

(1) “billiard ball” collides between particles of light

(X-ray photons) and weak-binding electrons in the

material, part of energy is transported to electron,

leads to the energy decrease of scattered photon,

hence the frequency, wavelength increases

(2) Qualitative Analysis

(2) photon collides with tight-binding electron,

without significant lost of energy, results in the

same wave-length in scattered light


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57

(3) Quantitative Calculation

x

y

00 e

c

h e

c

h

v

m

e

0e

v

mec

he

c

h

0

0

Momentum conservation

2200 mchcmhv

Energy conservation

cos2

20

2

2

22

2

20

222

c

h

c

h

c

hm v


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cos2

20

2

2

22

2

20

222

c

h

c

h

c

hm v

)(2)cos1(2)1( 02

00242

02

242 hcmhcm

ccm

v

)cos1(00

cm

hcc

2/1220 )/1( cmm v

0


Physics

59

Compton Wavelength

m 1043.2 12

0C

cm

h

2sin

2)cos1( 2

00

cm

h

cm

h

Compton Formula

)cos1()cos1( C0

cm

h


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Scattered light wave length change depends only on

0,0

Cmax 2)(,π

scattered photon energy decrease

00 ,

(4) Conclusions

x

y

00 e

c

h e

c

h

v

m

e

0e


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61

Eg-1. X-ray with wavelength

elastically collides with a electron at rest,

observing along the direction with respect

to scattering angle,

m 101.00 -100

90

(2) Kinetic energy bouncing electron gets?

(3) Energy that photon loses during collision?

(1) Change of the scattered wavelength ?


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62

(1) )cos1(C CC )90cos1(

eV 295)1( 0

00

20

2k

hchchc

cmmcE

m1043.2 12(2) bouncing electron kinetic energy

(3) Energy photon loses＝

Solution

kE


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63

(1) Experimental discoveries of atomic

hydrogen spectrum

1. Review of Modern View of Atomic Hydrogen Structure


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64

Light Bulb

Hydrogen Lamp

Quantized, not continuous


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65

(1) Experimental discoveries of atomic

hydrogen spectrum

Joseph Balmer (1885) first noticed that the

frequency of visible lines in the H atom

spectrum could be reproduced by:

,5,4,3,nm2

46.36522

2

nn

n


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66

Johann Rydberg (1890) extends the Balmer

model by finding more emission lines outside

the visible region of the spectrum:

)11

(1

22

if nnR

wave number

Rydberg const. 17 m10097.1 R

,,4,3,2,1 fn ,3,2,1 fffi nnnn


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Lyman ,3,2,)1

11

(1

22 nn

R

Ultraviolet

,4,3,)1

21

(1

22 nn

R

Balmer

Visible


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68

,5,4,)1

3

1(

122

nn

R

Paschen

,6,5,)1

41

(1

22 nn

R

Brackett

,7,6,)1

51

(1

22 nn

R

Pfund

,8,7,)1

61

(1

22 nn

R

Humphrey

Infrared


Physics

69

Balmer spectrum of H atom

656.3 nm

486.1 nm

434.1 nm

410.2 nm

364.6 nm


Physics

Hydrogen atom spectra

Visible lines in H atom spectrum are called the BALMER series.

En

erg

y

Ultra VioletLyman

InfraredPaschen

VisibleBalmer

65

3

2

1

4

n


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71

(2) Rutherford’s model of atomic structure

1897, J. J. Thomson discovered electron

1904, J. J. Thomson proposed“plum

pudding model” of atomic structure

the atom is composed of electrons surrounded by a soup of positive charge to balance the electrons' negative charges, like negatively-charged "plums" surrounded by positively-charged "pudding".


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72

Ernest Rutherford (1871 – 1937)

New Zealand-born British chemist

and physicist who became known as

the father of nuclear physics. He

discovered the concept of radioactive

half-life, differentiated and named α,

β radiation.

He was awarded Nobel prize of Chemistry in 1908 "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances"


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In 1911, he proposed the Rutherford model of the atom, through his gold foil experiment. He discovered and named the proton. This led to the first experiment to split the nucleus in a fully controlled manner.

He was honoured by being interred with the greatest scientists of the United Kingdom, near Sir Isaac Newton’s tomb in Westminster Abbey.

The chemical element rutherfordium (element 104) was named after him in 1997.


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Rutherford atomic model (Planetary model)

the atom is made up of a central charge (this is

the modern atomic nucleus, though Rutherford

did not use the term "nucleus" in his paper)

surrounded by a cloud of orbiting electrons.


