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Chapter 15 QUANTUM PHYSICS Chapter Index Physics 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 Chap 15 Quantum Physics 2 Chapter Index Physics 15-7 Uncertainty Principle 15-8 Introduction to Quantum Mechanics 15-9 Introduction to Quantum Mechanics of Hydrogen Atom *15-10 Electron Distributions of Multi-electron Atoms *15-11 Laser *15-12 Semiconductor *15-13 Superconductivity Chap 15 Quantum Physics 3 15-0 Basic Requirements Physics 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 Chap 15 Quantum Physics 4 15-0 Basic Requirements Physics 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. Chap 15 Quantum Physics 5 15-0 Basic Requirements Physics 4. Understand experimental results of Hydrogin atom spectra, and Bohr’s theory 5. Understand de Broglie hypothesis and electron diffraction experiment and waveparticle duality of particles; Understand the relation between physical quantities (wavelength, frequency) describing wave property and ones (energy, momentum) describing particle property. Chap 15 Quantum Physics 6 15-0 Basic Requirements Physics 6. Understand 1-dimension coordinate momentum uncertainty principle 7. Understand wave function and its statistical explanation. Understand 1dimension stationary Schrodinger equation, and the quantum mechanical method deal with 1 dimensional infinity potential well etc. Chap 15 Quantum Physics 7 Physics 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. Chap 15 Quantum Physics 8 Physics 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. Chap 15 Quantum Physics 9 Physics Chap 15 Quantum Physics 10 Physics 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 Chap 15 Quantum Physics 11 Physics 1. Thermal Radiation (1) Fundamental concepts and basic laws (1a) Monochromatic radiant emittance: the power of electromagnetic radiation whose M (T ) W m -2 Hz -1 frequency around (or M (T ) W m -3 wavelength ) per unit area and unit time radiated by a surface. Chap 15 Quantum Physics 12 Physics (2) Radiation emittance power emitted from a surface per unit time and unit area M (T ) M (T )d 0 M (T ) M (T )d 0 Chap 15 Quantum Physics 13 Physics SUN M(T )/(10 8 W m2 Hz1 ) Ti M(T )/(10 9 W m- 2 Hz1 ) Monochromatic radiation emittance of Sun and Ti 12 T 5 800 K SUN visible 10 8 6 4 2 0 Ti / 1014 Hz 2 4 Chap 15 6 8 10 12 Quantum Physics 14 Physics (3) Monochromatic absorption ratio and reflection ratio monochromatic absorption ratio (T) : The ratio of absorbed energy to the incident energy between wavelength and d absorption Incident Reflection Chap 15 transmission Quantum Physics 15 Physics monochromatic reflection ratio r(T ): the ratio of reflected energy to the incident energy between wavelength and d For opaque object (T ) + r(T )=1 absorption Incident Reflection Chap 15 transmission Quantum Physics 16 Physics (4) Black body 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 Chap 15 Quantum Physics 17 Physics (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 Chap 15 Quantum Physics 18 Physics 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 M (T ) M B ( , T ) (T ) Chap 15 Quantum Physics 19 Physics Chap 15 Quantum Physics 20 Physics 2. Experimental Observations of Blackbody Radiation M (T ) /(1014 W m 3 ) 1.0 Visible Region Exp. Curve 0.5 6 000 K 3 000 K 0 Chap 15 m 1 000 / nm 2 000 Quantum Physics 21 Physics 1. Stephan-Boltzmann Law M (T ) /(1014 W m 3 ) Total Radiation Emittance M (T ) M (T )d T 4 Visible region 1.0 0 0.5 where 5.670 108 W m 2 K 4 Stephan-Boltzmann const. Chap 15 6 000 K 3 000 K 0 1m000 Quantum Physics / nm 2 000 22 Physics 2. Wien’s Displacement Law M (T ) /(1014 W m 3 ) mT b Visible 1.0 region Peak wave length 6 000 K Const. b 2.898 103 m K 3 000 K 0 Chap 15 1m000 Quantum Physics / nm 2 000 23 Physics E.g-1 (1) Suppose a blackbody with temperature T= 20 C , what is the wavelength of its monochromatic peak?