Chapter 15 QUANTUM PHYSICS.

Chapter 15 QUANTUM PHYSICS

15-1 Blackbody Radiation, Planck Hypothesis
Chapter Index 15-0 Basic Requirements 15-1 Blackbody Radiation, Planck Hypothesis 15-2 Photoelectric Effect, Wave-particle Duality of Light 15-3 Compton Effect 15-4 Bohr’s Theory of Hydrogen Atom *15-5 Franck-Hertz Experiment 15-6 de Broglie Matter Wave, Wave-particle Duality of Particles

15-7 Uncertainty Principle
Chapter Index 15-7 Uncertainty Principle 15-8 Introduction to Quantum Mechanics 15-9 Introduction to Quantum Mechanics of Hydrogen Atom * Electron Distributions of Multi-electron Atoms * Laser * Semiconductor * Superconductivity

15-0 Basic Requirements 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

15-0 Basic Requirements 2. Understand difficulties of classic physics theory in explanation of experimental discoveries of photoelectronic effect. Understand Einstein photon hypothesis, grasp Einstein equation 3. Understand experimental laws of Compton effect, and its explanation by photon. Understand wave-particle duality of light.

15-0 Basic Requirements 4. Understand experimental results of Hydrogin atom spectra, and Bohr’s theory 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.

15-0 Basic Requirements 6. Understand 1-dimension coordinate momentum uncertainty principle 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. END

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. 8

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. 9

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 11

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. 12

(2) Radiation emittance
power emitted from a surface per unit time and unit area 13

Monochroma-tic radiation emittance of Sun and Ti visible
2 12 10 4 6 8 visible Ti 14

(3) Monochromatic absorption ratio and reflection ratio
monochromatic absorption ratio  (T) ： The ratio of absorbed energy to the incident energy between wavelength and Incident absorption Reflection transmission 15

monochromatic reflection ratio r(T ):
the ratio of reflected energy to the incident energy between wavelength and monochromatic reflection ratio r(T ): For opaque object (T ) + r(T )=1 Incident absorption Reflection transmission 16

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 Blackbody is an idealized model 17

(5) Kirchhoff’s Law 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 18

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 19

2. Experimental Observations of Blackbody Radiation
Visible Region 1.0 6 000 K Exp. Curve 0.5 3 000 K 21

1. Stephan-Boltzmann Law
1.0 Visible region 3 000 K 6 000 K Total Radiation Emittance 0.5 where Stephan-Boltzmann const. 22

2. Wien’s Displacement Law
1.0 Visible region 3 000 K 6 000 K Peak wave length Const. 23

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？ Solu: (1) From Wien’s displacement law 24

From Wien displacement law （2）
（3）From Stephan-Boltzmann law 25

3. Rayleigh-Jeans formula Failures of classical physics
6 2 4 Rayleigh-Jeans Rayleigh-Jeans Exp. Curve * T = K Violet Catastrophy 26

M. Planck ( ) 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. 27

4. Planck’s hypothesis and blackbody radiation formula
(1) Planck’s blackbody radiation formula Planck’s Constant 28

Reighley - Jeans Planck’s formula Exp. Data
Exp. Data vs. Planck theoretical curve 6 Planck’s formula Exp. Data * 4 2 T = K 29

2. Planck’s quantum hypothesis
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 Planck quantum hypothesis is the milestone of quantum mechanics 30

E.g-2 Suppose a tuning fork mass m = 0.05 kg ，frequency , amplitude .
(2) when quantum number increases from to ，how much does the amplitude change？ E.g-2 Suppose a tuning fork mass m = 0.05 kg ，frequency , amplitude . (1) quantum number of vibration； Solu: (1) 31

energy （2） 32

Macroscopically, the effect of energy quantization is not obvious, namely, the energy of macroscopic object is continuous END 33

1. Photoelectric Effect and Phenomenon
(1) Experimental Setup and Phenomenon V A 34

(2a) Current linearly proportional to the intensity.
(2) Discoveries (2a) Current linearly proportional to the intensity. 35

(2b) threshold frequency
For a given metal, electrons only emitted if frequency of incident light exceeds a threshold0. 0 is called threshold frequency Threshold frequency depends on type of metal, but not on intensity 36

(2d) Current appears with no delay
(2c) Stopping Voltage Applied reverse voltage that makes zero current is so-called stopping voltage , different metal has different O Stopping voltage linearly related incident light frequency (2d) Current appears with no delay 37