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2. Bohr’s Theory of Atomic Hydrogen

(1) Failures of Classical Atomic Models

According to the classical electro-

magnetic theory, electrons rotate

around atomic nucleus, accelerated

electrons radiate electro-magnetic

wave and hence lose energy


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76

electrons orbiting a nucleus – the laws of classical

mechanics, predict that the electron will release

electromagnetic radiation while orbiting a nucleus.

Hence would lose energy, it would gradually spiral

inwards, collapsing into the nucleus.

e

e


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77

v

F

r e

e

As the electron spirals inward,

the emission would gradually

increase in frequency as the orbit

got smaller and faster. This

would produce a continuous

smear, in frequency, of

electromagnetic radiation, one

should observe continuous light

spectra


Physics

Niels Bohr (1885 - 1962)

Danish theoretical physicist,

one of the founding fathers of

quantum mechanics.

He uses the emission spectrum

of hydrogen to develop a

quantum model for H atom

and explains H atom spectrum

1922 Nobel Prize in physics


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In 1913, N. Bohr uses the emission spectrum

of hydrogen to propose a quantum model for

H atom, with the following three assumptions

(2) Bohr’s Theory of H Atom

(b) Frequency condition

(a) Stationary hypothesis

(c) Quantization condition


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Electrons do not radiate EM wave if they

are on some specific circular trajectories,

they can keep staying on those stable

states, i.e., so-called stationary states

Energies corresponding

to stationary states are

E1, E2… , E1 < E2< E3

+E1

E3

(a) Stationary hypothesis


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81

fi EEh

Ef

Ei

emmision absorption

π2

hnrmL v

,3,2,1n

(b) Frequency condition

(c) Quantization condition

Principal quantum number


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82

(3) Calculate H-atom energy and orbital radii

(a) Orbital radii

π2

hnrm nn vQuantization

condition

n

n

n rm

r

e 2

20

2

π4

v

Classical mechanics

21

22

20

πnrn

me

hrn

),3,2,1( n

+

rn


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83

, Bohr radius m 1029.5π

112

20

1

me

hr

1n

21

22

20

πnrn

me

hrn

),3,2,1( n

(b) Energy

nnn r

emE

0

22

π42

1

v

The nth orbital electron’s energy：


Physics

84

21

2220

4 1

8 n

E

nh

meEn

(Ionized energy)22

0

4

1 8 h

meE

eV 6.13

ground state energy )1( n

21 nEEn

Excited state energy )1( n


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85

Energy level transition and Spectrum of H-atom

Lyman

Balmer

Brackett

Paschen

-13.6 eV

-3.40 eV

-1.51 eV

-0.85 eV

-0.54 eV 0

n=1

n=2

n=3

n=4n=5n=


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(4) Explanations of Bohr’s Theory on H-atom Spectrum

fi EEh

fiif

nnnnch

me

c ,)

11(

8

12232

0

4

2220

4 1

8 nh

meEn

17 m10097.1 (Rydberg const.)Rch

me32

0

4

8


Physics

87

(a) Correctly predicted the existence of atom

energy level and energy quantization

3. The Successes and Failures of Bohr Theory(1) Successes

(c) Correctly explained H-atom and H-like-atom

spectrum

(b) Correctly proposed the concepts of stationary

state and angular momentum quantization.


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88

(c) Can not deal with the widths, intensity etc. of

spectrum.

(d) Half classical half quantum theory: on one

hand microscopic particles have classical

properties, on the other hand, quantum nature

(2) Failures

(a) Does not work for multi-electron atoms

(b) Microscopic particles do not have certain trajectory


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89

1. de Broglie Hypothesis

In 1923, de Broglie postulated that ordinary matter can have wave-like properties, with the wavelength λ related to momentum p in the same way as for light

/

2

hmP

hmcE

vWave natureParticle nature


Physics

90

French physicist and a Nobel laureate in 1929. His 1924 Recherches sur la théorie des quanta (“Research on the Theory of the Quanta”), introduced his theory of electron waves, thus set the basis of wave mechanics, uniting the physics of energy (wave) and matter (particle).