(2) the monochromatic emittance peak wave length m 483 nm , estimate the surface temperature of the sun; (3) 上what is the ratio of above two? Solu: (1) From Wien’s displacement law 3 b 2.898 10 m nm 9 890 nm T1 293 Chap 15 Quantum Physics 24 Physics (2) From Wien displacement law 3 2.898 10 T2 K 6 000 K 9 m 483 10 b (3)From Stephan-Boltzmann law M (T2 ) M (T1 ) (T2 T1 ) 4 1.76 105 Chap 15 Quantum Physics 25 Physics 3. Rayleigh-Jeans formula Failures of classical physics M (T ) /(10 9 W m -2 Hz -1 ) Rayleigh-Jeans 6 * * 4 * * 2 * 0 1 ** Rayleigh-Jeans * Exp. Curve * * T = 2 000 K * * * * * * 2 3 Chap 15 2 π 2 M (T ) 2 kT c Violet Catastrophy / 1014 Hz Quantum Physics 26 Physics 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. Chap 15 Quantum Physics 27 Physics 4. Planck’s hypothesis and blackbody radiation formula (1) Planck’s blackbody radiation formula 2 π h d M (T )d 2 hν / kT c e 1 3 Planck’s Constant h 6.63 10 34 J s Chap 15 Quantum Physics 28 Physics M (T ) /(10 9 W m-2 Hz1 ) Reighley - Jeans Exp. Data vs. Planck theoretical curve 6 * 4 ** * * Planck’s formula * * * * * Exp. Data * T = 2 000 K * * * * * / 1014 Hz 0 1 2 3 2 Chap 15 Quantum Physics 29 Physics 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 ε nh (n 1,2,3,) Planck quantum hypothesis is the milestone of quantum mechanics Chap 15 Quantum Physics 30 Physics E.g-2 Suppose a tuning fork mass m = 0.05 kg , frequency 480 Hz , amplitude A 1.0 mm (1) quantum .number of vibration; (2) when quantum number increases from n to n 1,how much does the amplitude change? Solu: (1) 1 1 2 2 2 2 E m A m(2 π ) A 0.227 J 2 2 Chap 15 Quantum Physics 31 Physics E nh energy E 29 n 7.13 10 h h 3.18 10 31 J (2) E nh E nh A 2 2 2π m 2π 2 m 2 h 2 AdA dn 2 2π m Chap 15 Quantum Physics 32 Physics n A A n 2 n 1 A 7.0110 34 m Macroscopically, the effect of energy quantization is not obvious, namely, the energy of macroscopic object is continuous Chap 15 Quantum Physics 33 Physics 1. Photoelectric Effect and Phenomenon (1) Experimental Setup and Phenomenon V Chap 15 Quantum Physics A 34 Physics (2) Discoveries (2a) Current linearly proportional to the intensity. i im2 im1 I2 I1 I 2 I1 U0 o U Chap 15 Quantum Physics 35 Physics (2b) threshold frequency 0 For a given metal, electrons only emitted if frequency of incident light exceeds a threshold0. 0 is called threshold frequency Threshold frequency depends on type of metal, but not on intensity Chap 15 Quantum Physics 36 Physics (2c) Stopping Voltage U 0 Applied reverse voltage that makes zero current is socalled stopping voltage U 0 , different metal has different U0 O U0 C s Z n Pt 0 Stopping voltage linearly related incident light frequency (2d) Current appears with no delay Chap 15 Quantum Physics 37 Physics (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 Chap 15 Quantum Physics 38 Physics 2. “Photon”, Einstein Equation (1) “light quanta” hypothesis Light comes in chunks (composed of particle-like “photon”), each light quanta has energy ε h (2) Einstein photoelectric equation 1 2 h mv W 2 Chap 15 Escape work depends on material Quantum Physics 39 Physics 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 Theoretical Explanation: the greater the intensity, the more photons, the more photo-electrons, and hence the larger current ( 0) Chap 15 Quantum Physics 40 Physics Stopping voltage Applied reverse stopping voltage U 0 stops electrons V A 1 2 eU 0 mv 2 Chap 15 Quantum Physics 41 Physics Threshold frequency: W h 0 0 0 W h thrshold frequency No lag: photon energy h ( 0 ) 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)! Chap 15 Quantum Physics 42 Physics (3) Measurement of Planck const. 