(3) Failures of Classical Theory
Threshold frequency Electrons should be emitted whatever the frequency ν of the light, so long as electric field E is sufficiently large No time delay For very low intensities, expect a time lag between light exposure and emission, while electrons absorb enough energy to escape from material 38

2. “Photon”, Einstein Equation
(1) “light quanta” hypothesis Light comes in chunks (composed of particle-like “photon”), each light quanta has energy (2) Einstein photoelectric equation Escape work depends on material 39

Approximate escape work value of different metals (in eV)
Na Al Zn Cu Ag Pt Theoretical Explanation: the greater the intensity, the more photons, the more photo-electrons, and hence the larger current ( ) 40

Applied reverse stopping voltage stops electrons
41

Threshold frequency: thrshold frequency
No lag: photon energy ( ) is absorbed by a electron and the electron then emits without time delay Einstein’s theory successfully explained the photoelectric effect and won 1921 Nobel prize of physics (not for relativity)! 42

Stopping voltage vs. frequency
(3) Measurement of Planck const. Stopping voltage vs. frequency O 43

E. g. -1. Consider a thin circular plane with radius , 1
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. 44

Solution: 45

3. Applications in Modern Technology
Photo-relay circuit, Automatic counter, measuring device etc. Amplifier Controller light Demo. of photo-relay Photomultiplier 46

4. Wave-particle Duality of Light
(1) wave:diffraction and interference (2) particle: , photo-electric effect etc. Relativistic energy-momentum relation photon 48

photon Particle character Wave character END 49

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 50

1. Experimental Asparatus
51

2. Experimental Results (1) shift in wave length depends on
Relative Intensity (1) shift in wave length depends on (2) is indep. of targets 52

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! 53

4. Quantum Explanations (1) Physical model
Photon Electron Electron Photon Incident photon (X-ray or -ray) with higher energy 54

electrons near the surface of solids with weak binding, quasi-free
electron energy of thermal motion , so we can treat electron as at-rest approximately phton electron electron photon electrons near the surface of solids with weak binding, quasi-free electron with large bouncing velocity, use relativistic mechanics 55

(2) Qualitative Analysis
(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) photon collides with tight-binding electron, without significant lost of energy, results in the same wave-length in scattered light 56

(3) Quantitative Calculation
Energy conservation Momentum conservation 57

Compton Wavelength Compton Formula 59

(4) Conclusions Scattered light wave length change depends only on
scattered photon energy decrease 60

(1) Change of the scattered wavelength ?
Eg-1. X-ray with wavelength elastically collides with a electron at rest, observing along the direction with respect to scattering angle, (1) Change of the scattered wavelength ? (2) Kinetic energy bouncing electron gets? (3) Energy that photon loses during collision? 61

END Solution (1) (2) bouncing electron kinetic energy
(3) Energy photon loses＝ END 62

1. Review of Modern View of Atomic Hydrogen Structure
(1) Experimental discoveries of atomic hydrogen spectrum 63

Light Bulb Hydrogen Lamp Quantized, not continuous 64

(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: 65

Johann Rydberg (1890) extends the Balmer model by finding more emission lines outside the visible region of the spectrum: wave number Rydberg const. 66

Lyman Ultraviolet Balmer Visible 67

Infrared Paschen Brackett Pfund Humphrey 68

Balmer spectrum of H atom
656.3 nm 486.1 nm 434.1 nm 410.2 nm 364.6 nm 69

n Hydrogen atom spectra
Visible lines in H atom spectrum are called the BALMER series. 6 5 3 2 1 4 n Energy Ultra Violet Lyman Visible Balmer Infrared Paschen

(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". 71

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" 72

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. 73

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. 74

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 75

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. 76

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 77

Niels Bohr ( ) 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

(2) Bohr’s Theory of H Atom
In 1913, N. Bohr uses the emission spectrum of hydrogen to propose a quantum model for H atom, with the following three assumptions (a) Stationary hypothesis (b) Frequency condition (c) Quantization condition 79

(a) Stationary hypothesis
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 + E1 E3 Energies corresponding to stationary states are E1, E2… , E1 < E2< E3 80

(b) Frequency condition
Ef Ei emmision absorption (c) Quantization condition Principal quantum number 81

(3) Calculate H-atom energy and orbital radii
(a) Orbital radii + rn Classical mechanics Quantization condition 82

The nth orbital electron’s energy：
, Bohr radius (b) Energy The nth orbital electron’s energy： 83

ground state energy (Ionized energy) Excited state energy 84

Energy level transition and Spectrum of H-atom
n=1 n=2 n=3 n=4 n=5 n= Balmer Paschen Brackett Lyman 85