L. de Broglie (1892 – 1987)


Physics

91

vm

h

p

h de Broglie relation

de Broglie wave or Matter wave

Note

0mmc vif then

(1)if thencv 0mm

h

mc

h

E 2


Physics

92

2. de Broglie wavelength of a macroscopic

object is too tiny to be measured, this is why a

macroscopic object behaves particle-like nature

E.g.-1 In a beam of electron, the kinetic

energy of electron is , Calculate its de

Broglie wavelength.

eV200

20k 2

1, vv mEc

0

k2

m

EvSolution:


Physics

93

1-6131

19

sm 104.8sm101.9

106.12002

v

nm 1067.8 2

nm 104.8101.9

1063.6631

34

0

vm

hcv

Roughly the order of X-ray wavelength


Physics

94

E.g.-2 Derive quantization condition of angular

momentum in Bohr’s theory of hydrogen atom

nr π2 ,4,3,2,1n

Solution: Consider a string

with two ends fixed, if its

length equals wave-length

then a stable standing wave

can form

rπ2to form a circle


Physics

95

nhrm vπ2

vm

h

Electron’s de Broglie wavelength

π2

hnrmL v

We get quantization condition of angular momentum


Physics

96

2. Experimental confirmation of de Broglie matter wave

Quantum Corral: 48 iron atoms form a circular quantum corral (radius 7.13nm) on the Cu (111) surface


Physics

Davisson G.P. Thomson

C. J. Davisson, "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928)

The Davisson-Germer experiment: scattering a beam of electrons from a Ni crystal. Davisson got the 1937 Nobel prize.

At fixed accelerating voltage (fixed electron energy) find a pattern of sharp reflected beams from the crystal

At fixed angle, find sharp peaks in intensity as a function of electron energy

G.P. Thomson performed similar interference experiments with thin-film samples

θi

θi

ELECTRON DIFFRACTIONELECTRON DIFFRACTIONThe Davisson-Germer experiment (1927)The Davisson-Germer experiment (1927)


Physics

98

2. Experimental Confirmation of de Broglie Wave(1) Davisson-Germer Diffraction Exp.

I

35 54 75 V/U

50

Current vs. Acceleration voltage, 50

检测器

Electron beam Scatter

ing beam

Ni-crystal diffraction

M

UK

G

Electron gun


Physics

99

The exp. results of single crystal diffraction by

electron beam agree with “Bragg’s law” in X-

ray diffraction Interference condition:

kd

2cos

2sin2

kd sin

50,1 k

. . . . . . . .

. . . . . . . .

. . . . . . . .

d2

2

2

2sin

d


Physics

100

m1015.2 10dFor Ni crystal

m1065.1sin 10 d

m1067.12

10

kee

Em

h

m

h

v

Wavelength of electron wave

emUkhd

2

1sin


Physics

101

k777.0sin

when , agree well

with experimental results.

51777.0arcsin1 k

emUd

kh

2

1sin

emUkhd

2

1sin


Physics

102

U MD

P

Diffraction of electron beam from polycrystalline foil

K

(2) G. P. Thomson electron diffraction exp.

Electron beam from polycrystalline foil

generates diffraction fringe similar to the

X-ray diffraction fringe


Physics

103

3. Applications

Scanning Tunneling Microscopy (STM) Developed by Gerd Binnig and Heinrich Rohrer at the IBM Zurich Research Laboratory in 1982.

The two shared half of the 1986 Nobel Prize in physics for developing STM.

Binnig Rohrer


Physics

104

Classical particle undividable unity, with certain momentum and trajectory 　Classical wave periodic spatial distribution

of some physical quantity, with property of

interference

Wave-particle Duality United wave and

particle natures within one unity

4. Statistical interpretation of de Broglie wave


Physics

105

Single particle randomly appears, but large number of

particles show a statistical regularity. The probability

that a particle appear at different position is different

(1) Explanation by particle nature

Electron beam

slit

single-slit diffraction


Physics

106

The more intense the electrons at some

place, the higher intensity of wave; or

vice versa.

(2) Explanation by wave nature

Electron beam

slit

single-slit diffraction


Physics

107

At some place the intensity of de Broglie

wave proportioned to the probability that

the particle appears around that place

(3) Statistical Interpretation

M. Born (1926) pointed out , de Broglie wave is probability wave.


Physics

108

1. Heisenberg Uncertainty principle of Coordinate and Momentum

b sin

the 1st order min.

diffraction angle

Position uncertainty

of the electron bx

Electron diffraction

y

x

hp

hp b

Electron Single-slit Diffraction Exp.

o


Physics

109

W. Heisenberg (1901 – 1976)

German theoretical physicist, who

made foundational contributions to

quantum mechanics and proposed the

uncertainty principle (1927). He also

made important contributions to

nuclear physics, quantum field

theory, and particle physics.