1 2 h mv W 2 Stopping voltage vs. frequency U0 h eU 0 W h W U0 e e U 0 h e Chap 15 O 0 U 0 h e Quantum Physics 43 Physics E.g.-1. Consider a thin circular plane with radius 1.0 10 3 m , 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. Chap 15 Quantum Physics 44 Physics Solution: 3 6 S π (1.0 10 m) π 10 m 2 2 S 7 1 EP 2.5 10 J s 2 4πr E E 11 1 N 7.4 10 s h hc Chap 15 Quantum Physics 45 Physics 3. Applications in Modern Technology Photo-relay circuit, Automatic counter, measuring device etc. Demo. of photo-relay light Amplifier Controller Chap 15 Photomultiplier Quantum Physics 46 Physics Chap 15 Quantum Physics 47 Physics 4. Wave-particle Duality of Light (1) wave:diffraction and interference (2) particle: E h , photo-electric effect etc. Relativistic energymomentum relation photon Chap 15 E p c E 2 E0 0 , 2 2 2 0 E pc Quantum Physics 48 Physics photon E0 0 , E pc E h h p c c Particle character E h h p Chap 15 Wave character Quantum Physics 49 Physics Compton (1923) measured intensity of scattered Xrays from solid target, as function of wave- length 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. Chap 15 A.H. Compton, Phys. Rev. 22 (1923) 409 Quantum Physics 50 Physics 1. Experimental Asparatus Chap 15 Quantum Physics 51 Physics 2. Experimental Results (1) shift in wave length 0 depends on (2) is indep. I Relative Intensity 0 0 45 90 of targets 0 Chap 15 Quantum Physics 135 52 Physics 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! Chap 15 Quantum Physics 53 Physics 4. Quantum Explanations (1) Physical model Photon 0 y y Electron v0 0 x Photon x Electron Incident photon (X-ray or -ray) with higher energy 4 5 10 ~ 10 eV E h Chap 15 Quantum Physics 54 Physics electron energy of thermal motion h , so we can treat electron as at-rest approximately phton y 0 electron v0 0 photon y x x electron electrons near the surface of solids with weak binding, quasi-free electron with large bouncing velocity, use relativistic mechanics Chap 15 Quantum Physics 55 Physics (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 Chap 15 Quantum Physics 56 Physics (3) Quantitative Calculation Energy conservation hv0 m0 c h mc 2 2 Momentum conservation h y e h 0 c e0 e c e0 x h 0 h mv e0 e mv c c 2 2 2 2 2 h 0 h h 0 2 2 m v 2 2 2 2 cos c c c Chap 15 Quantum Physics 57 Physics h h 0 h m v 2 2 2 2 cos c c c 2 2 2 2 0 2 2 2 v 2 4 2 2 m c (1 2 ) m0 c 2h 0 (1 cos ) 2m0c h( 0 ) c 2 2 4 2 1/ 2 m m0 (1 v / c ) 2 c c h (1 cos ) 0 0 m0c Chap 15 Quantum Physics 58 Physics h 2h 2 (1 cos ) sin m0 c m0 c 2 Compton Wavelength h 12 C 2.43 10 m m0 c Compton Formula h (1 cos ) C (1 cos ) m0 c Chap 15 Quantum Physics 59 Physics (4) Conclusions Scattered light wave length change depends only on 0, 0 π, ( ) max 2C scattered photon energy decrease h y e h 0 c e0 e c e0 0 , 0 Chap 15 Quantum Physics x mv 60 Physics -10 1.00 10 m Eg-1. X-ray with wavelength 0 elastically collides with a electron at rest, 90 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? Chap 15 Quantum Physics 61 Physics Solution (1) C (1 cos ) C (1 cos 90 ) C 2.43 10 12 m (2) bouncing electron kinetic energy 0 hc hc hc Ek mc m0c (1 ) 295 eV 0 0 2 2 (3) Energy photon loses= Ek Chap 15 Quantum Physics 62 Physics 1. Review of Modern View of Atomic Hydrogen Structure (1) Experimental discoveries of atomic hydrogen spectrum Chap 15 Quantum Physics 63 Physics Light Bulb Hydrogen Lamp Quantized, not continuous