(4) Explanations of Bohr’s Theory on H-atom Spectrum
(Rydberg const.) 86

3. The Successes and Failures of Bohr Theory
(a) Correctly predicted the existence of atom energy level and energy quantization (b) Correctly proposed the concepts of stationary state and angular momentum quantization. (c) Correctly explained H-atom and H-like-atom spectrum 87

END (2) Failures (a) Does not work for multi-electron atoms
(b) Microscopic particles do not have certain trajectory (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 END 88

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 Particle nature Wave nature 89

L. de Broglie (1892 – 1987) French physicist and a Nobel laureate in 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). 90

de Broglie wave or Matter wave
de Broglie relation de Broglie wave or Matter wave Note (1)if then if then 91

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. Solution: 92

Roughly the order of X-ray wavelength
93

E.g.-2 Derive quantization condition of angular momentum in Bohr’s theory of hydrogen atom
Solution: Consider a string with two ends fixed, if its length equals wave-length then a stable standing wave can form to form a circle 94

Electron’s de Broglie wavelength
We get quantization condition of angular momentum 95

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 96

ELECTRON DIFFRACTION The Davisson-Germer experiment (1927)
The Davisson-Germer experiment: scattering a beam of electrons from a Ni crystal. Davisson got the 1937 Nobel prize. Davisson G.P. Thomson θi At fixed angle, find sharp peaks in intensity as a function of electron energy C. J. Davisson, "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) At fixed accelerating voltage (fixed electron energy) find a pattern of sharp reflected beams from the crystal G.P. Thomson performed similar interference experiments with thin-film samples

Ni-crystal diffraction
2. Experimental Confirmation of de Broglie Wave (1) Davisson-Germer Diffraction Exp. 检测器 Electron beam Scattering beam Ni-crystal diffraction M K Electron gun Current vs. Acceleration voltage, 98

The exp. results of single crystal diffraction by electron beam agree with “Bragg’s law” in X-ray diffraction Interference condition: 99

Wavelength of electron wave
For Ni crystal Wavelength of electron wave 100

when , agree well with experimental results.
101

Diffraction of electron beam from polycrystalline foil
(2) G. P. Thomson electron diffraction exp. Electron beam from polycrystalline foil generates diffraction fringe similar to the X-ray diffraction fringe Diffraction of electron beam from polycrystalline foil K 102

3. Applications Scanning Tunneling Microscopy (STM) Developed by Gerd Binnig and Heinrich Rohrer at the IBM Zurich Research Laboratory in 1982. Binnig Rohrer The two shared half of the 1986 Nobel Prize in physics for developing STM. 103

4. Statistical interpretation of de Broglie wave
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 104

(1) Explanation by particle nature
Single particle randomly appears, but large number of particles show a statistical regularity. The probability that a particle appear at different position is different Electron beam slit single-slit diffraction 105

(2) Explanation by wave nature
The more intense the electrons at some place, the higher intensity of wave; or vice versa. Electron beam slit single-slit diffraction 106

END (3) Statistical Interpretation
At some place the intensity of de Broglie wave proportioned to the probability that the particle appears around that place M. Born (1926) pointed out , de Broglie wave is probability wave. END 107

Electron Single-slit Diffraction Exp.
1. Heisenberg Uncertainty principle of Coordinate and Momentum Electron Single-slit Diffraction Exp. Electron diffraction Position uncertainty of the electron the 1st order min. diffraction angle 108

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 109

x-direction momentum uncertainty after passing the slit
110

the 2nd order diffraction
Heisenberg proposed uncertainty principle in 1927 Microscopic particles can not be described by simultaneous coordinate and momentum Uncertainty Relation 111

Implications (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 (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 112

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. 113

Solution: Bullet’s momentum
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? Solution: Bullet’s momentum Uncertainty of momentum 114

Uncertain range of the position
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? 115

END Solution: electron’s momentum Uncertain range of the momentum
Uncertain range of the position END 116

1. Wave Function and Its Statistical Explanation
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 117

(1a) Classical wave and wave function
mechanical wave em wave classical wave is a real function 118

(1b) QM wave function (complex function )
Wave function that descibe the motion of the microscopic particle 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 119

(2) The statistical interpretation of wave function
Free particle plane wave function (2) The statistical interpretation of wave function Probability Density: the probability that the particle appears in unit (spatial) volume Positive Real number 120

Probability that the particle appears at some moment in a volume element
Hence de Broglie wave (or matter wave) is a probability wave, it is very different with electromagnetic wave 121

Normalization Condition
At some moment the probability one finds the particle in entire space is Normalization Condition (Bound State) Standard Condition Wave function is single-valued, real, finite function 122