Awarded the 1932 Nobel Prize in Physics for the creation of quantum mechanics, and its application especially to the discovery of the allotropic forms of hydrogen


Physics

110

p

hb

hpx

hpx x

bpppx sin

x-direction momentum uncertainty after passing the slit

b sin

y

x

hp

hp b

o


Physics

111

Heisenberg proposed uncertainty principle in 1927

Microscopic particles can not be described

by simultaneous coordinate and momentum

hpy y hpx x

hpz z

Uncertainty Relation

hpx x the 2nd order diffraction


Physics

112

(2) this uncertainty deeply roots in the wave-particle

duality, which is the fundamental property of particles

(3) for macroscopic particles, since is extremely

small, , hence in macroscopic limit, the

momentum and position can be simultaneously

determined

h0 xpx

(1) a fundamental limit on the accuracy with which

certain pairs of physical properties of a particle, such as

position and momentum, can be simultaneously known

Implications


Physics

113

For microscopic particles, h can not be

ignored and x px can not be simultaneously

determined. To describe their motion one has

to borrow the concept of probability. In

quantum mechanics, wave function is used to

describe particle’s states.

The uncertainty principle is one of the foundational postulates of quantum mechanics.


Physics

114

1smkg 2 vmpSolution: Bullet’s momentum

14 smkg 102%01.0 ppUncertainty of momentum

%01.01sm 200 E.g.-1. The mass of a bullet is 10 g, speed

. Momentum

uncertainty is of its momentum (this

is good enough in macroscopic world),

What is the position uncertainty of the

bullet?


Physics

115

m 103.3m102

1063.6 304

34

p

hx

Uncertain range of the position

-1sm 200 E.g.-2. An electron’s speed is . The

degree of momentum uncertainty is 0.01%

of the momentum, what is the uncertainty

of position of the electron?

14 smkg102%01.0 pp


Physics

116

128 smkg108.1 p

Solution: electron’s momentum

131 smkg 200109.1 vmp

132 smkg108.1%01.0 pp

Uncertain range of the momentum

m107.3m108.1

1063.6 232

34

p

hx

Uncertain range of the position


Physics

117

Due to the wave-particle duality of

microscopic particle, one can not determine

its position and momentum spontaneously,

the classical way of description of its states

breaks down, we use wave function

1. Wave Function and Its Statistical

Explanation

(1) Wave Function


Physics

118

(1a) Classical wave and wave function

)(π2cos),( 0 x

tEtxE

)(π2cos),( 0 x

tHtxH em wave

)(π2cos),(

xtAtxy mechanical wave

]eRe[),()(π2i

x

tAtxy

classical wave is a real function


Physics

119

(1b) QM wave function (complex function )

),,,( tzyxΨWave function that descibe the motion of the microscopic particle

p

h

h

E ,Wave-particle duality of

microscopic particles

The energy and momentum of free particle are of

certain values, its de Broglie wave length and

frequency are invariant, so it is plane wave with

infinity wave train, the x-position of the particle is

fully uncertain due to the uncertainty principle


Physics

120

Free particle plane wave function

(2) The statistical interpretation of wave function

*2 Ψ

Probability Density: the probability that the

particle appears in unit (spatial) volume

Positive Real number

)(π2

i

0),(pxEt

h

etxΨ


Physics

121

Probability that the particle appears at some

moment in a volume element Vd

VΨVΨ dd *2 Ψ

Hence de Broglie wave (or matter wave) is a

probability wave, it is very different with

electromagnetic wave


Physics

122

Standard Condition

Wave function is single-valued, real, finite

function

1d2

VΨNormalization

Condition(Bound State)

At some moment the probability one finds the particle in entire space is


Physics

123

Austrian theoretical physicist

Proposed the famous wave

equation with his name,

founded wave mechanics, and

its approximation methods.

1933 Nobel Prize for Physics

(with P. Dirac)

Erwin Schrodinger ， 1887 - 1961


Physics

124

2. Schrodinger Equation(1) free particle Schrodinger equation

Free particle plane wave function

taking 2nd order partial derivative with respect to x and 1st order partial derivative with respect to t