Chap 15 Quantum Physics 64 Physics (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: n2 365.46 2 nm , 2 n 2 Chap 15 n 3,4,5, Quantum Physics 65 Physics Johann Rydberg (1890) extends the Balmer model by finding more emission lines outside the visible region of the spectrum: 1 1 1 R( 2 2 ) wave number n f ni n f 1,2,3,4,, ni n f 1, n f 2, n f 3, Rydberg const. R 1.097 10 7 m 1 Chap 15 Quantum Physics 66 Physics Ultraviolet Lyman 1 1 1 R( 2 2 ) , n 2,3, 1 n Visible Balmer 1 1 1 R( 2 2 ) , n 3,4, 2 n Chap 15 Quantum Physics 67 Physics Infrared 1 1 1 Paschen R( 2 2 ) , n 4,5, 3 n 1 1 1 Brackett R( 2 2 ) , n 5,6, 4 n 1 1 1 R( 2 2 ) , n 6,7, Pfund 5 n 1 1 1 Humphrey R( 2 2 ) , n 7,8, 6 n Chap 15 Quantum Physics 68 Physics Balmer spectrum of H atom 364.6 nm 410.2 nm 434.1 nm 486.1 nm 656.3 nm Chap 15 Quantum Physics 69 Physics Hydrogen atom spectra Visible lines in H atom spectrum are called the BALMER series. 6 5 4 Energy 3 2 1 Ultra Violet Lyman Chap 15 Visible Balmer Quantum Physics Infrared Paschen n Physics (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". Chap 15 Quantum Physics 71 Physics 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" Chap 15 Quantum Physics 72 Physics 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. Chap 15 Quantum Physics 73 Physics 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. Chap 15 Quantum Physics 74 Physics 2. Bohr’s Theory of Atomic Hydrogen (1) Failures of Classical Atomic Models According to the classical electromagnetic theory, electrons rotate around atomic nucleus, accelerated electrons radiate electro-magnetic wave and hence lose energy Chap 15 Quantum Physics 75 Physics 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 Chap 15 Quantum Physics 76 Physics 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 Chap 15 Quantum Physics v e F r e 77 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 Chap 15 Quantum Physics 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 Chap 15 Quantum Physics 79 Physics (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 Energies corresponding to stationary states are E1, E2… , E1 < E2< E3 Chap 15 E1 + E3 Quantum Physics 80 Physics (b) Frequency condition h Ei E f Ei emmision absorption Ef (c) Quantization condition h L mvr n 2π n 1,2,3, Principal quantum number Chap 15 Quantum Physics 81 Physics (3) Calculate H-atom energy and orbital radii (a) Orbital radii 2 v n Classical m mechanics 4π 0 rn2 rn h Quantization mvn rn n condition 2π e2 rn 0h 2 π me 2 + rn n 2 r1n 2 (n 1,2,3,) Chap 15 Quantum Physics 82 Physics rn 0h 2 π me n r1n (n 1,2,3,) 2 2 2 0h 2 n 1 , Bohr radius r1 π me2 5.29 10 11 m (b) Energy The nth orbital electron’s energy: 2 1 e En mvn2 2 4π 0 rn Chap 15 Quantum Physics 83 Physics me4 1 E1 En 2 2 2 2 8 0 h n n ground state energy (n 1) 4 me (Ionized E1 2 2 13.6 eV 8 0 h energy) Excited state energy (n 1) En E1 n Chap 15 2 Quantum Physics 84 Physics Energy level transition and Spectrum of H-atom n= n=5 n=4 0 -0.54 eV Brackett Paschen n=3 -0.85 eV -1.51 eV Balmer -3.40 eV n=2 Lyman n=1 -13.6 eV Chap 15 Quantum Physics 85 Physics (4) Explanations of Bohr’s Theory on H-atom Spectrum 4 me 1 h Ei E f En 2 2 2 8 0 h n 1 4 me 1 1 2 3 ( 2 2 ), c 8 0 h c n f ni me4 7 1 1 . 097 10 m 2 3 8 0 h c Chap 15 ni n f R (Rydberg const.) Quantum Physics 86 Physics 3. The Successes and Failures of Bohr Theory (1) Successes (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 Chap 15 Quantum Physics 87 Physics (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 Chap 15 Quantum Physics 88 Physics 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 E mc 2 h Particle nature Wave nature P mv h / Chap 15 Quantum Physics 89 Physics L. de Broglie (1892 – 1987) 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). Chap 15 Quantum Physics 90 Physics h h de Broglie relation p mv 2 E mc h h de Broglie wave or Matter wave Note (1)if v c then m m0 if v c then m m0 Chap 15 Quantum Physics 91 Physics 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 200 eV , Calculate its de Broglie wavelength. 