Austrian theoretical physicist
Erwin Schrodinger， 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) 123

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 124

1-dimension free particle time-dependent Schrodinger equation
One gets Free particle　 1-dimension free particle time-dependent Schrodinger equation 125

(2) Particle in potential field with potential energy :
1-dimensional time-dependent Schrodinger equation (3) particle in stationary potential time-indep. 126

1-dimensional stationary Schrodinger equation in any potential field
127

Stationary Schrodinger equation in 3-dimensional potential field
Lapalce operator Stationary wave function 128

e.g., stationary Schrodinger equation for hydrogen atom
Properties of stationary wave function (1)E is time-independent (2) is time-independent 129

(3) is finite, single-valued
wave function single-valued, finite, continuous (1) normalization (2) 和 continuous (3) is finite, single-valued 130

3. 1-dim. Potential Well Particle potential energy satisfies boundary condition Simplified model for free electron gas model of metal in solid physics Demonstrate QM basic concepts and principles with simple math 131

Wave function single-value, finite, and continuous
133

quantum number 134

Normalization 135

hence wave equation 136

wave function Prob. density Energy 137

Discussions： 1. energy quantization g.s. Energy excited state Energy
the particle’s energy in 1-dim. infinity square well is quantized. 138

(2) the prob. density that particle appears in the well is different
Wave function Prob. density e.g., when n =1, the maximum probability is at the place x = a /2 139

(3) wave function is standing wave, the nodes locate at the wall, the No. of valley equals quantum number n 16E1 9E1 4E1 E1 140

4. 1-dim. Square Well, Tunneling Effect
Particle’s Energy 141

Wave functions in different regions
Tunneling Effect Wave functions in different regions 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 142

Single atom lithography
Applications STM (1981) Scanning Tunneling Microscopy AFM (1986) Atom Force Microscopy Xenon on Nickel Single atom lithography END 143

Quantum Corrals Iron on Copper Imaging the standing wave created by interaction of species

1. Schrodinger Equation of Hydrogen Atom
Potential energy of electron in H-atom Stationary Schrodinger equation: 145

Spherical Coordinates
Transform to spherical polar coordinates because of the radial symmetry 146

In Spherical coordinates:
Separable solution, let 147

We get 148

(1) Energy quantization and principal quantum number
2. Quantization condition and quantum number Solve Schrodinger equation we get the following quantum number and quantization properties: (1) Energy quantization and principal quantum number n =1,2,3,... Principal quantum number 149

(2) Angular momentum quantization and angular quantum number
Orbital angular quantum number E.g.，n =2， = 0,1 corresponds to 150

(3) Angular momentum spatial quantization and magnetic quantum number
In applied magnetic field, angular momentum L can only take some specific directions, projection of L along magnetic field satisfies magnetic quantum number reduced Planck const. 151

magnetic quantum number ml =0, 1 and
e.g., when magnetic quantum number ml =0, 1 and z o ħ L z 152

(4) Spin and spin quantum number
Spin angular momentum where spin quantum number Spin angular momentum takes only two components along applied magnetic field: ms spin magnetic quantum number 153

Spin angular momentum and spin magnetic quantum number of electron
z Sz Sz S 154

Principal qn. n determines energy
(5) Summary The states of electron in hydrogen atom can be represented by 4 quantum numbers (qn.), (n, l ,ml , ms) Principal qn. n determines energy 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 155

3. Ground state radial wave function and distribution probability
(1) Ground state energy Ground state n = l = 0 Radial wave function equation: solution 156

where Substitute into get 157

(2) Ground state radial wave function
the probability that electron appears in volume element dV: let the prob. density along radial vector p, the prob. that the electron appears in (r , r+dr) 158

g.s. radial wave function is
from normalization  g.s. radial wave function is 159

(3) Probability Density Distribution of Electron
END 160

Light Amplification by Stimulated Emission of Radiation 161

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 162

Spontaneous Radiation
. Before Radiation . 。 After Radiation 163

. . 。 (2) Absorption of light
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 After Absorption 。 . Before Absorption . Excited Absorption 164

(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 165

. 。 Stimulated Radiation Amplification of stimulated radiation
Before After Amplification of stimulated radiation when a population inversion is present, the rate of stimulated emission exceeds that of absorption, results in a coherent amplification  laser 166

2. The principle of laser (1) Normal and inverse distribution of population known shows that the electron population at lower energy level greater than that at higher level, this is normal distribution 167

is instead inverse distribution of population, or simply population inversion
Population normal distribution and inversion Inversion . 。 Normal . 。 168