)(π2

i

0),(pxEt

h

etxΨ


Physics

125

One gets

Ψh

p

x

Ψ2

22

2

2 π4

EΨht

Ψ π2i

Free particle 　 )c(v kEE k2 2mEp

tΨh

xΨ

mh

π2i

π8 2

2

2

2

1-dimension free particle time-dependent Schrodinger equation


Physics

126

t

ΨhΨtxV

x

Ψ

m

h

π2

i),(π8 p2

2

2

2

1-dimensional time-dependent Schrodinger equation

pk VEE (2) Particle in potential field with potential energy :

pV

(3) particle in stationary potential

p

2

2V

m

pE time-indep.)(p xV


Physics

127

hpxEtetxΨ /)(π2i0),(

hEthpx ee /π2i/π2i0

)()( tx hpxex /π2i

0)(

0)()(π8

d

dp2

2

2

2

xVEh

m

x

1-dimensional stationary Schrodinger equation in any potential field


Physics

128

0)(π8

p2

2

2

2

2

2

2

2

VEh

m

zyx

Stationary Schrodinger equation in 3-dimensional potential field

Lapalce operator 2

2

2

2

2

22

zyx

Stationary wave function ),,( zyx

0)(π8

p2

22 VE

h

m


Physics

129

e.g., stationary Schrodinger equation for hydrogen atom

20

2

p π4 rε

eV

(1)E is time-independent

(2) is time-independent 2

Properties of stationary wave function

0)π4

(π8

20

2

2

22

rε

eE

h

m


Physics

130

zyx

,,(2) 和 continuous

),,( zyx(3) is finite, single-valued

wave function single-valued, finite, continuous

1ddd,,

2 zyxzyx(1) normalization


Physics

131

3. 1-dim. Potential Well

Particle potential energy satisfies boundary condition

pV

pVaxxV

ax

,0,

0,0

p

(1)Simplified model for free electron gas

model of metal in solid physics

(2)Demonstrate QM basic concepts and

principles with simple math


Physics

132

),0(,0 axx

axxE ,0,p

2

2π8

h

mEk

axE 0,0p

0π8

dd

2

2

2

2

h

mEx

0dd 2

2

2

k

x

pE

a xo


Physics

133

kxBkxAx cossin)(

0dd 2

2

2

k

x

Wave function single-value, finite, and continuous

0,0,0 Bx

kxAx sin)(

pE

a xo


Physics

134

π,0sin nkaka

2

2π8

h

mEk 2

22

8ma

hnE

,3,2,1,π

na

nk quantum number

0sin, kaAax

0sin ka

pE

a xo


Physics

135

kxAx sin)(

xa

nAx

πsin)(

,3,2,1,π

na

nk

Normalization 1dd0

*2

xx

a

1dπ

sin0

22 xxa

nA

a

aA

2

pE

a xo


Physics

136

)0(,π

sin2

)( axxa

n

ax

kxAx sin)(

a

nk

π

aA

2

hence

0π8

d

d2

2

2

2

h

mE

x wave equation

pE

a xo


Physics

137

xa

n

ax

πsin

2)( 22 Prob. density

2

22

8ma

hnEn Energy

)0(,π

sin2

axxa

n

a

)(x),0(,0 axx

wave function pE

a xo


Physics

138

1. energy quantization

Discussions ：

g.s. Energy )1(,8 2

2

1 nma

hE

2

22

8ma

hnEn Energy

excited state ),3,2(,8

12

2

22 nEn

ma

hnEn

the particle’s energy in 1-dim. infinity square well is quantized.

pE

a xo


Physics

139

(2) the prob. density that particle appears

in the well is different

Prob. density )π

(sin2

)( 22x

a

n

ax

xa

n

ax

πsin

2)( Wave function

e.g., when n =1, the maximum probability is

at the place x = a /2


Physics

140

(3) wave function is standing wave, the nodes

locate at the wall, the No. of valley equals

quantum number n

0x a1n

2n

3n

4n

n 2n

xa

nAx

πsin)( x

a

n

ax

πsin

2)( 22

0p Ea

16E1

9E1

4E1

E1

0x


Physics

141

4. 1-dim. Square Well, Tunneling Effect

)(p xVaxx ,0,0

axV 0,p0

1-dim. Square Well

0pVE Particle’s Energy

0pV

)(p xV

ao x


Physics

142

When particle’s energy E < Vp0 , the region

x > a is classically forbidden, however in

quantum mechanics, particle can penitrate

in the region with a non-zero probability

Tunneling Effect

Wave

functions in

different

regions

1 2 3)(x

a xo


Physics

143

STM (1981)

Scanning Tunneling

Microscopy

AFM (1986) Atom

Force Microscopy

Applications

Xenon on Nickel

Single atom lithography


Physics

Iron on Copper

Quantum Corrals

Imaging the standing wave created by interaction of species


Physics

145

1. Schrodinger Equation of Hydrogen Atom

Potential energy of electron in H-atom

Stationary Schrodinger equation:

0)π4

(π8

0

2

2

22

rε

eE

h

m

rε

eV

0

2

p π4


Physics

Spherical CoordinatesTransform to spherical polar coordinates because of the radial symmetry

2 2 2

1

1

sin cos

sin sin

cos

cos

tan

x r

y r

z r

r x y z

z

ry

x

Polar angle

Azimuthal angle


Physics

147

0)π4

(π8

sin

1)(sin

sin

1)(

1

0

2

2

2

2

2

2222

2

rε

eE

h

m

rrrr

rr

In Spherical coordinates:

)()()(),,( ΦΘrRr

Separable solution, let


Physics

148

We get

0d

d 22

2

ΦmΦ

l

)1()d

d(sin

d

d

sin

1

sin 2

2

llΘ

Θ

ml

)1()π4

(π8

)d

d(

d

d1

0

2

2

222 ll

rε

eE

h

mr

r

Rr

rR


Physics

149

2. Quantization condition and quantum

number

12

1E

nEn n =1,2,3,... Principal quantum

number

Solve Schrodinger equation we get the following

quantum number and quantization properties:

(1) Energy quantization and principal quantum number

)eV( 6.138 22

0

4

1 h

meE


Physics

150

π2)1(

hllL

(2) Angular momentum quantization and angular quantum number

Angular momentum ：

π220

hLL

Orbital angular

quantum number)1(210 nl ，，，，

E.g. ， n =2 ， = 0,1 corresponds

to

l


Physics

151

In applied magnetic field, angular momentum L

can only take some specific directions,

projection of L along magnetic field satisfies

llz mh

mL π2

(3) Angular momentum spatial quantization and magnetic quantum number

π2/h reduced Planck const.

magnetic quantum number

lml ,,2,1,0


Physics

152

L

zZL

z

o

ħ

ħ

2L

2π2

2π2

)1( hh

llL

magnetic quantum number ml =0, 1 and

π2,

π2,0

hhLz

e.g., when 1l


Physics

153

(4) Spin and spin quantum number

sz mS

Spin angular momentum takes only two

components along applied magnetic field:

)1( ssSSpin angular momentum

2

1sm

where spin quantum number 2

3S

2

1s

ms spin magnetic quantum number


Physics

154

SzS

Spin angular momentum and spin magnetic quantum number of electron

2/2

1 zs Sm

o

z

2

1

2

1

2

3S

Sz

2

1sm

2

1sm


Physics

155

(5) Summary

The states of electron in hydrogen atom can

be represented by 4 quantum numbers (qn.),

(n, l ,ml , ms)

Angular qn. l determines orbital angular momentum

Magnetic qn. ml determines direction of

orbital angular momentum Spin qn. ms determines direction of spin

angular momentum

Principal qn. n determines energy


Physics

156

3. Ground state radial wave function and distribution probability(1) Ground state energy

Ground state n = 1 l = 0

Radial wave function equation:

0)π4

(π8

)d

d(

d

d1

0

2

2

222

rε

eE

h

mr

r

Rr

rR

1/e rrCR solution


Physics

157

where )π8/( 2221 mEhr

Substitute into 02

π4

π8

12

0

22

r

rhε

me

get nm 9 052.0π 2

20

1 me

hεr

eV 6.13π8 2

12

2

mr

hE


Physics

158

(2) Ground state radial wave function

1/e rrCR the probability that electron appears in volume element dV:

dddsind 22222rrΦΘRVΨ

let the prob. density along radial vector p, the

prob. that the electron appears in (r , r+dr)

ddsindd2π2

0

2π

0

22 ΦΘrrRrp


Physics

159

from normalization rrRrp dd 22

1dd 22

00

rrRrp

1de 2/22

0

1

rrC rr2/1

31

4

rC

1/

2/1

31

e4

)( rr

rrR

g.s. radial wave function is

1/e rrCR


Physics

160

p(r)

o

22)( rrp

r

(3) Probability Density Distribution of Electron

r1


Physics

161

LLight

AAmplification by

SStimulated

EEmission of

RRadiation


Physics

162

1. Spontaneous and stimulated radiations

(1) Spontaneous radiation

the process by which an atom in an excited state

with higher energy undergoes a (spontaneous)

transition to a state with a lower energy , e.g., the

ground state, and emits a photon, the frequency of

the radiation is determined by

2E

1E

h

EE 12


Physics

163

.

1E

2E

Before Radiation

.