1 Solution: v c, Ek m0 v 2 v 2 Chap 15 Quantum Physics 2 Ek m0 92 Physics 2 200 1.6 10 v 9.11031 v c 19 1 m s 8.4 10 m s 6 -1 34 h 6.63 10 nm 31 6 m0 v 9.110 8.4 10 8.67 10 2 nm Roughly the order of X-ray wavelength Chap 15 Quantum Physics 93 Physics 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 2π r 2π r n n 1,2,3,4, Chap 15 Quantum Physics 94 Physics Electron’s de Broglie wavelength h mv 2π rmv nh We get quantization condition of angular momentum h L mvr n 2π Chap 15 Quantum Physics 95 Physics 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 Chap 15 Quantum Physics 96 Physics ELECTRON DIFFRACTION The Davisson-Germer experiment (1927) θi θi The Davisson-Germer Davisson G.P. Thomson 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 C. J. Davisson, "Are Electrons Waves?," Franklin Institute Journal 205, 597 (1928) G.P. Thomson performed similar interference experiments with thin-film samples Chap 15 Quantum Physics Physics 2. Experimental Confirmation of de Broglie Wave (1) Davisson-Germer Diffraction Exp. U K Electron gun 50 I 检测器 Electron beam Scatter ing M G 35 54 75 U /V beam Ni-crystal diffraction Chap 15 Current vs. Acceleration voltage, 50 Quantum Physics 98 Physics The exp. results of single crystal diffraction by electron beam agree with “Bragg’s law” in Xray diffraction Interference condition: d 2 2 2d sin 2 cos . . . . . . . . . . . . . . . . d sin k . . . . . . . . 2 k 2 d sin 2 Chap 15 k 1, 50 Quantum Physics 99 Physics For Ni crystal d 2.15 10 10 m d sin 1.65 1010 m Wavelength of electron wave h h 1.67 1010 m me v 2me Ek 1 d sin kh 2emU Chap 15 Quantum Physics 100 Physics 1 d sin kh 2emU kh 1 sin d 2emU sin 0.777k when k 1 arcsin 0.777 51 , agree well with experimental results. Chap 15 Quantum Physics 101 Physics (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 U D M Chap 15 P Quantum Physics 102 Physics 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. Chap 15 Quantum Physics 103 Physics 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 Chap 15 Quantum Physics 104 Physics (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 Chap 15 Quantum Physics 105 Physics (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 Chap 15 Quantum Physics 106 Physics (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. Chap 15 Quantum Physics 107 Physics 1. Heisenberg Uncertainty principle of Coordinate and Momentum x Electron diffraction b ph Position uncertainty of the electronx b the 1st order min. diffraction angle sin b Chap 15 ph y o Electron Single-slit Diffraction Exp. Quantum Physics 108 Physics 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 Chap 15 Quantum Physics 109 Physics x-direction momentum uncertainty after passing the slit x sin b p x p sin p h p h p x b b ph b ph y o xpx h Chap 15 Quantum Physics 110 Physics the 2nd order diffraction xpx h Heisenberg proposed uncertainty principle in 1927 Microscopic particles can not be described by simultaneous coordinate and momentum Uncertainty Relation Chap 15 xpx h yp y h zp z h Quantum Physics 111 Physics 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 h is extremely small, xpx 0 , hence in macroscopic limit, the momentum and position can be simultaneously determined Chap 15 Quantum Physics 112 Physics 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. Chap 15 Quantum Physics 113 Physics E.g.