Energy level of ion Cr in Ruby laser
T. H. Maiman (U.S. physicist) made the first functional ruby laser in sept., 1960 Energy level of ion Cr in Ruby laser . 。 Ground state Metastable state Excited state 169

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 condition 170

Optical resonator . Laser beam HRM Demonstration of O.R. PTM 171

PTM: partially transmissive mirror HRM: highly reflectance mirror
3. Laser (1) Helium-Neon Gas Laser PTM: partially transmissive mirror HRM: highly reflectance mirror He-Ne Laser HRM PTM A K Energy levels of He and Ne Ground state Metastable He Ne 632.8 nm 2 3 1 172

HELIUM-NEON GAS LASER 173

(2) Ruby (CrAlO3) laser Its active medium is ruby crystal rod, generates pulse laser with wavelength nm. 。 High reflectance mirror Partialy transmissive mirror Ruby rod Pulse Demo. of Ruby Laser 174

Safety Shutter Polarizer Assembly (optional)
NEODYMIUM YAG LASER Rear Mirror Adjustment Knobs Safety Shutter Polarizer Assembly (optional) Coolant Beam Tube Adjustment Knob Output Mirror Beam Tube Harmonic Generator (optional) Laser Cavity Pump Cavity Flashlamps Nd:YAG Laser Rod Q-switch (optional) Courtesy of Los Alamos National Laboratory 175

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 (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 176

Incandescent vs. Laser Light
Many wavelengths Multidirectional Incoherent Monochromatic Directional Coherent 177

Fully Separated Energy Levels of Two H-atom
1. Energy Gap of Solids + Fully Separated Energy Levels of Two H-atom 178

Six closed H-atom’s energy level split
Two closed H-atom’s energy level split Six closed H-atom’s energy level split Energy Band of Solids 179

quantum states per energy level
Electron distribution of different energy bands in Na quantum states per energy level electrons per energy level electrons per energy band 180

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 181

Energy band of crystals
Cond-uction band Cond-uction band Valence band (not full) Empty band Valence band (full) Forbi-dden band Forbi-dden band 182

Comparison between Conductor, Semi-conductor and Insulator
Conductor Semiconductor Insulator Resistance Temp. Coeff. F-band V-band Pos. + Not full Neg. - Small Full Neg. - Large Full 183

Typical Semiconductors
Silicon Diamond Cubic Structure 4 atoms at (0,0,0)+ FCC translations 4 atoms at (¼,¼,¼)+FCC translations Bonding: covalent GaAs ZnS (Zinc Blende) Structure 4 Ga atoms at (0,0,0)+ FCC translations 4 As atoms at (¼,¼,¼)+FCC translations Bonding: covalent, partially ionic 184

2. Intrinsic and Extrinsic semi-conductor
(1) Intrinsic: pure, no dopants C-band F-band Full band hole electron Normal Bond in Ge Electrons are excited, Holes appear 185

(2) Extrinsic semiconductor)
Electron type (n-type) Phosphorus atom are dopant Si atoms are hosts， V-band C-band Donor level Donor Level 186

Boron atom doping into Ge atom lattice
p-type semiconductor Hole Boron atom doping into Ge atom lattice V band C band Acceptor level 187

Current-Volt Characteristics of pn Junction
188

- - p n p n + + Voltage variation between p-layer and n-layer Hole
Electron Voltage variation between p-layer and n-layer 189

4. Photovoltaic effect END
Light Photovoltaic effect is the creation of voltage or electric current in pn upon exposure to light END 190

The transition temperature of superconductor
0.150 0.100 0.050 0.000 * around T=4.20K risistance is ZERO : the critical temperature 191

2. Major Properties of Superconductors
(1) Null resistance When (critical electric flow) conductance resistance (2) Critical magnetic field The critical point of applied magnetic fields that breaks the superconducting states 192

Super- conductor Normal o (3) Meissner effect 193

in super-conductor when S N 194

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 Interestingly, this theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus. 195

BCS=Bardeen, Cooper, Schrieffer
196

deformation of local area
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" deformation of local area Normal location of Lattice deformation of lattice 197

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. Cooper Pair: two electrons couple by exchanging phonon, and form the coupled electron pair called Copper pair The distance between two electrons is about their spins and momenta are opposite, the total momenta is zero. 199

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 END 200

Chapter 15 QUANTUM PHYSICS.

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Chapter 15 QUANTUM PHYSICS.

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Presentation on theme: "Chapter 15 QUANTUM PHYSICS."— Presentation transcript:

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