。2E

1E

h

After Radiation

h

EE 12

Spontaneous Radiation


Physics

164

(2) Absorption of light

1E

2E hEE 12

the process by which an atom in a state with lower

energy , e.g., the ground state, absorb a photon

energy , spontaneously transit to a state with a

higher energy , and

h

After Absorption

。

.2E

1E

h

Before Absorption

.1E

2E

Excited Absorption


Physics

165

(3) Stimulated radiation

the process by which an atomic electron at energy

level , interacting with an electromagnetic wave

of a certain frequency may drop to a lower energy

level , transferring its energy to that field. A

photon created in this manner has the same phase,

frequency, polarization, and direction of travel as the

photons of the incident wave, and satisfies

2E

1E

12 EEh


Physics

166

.

1E

2E

.

。2E

1E

Before After

hh

h

Amplification of stimulated radiation

Stimulated Radiation

when a population inversion is present, the rate of stimulated emission exceeds that of absorption, results in a coherent amplification laser


Physics

167

2. The principle of laser

(1) Normal and inverse distribution of population

kTEi

iCN /e 2211 ENEN

kTEENN /)(21

21e/ 12 EE known

shows that the electron population

at lower energy level greater than that at

higher level, this is normal distribution

21 NN


Physics

168

is instead inverse distribution of

population, or simply population inversion12 NN

Normal

1E

2E . ... .。。。。。。。。。。。。。 1N

2N12 EE

Inversion

2E

1E

....... ........

。。。。。2N

1N12 EE

Population normal distribution and inversion


Physics

169

T. H. Maiman (U.S. physicist) made the

first functional ruby laser in sept., 1960

Energy level of ion Cr in Ruby laser

1E

2E.

。

。

Ground state

Metastable stateExcited state

3E

.


Physics

170

(2) Optical resonant cavity Formation of laser light

Light confined in the cavity reflect multiple times

producing standing waves for certain resonance

frequencies. When the standing wave condition is

satisfied the light is amplified, one obtains laser

standing wave condition2

kl


Physics

171

.. .

..

.. .. .

Laser beam

HRM

l

Demonstration of O.R.

PTM

Optical resonator


Physics

172

3. Laser

(1) Helium-Neon Gas Laser

Energy levels of He and Ne

Ground state

Metastable

He Ne

632.8 nm2

3

1

He-Ne Laser

HRMPTM

A K

PTM: partially transmissive mirror

HRM: highly reflectance mirror


Physics

173

HELIUM-NEON GAS LASER


Physics

174

(2) Ruby (CrAlO3) laser

Its active medium is ruby crystal rod, generates pulse laser with wavelength 694.3 nm.

。。

High reflectance mirror

Partialy transmissive mirrorRuby rod

。

Pulse

0U

U

Demo. of Ruby Laser

。


Physics

175

Rear Mirror

Adjustment Knobs

Safety Shutter Polarizer Assembly (optional)

CoolantBeamTube

AdjustmentKnob

OutputMirror

Beam

Beam Tube

HarmonicGenerator (optional)

Laser Cavity

PumpCavity

Flashlamps

Nd:YAGLaser Rod

Q-switch(optional)

Courtesy of Los Alamos National Laboratory

NEODYMIUM YAG LASER


Physics

176

4. Characteristics and Applications of Laser

(1) highly-directional, a laser collimator can reach accuracy of 16 nm/2.5 km.

(2) highly-monochromatic, better than ordinary light

1010

(3) focusing, laser light focuses 100 times better than ordinary light

(4) coherent, ordinary light source generates incoherent light, while laser light is highly coherent


Physics

177

Incandescent vs. Laser Light

1. Many wavelengths

2. Multidirectional

3. Incoherent

1. Monochromatic

2. Directional

3. Coherent


Physics

178

1. Energy Gap of Solids

ee+

s1

s2p2

Fully Separated Energy Levels of Two H-atom

ee+

s1

s2p2

BA


Physics

179

O

E

r

Two closed H-atom’s energy level split

s1s2

p2

O

E

rs1

s2

Six closed H-atom’s energy level split

O

E

r

Energy Band of Solids s2


Physics

180

quantum states

per energy level

)12(2 l

electrons per

energy level

)12(2 l

electrons per

energy band

Nl )12(2

Electron distribution of different energy bands in Na

s1

s2

p2

s3

p3

N2

N2

N6

N


Physics

181

Experiments show that:

The interval between the highest and the lowest

energy level in a energy band is less than the

order of , the number of atoms is of

order , hence the distance of the

neighboring energy levels is about

eV102 N319 mm10

eV 10eV/1010 17192


Physics

182

Energy band of crystals

E

gEForbi-dden band

gEForbi-dden band

Cond-uction band

Valence band (not full)

Empty band

Valence band (full)

Cond-uction band


Physics

183

Conductor Semiconductor Insulator

Resistancem)(Ω

Temp. Coeff.