-1. The mass of a bullet is 10 g, speed 1 200 m s . Momentum uncertainty is 0.01% of its momentum (this is good enough in macroscopic world), What is the position uncertainty of the bullet? Solution: Bullet’s momentum p mv 2 kg m s Uncertainty of momentum 1 4 p 0.01% p 2 10 kg m s Chap 15 Quantum Physics 1 114 Physics 4 p 0.01% p 2 10 kg m s 1 Uncertain range of the position h 6.63 1034 30 x m 3.3 10 m 4 p 2 10 E.g.-2. An electron’s speed is 200 m s -1 . The degree of momentum uncertainty is 0.01% of the momentum, what is the uncertainty of position of the electron? Chap 15 Quantum Physics 115 Physics Solution: electron’s momentum p mv 9.11031 200 kg m s 1 28 p 1.8 10 kg m s 1 Uncertain range of the momentum p 0.01% p 1.8 1032 kg m s 1 Uncertain range of the position h 6.63 1034 2 x m 3 . 7 10 m 32 p 1.8 10 Chap 15 Quantum Physics 116 Physics 1. Wave Function and Its Statistical Explanation (1) Wave Function 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 Chap 15 Quantum Physics 117 Physics (1a) Classical wave and wave function x mechanical wave y ( x, t ) A cos 2π (t ) E ( x, t ) E0 cos 2π (t em wave H ( x, t ) H 0 cos 2π (t x ) ) x classical wave is a real function y ( x, t ) Re[ Ae Chap 15 Quantum Physics x i 2 π (t ) ] 118 Physics (1b) QM wave function (complex function ) Wave function that descibe the Ψ( x, y, z, t ) motion of the microscopic particle Wave-particle duality of E , h h p 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 Chap 15 Quantum Physics 119 Physics Free particle plane wave function Ψ ( x, t ) 0e i 2π ( Et px ) h (2) The statistical interpretation of wave function Probability Density: the probability that the particle appears in unit (spatial) volume Ψ * 2 Chap 15 Positive Real number Quantum Physics 120 Physics Probability that the particle appears at some moment in a volume element dV 2 Ψ dV ΨΨ dV * Hence de Broglie wave (or matter wave) is a probability wave, it is very different with electromagnetic wave Chap 15 Quantum Physics 121 Physics At some moment the probability one finds the particle in entire space is Normalization Condition Ψ 2 dV 1 (Bound State) Standard Condition Wave function is single-valued, real, finite function Chap 15 Quantum Physics 122 Physics Erwin Schrodinger,1887 - 1961 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) Chap 15 Quantum Physics 123 Physics 2. Schrodinger Equation (1) free particle Schrodinger equation Free particle plane wave function Ψ ( x, t ) 0e i 2π ( Et px ) h taking 2nd order partial derivative with respect to x and 1st order partial derivative with respect to t Chap 15 Quantum Physics 124 Physics One gets Ψ 4π p Ψ 2 2 x h 2 2 Free particle 2 Ψ i 2π EΨ t h (v c) E Ek p 2mEk 2 1-dimension free particle time-dependent Schrodinger equation h Ψ h Ψ 2 i 2 8π m x 2 π t 2 2 Chap 15 Quantum Physics 125 Physics (2) Particle in potential field with potential energy V p : E Ek Vp 1-dimensional time-dependent Schrodinger equation h 2 2Ψ h Ψ 2 Vp ( x, t )Ψ i 2 8π m x 2π t (3) particle in stationary potential 2 p Vp ( x ) time-indep. E Vp 2m Chap 15 Quantum Physics 126 Physics Ψ ( x, t ) 0 e i 2 π ( Et px ) / h 0e e i 2 πpx / h ( x) (t ) ( x) 0 e i 2 πpx / h i 2 πEt / h 1-dimensional stationary Schrodinger equation in any potential field d 8π m ( E Vp ) ( x) 0 2 2 dx h 2 2 Chap 15 Quantum Physics 127 Physics Stationary Schrodinger equation in 3dimensional potential field 8π m 2 2 2 ( E Vp ) 0 2 x y z h 2 2 2 2 2 2 2 2 2 2 2 Lapalce operator x y z 2 8 π m 2 2 ( E Vp ) 0 h Stationary wave function Chap 15 ( x, y , z ) Quantum Physics 128 Physics e.g., stationary Schrodinger equation for hydrogen atom 2 2 2 e 8 π m e 2 Vp 2 (E ) 0 2 2 4πε0 r h 4πε0 r Properties of stationary wave function (1)E is time-independent 2 (2) is time-independent Chap 15 Quantum Physics 129 Physics wave function single-valued, finite, continuous (1) x, y, z 2 dxdydz 1 normalization , , (2) 和 continuous x y z (3) ( x, y, z ) is finite, single-valued Chap 15 Quantum Physics 130 Physics 3. 1-dim. Potential Well Particle potential energy V p satisfies boundary condition 0, 0 x a Vp Vp , x 0, x a (1)Simplified model for free electron gas model of metal in solid physics (2)Demonstrate QM basic concepts and principles with simple math Chap 15 Quantum Physics 131 Physics Ep , x 0, x a Ep 0, ( x 0, x a) Ep 0, 0 xa d 2 8π 2 mE 0 2 2 dx h o a x 8 π 2 mE k 2 h d 2 2 k 0 2 dx Chap 15 Quantum Physics 132 Physics d 2 k 0 2 dx Ep 2 ( x) A sin kx B cos kx o a x Wave function single-value, finite, and continuous x 0, 0, B 0 ( x) A sin kx Chap 15 Quantum Physics 133 Physics x a, A sin ka 0 Ep sin ka 0 sin ka 0, ka nπ nπ k , n 1,2,3, a 2 8π mE k h2 o a x quantum number 2 h E n2 8ma 2 Chap 15 Quantum Physics 134 Physics ( x) A sin kx Ep nπ k , n 1,2,3, a nπ ( x) A sin x a Normalization 2 a 2 A sin 0 nπ xdx 1 a Chap 15 o 2 a a x dx dx 1 * 0 2 A a Quantum Physics 135 Physics hence Ep nπ 2 k A a a ( x) A sin kx o a x 2 nπ ( x) sin x , (0 x a) a a d 8 π mE 0 2 2 dx h 2 wave equation Chap 15 2 Quantum Physics 136 Physics Ep wave function (x) 0 , ( x 0, x a) 2 nπ sin x , (0 x a) a a o a x 2 2 nπ Prob. density ( x) sin x a a 2 Energy Chap 15 2 h 2 En n 8ma 2 Quantum Physics 137 Physics Discussions: Ep 1. energy quantization Energy 2 h En n 2 8ma 2 o h E1 , ( n 1) 2 8ma 2 g.s. Energy a x 2 h 2 excited state En n 2 n E1 , (n 2,3,) 2 8ma the particle’s energy in 1-dim. infinity square well is quantized. Chap 15 Quantum Physics 138 Physics (2) the prob. density that particle appears in the well is different Wave function ( x) 2 nπ sin x a a 2 2 nπ Prob. density ( x) sin ( x) a a 2 e.g., when n =1, the maximum probability is at the place x = a /2 Chap 15 Quantum Physics 139 Physics (3) wave function is standing wave, the nodes locate at the wall, the No. of valley equals quantum number n ( x) A sin nπ x a ( x) 2 n 2 nπ sin 2 x a a n 2 n4 16E1 n3 9E1 n2 n 1 x0 a Chap 15 x0 Quantum Physics 4E1 E1 a Ep 0 140 Physics 4. 1-dim. Square Well, Tunneling Effect 1-dim. Square Well Vp ( x) 0, x 0, x a Vp0 , 0 x a Particle’s Energy Vp ( x) Vp 0 E Vp 0 Chap 15 Quantum Physics o a x 141 Physics Tunneling Effect Wave functions in different regions (x) 2 1 o a 3 x 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 Chap 15 Quantum Physics 142 Physics Applications STM (1981) Scanning Tunneling Microscopy AFM (1986) Atom Force Microscopy Xenon on Nickel Single atom lithography Chap 15 Quantum Physics 143 Physics Quantum Corrals Iron on Copper Chap 15 Imaging the standing wave created by interaction of species Quantum Physics Physics 1. Schrodinger Equation of Hydrogen Atom Potential energy of electron in H-atom 2 e Vp 4πε0 r Stationary Schrodinger equation: 2 2 8 π m e 2 2 (E ) 0 h 4πε0 r Chap 15 Quantum Physics 145 Physics Spherical Coordinates Transform to spherical polar coordinates because of the radial symmetry x r sin cos y r sin sin z r cos r x2 y 2 z 2 z r y tan 1 x cos 1 Polar angle Azimuthal angle Chap 15 Quantum Physics Physics In Spherical coordinates: 1 2 1 1 2 (r ) 2 (sin ) 2 2 2 r r r r sin r sin 2 8π m e 2 (E ) 0 h 4πε0 r 2 2 Separable solution, let (r, , ) R(r )Θ( )Φ( ) Chap 15 Quantum Physics 147 Physics We get d 2Φ 2 ml Φ 0 2 d 2 ml 1 d dΘ (sin ) l (l 1) 2 d sin Θ sin