F-band

V-band

48 10~10

Pos. +

Not full

84 10~10

Neg. -

Small

Full

208 10~10

Neg. -

Large

Full

Comparison between Conductor,

Semi-conductor and Insulator


Physics

GaAsZnS (Zinc Blende) Structure

4 Ga atoms at (0,0,0)+ FCC translations4 As atoms at (¼,¼,¼)+FCC translations

Bonding: covalent, partially ionic

SiliconDiamond Cubic Structure

4 atoms at (0,0,0)+ FCC translations4 atoms at (¼,¼,¼)+FCC translations

Bonding: covalent

Typical Semiconductors


Physics

185

2. Intrinsic and Extrinsic semi-conductor

(1) Intrinsic: pure, no dopants

gE

C-band

F-band

Full band

e

e

hole

electronNormal Bond in Ge

eG eG eG eG eG

eG eG eG eG

Electrons are excited, Holes appear e

eeG eG eG eG eG

eG eG eG eG


Physics

186

(2) Extrinsic semiconductor)

Electron type (n-type)

V-band

C-band

Donor level

Donor Level

iS

sA

e

Phosphorus atom are dopantSi atoms are hosts ，

sA

iS iS iS

iS iS iS iS

iS iS iS iS

iS iS iS iS

iS iS iSiS

iS iS iS iS


Physics

187

p-type semiconductor

eG eG eG eG

eG B eG

eG eG eG eG

Hole

Boron atom doping into Ge atom lattice

V band

C band

Acceptor level

Acceptor level


Physics

188

3. PN Junction

p n

UI

p n

UI

U

I

Current-Volt Characteristics of pn Junction


Physics

189

e eee

p n p n- -- -- -- -- -

+ ++ ++ ++ ++ +0xHole Electron

0U x0x

Voltage variation between p-layer and n-layer


Physics

190

4. Photovoltaic effect

P

ne

e

e

e

Light

eeγ

Photovoltaic effect is the creation of voltage or

electric current in pn upon exposure to light


Physics

191

4.00 4.20 4.40

0.150

0.100

0.050

0.000

K/T

)(R

1. The transition temperature of superconductor

***

cT : the critical temperature

around T=4.20K

risistance is ZERO


Physics

192

2. Major Properties of Superconductors

(1) Null resistance

When (critical electric flow)cc , IITT conductanceresistance 0

(2) Critical magnetic field

The critical point of applied magnetic fields that breaks the superconducting states


Physics

193

2

c0c )(1

T

THH

0c,K 0 HHT

T

H)(c TH

Super-

conductor

Normal

oCT(3) Meissner effect

t

SB

t

ΦlE

d

)(d

d

dd


Physics

194

0E 0d/d tB

cHH applied when 0inH

0inH

H

S

N

H

I


Physics

195

3. BCS Theory of Superconductivity

BCS Theory: proposed by Bardeen, Cooper,

and Schrieffer (BCS) in 1957, is the first

microscopic theory of superconductivity

since its discovery in 1911. Interestingly, this

theory is also used in nuclear physics to

describe the pairing interaction between

nucleons in an atomic nucleus.


Physics

196

BCS=Bardeen, Cooper, Schrieffer


Physics

197

deformation of local area

e

Normal location of Lattice

An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite "spin", to move into the region of higher positive charge density and to be correlated. A lot of such electron pairs overlap very strongly, forming a highly collective "condensate"


Physics

198


Physics

199

Phonon: a collective excitation in a periodic lattice

of atoms, such as solids. It represents an excited

state in the quantum mechanical quantization of

the modes of vibrations of elastic structures of

interacting particles.

The distance between two electrons is about

their spins and momenta are opposite, the total

momenta is zero.

m10 6

Cooper Pair: two electrons couple by exchanging

phonon, and form the coupled electron pair called

Copper pair


Physics

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4. The Perspectives of Superconductor

(1) Create strong magnetic field

(2) Energy & power industry, e.g., power storage etc.

(3) Magnetic levitated high-speed train

(4) Medical applications, e.g., nuclear magnetic resonance imaging

QUANTUM PHYSICS Chapter 15. Chap 15 Quantum Physics Physics 2 15-1 Blackbody Radiation, Planck Hypothesis 15-4 Bohr’s Theory of Hydrogen Atom 15-3 Compton.

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