d 1 d 2 dR 8π 2 mr 2 e2 (r ) (E ) l (l 1) 2 R dr dr h 4πε0 r Chap 15 Quantum Physics 148 Physics 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 1 En 2 E1 n =1,2,3,... Principal quantum n 4 number me E1 2 2 13.6 (eV) 8 0 h Chap 15 Quantum Physics 149 Physics (2) Angular momentum quantization and angular quantum number h Angular momentum: L l (l 1) 2π l 0, 1, 2, , (n 1) Orbital angular quantum number E.g.,n =2,l = 0,1 corresponds to h L0 L 2 2π Chap 15 Quantum Physics 150 Physics (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 h Lz ml ml 2π ml 0,1,2,,l magnetic quantum number h / 2 π reduced Planck const. Chap 15 Quantum Physics 151 Physics h h e.g., when l 1 L l (l 1) 2 2 2π 2π magnetic quantum number ml =0, 1 and Lz 0, h , h 2π z z LZ 2π L ħ o L 2 ħ Chap 15 Quantum Physics 152 Physics (4) Spin and spin quantum number Spin angular momentum S s( s 1) 1 3 where spin quantum number s S 2 2 Spin angular momentum takes only two components along applied magnetic field: 1 S z ms ms 2 ms spin magnetic quantum number Chap 15 Quantum Physics 153 Physics 1 ms 2 S z / 2 Spin angular momentum and spin magnetic quantum number of electron z Sz S Sz 1 2 o 1 2 Chap 15 Quantum Physics 1 ms 2 S 3 2 1 ms 2 154 Physics (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 Chap 15 Quantum Physics 155 Physics 3. Ground state radial wave function and distribution probability (1) Ground state energy Ground state n=1 l=0 Radial wave function equation: 1 d 2 dR 8π mr e (r ) (E )0 2 R dr dr h 4πε0 r 2 2 2 solution R Ce r / r1 Chap 15 Quantum Physics 156 Physics where 2 r1 h /(8π mE ) 2 2 8π 2 me2 2 r 0 2 r1 4πε0 h Substitute into ε0 h r1 0.052 9 nm 2 πme 2 get 2 h E 2 2 13.6 eV 8π mr1 Chap 15 Quantum Physics 157 Physics (2) Ground state radial wave function R Ce r / r1 the probability that electron appears in volume element dV: Ψ dV R Θ Φ r sin drdd 2 2 2 2 2 let the prob. density along radial vector p, the prob. that the electron appears in (r , r+dr) 2 pdr R r dr 2 Chap 15 π 0 2 Θ sin d Quantum Physics 2π 0 2 Φ d 158 Physics 2 2 from normalization pdr R r dr 2 0 pdr 0 R r dr 1 R Ce 0 2 C e 2 r / r1 1/ 2 2 r / r1 r dr 1 2 4 C 3 r1 1/ 2 4 r / r1 g.s. radial wave function is R (r ) 3 e r1 Chap 15 Quantum Physics 159 Physics (3) Probability Density Distribution of Electron p(r) p(r ) r 2 2 o r1 r Chap 15 Quantum Physics 160 Physics Light Amplification by Stimulated Emission of Radiation Chap 15 Quantum Physics 161 Physics 1. Spontaneous and stimulated radiations (1) Spontaneous radiation the process by which an atom in an excited state with higher energy E2 undergoes a (spontaneous) transition to a state with a lower energy E1 , e.g., the ground state, and emits a photon, the frequency of the radiation is determined by E2 E1 h Chap 15 Quantum Physics 162 Physics Spontaneous Radiation E2 . E2 。 h . E1 E1 Before Radiation After Radiation E2 E1 h Chap 15 Quantum Physics 163 Physics (2) Absorption of light the process by which an atom in a state with lower energy E1 , e.g., the ground state, absorb a photon energy h , spontaneously transit to a state with a higher energy E 2 , and E2 E1 h E2 Excited Absorption E1 . E2 h E1 Before Absorption Chap 15 . 。 After Absorption Quantum Physics 164 Physics (3) Stimulated radiation the process by which an atomic electron at energy level E2 , interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level E1 , 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 h E2 E1 Chap 15 Quantum Physics 165 Physics Stimulated Radiation E2 . h E1 Before E2 。 E1 . After h h 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 Chap 15 Quantum Physics 166 Physics 2. The principle of laser (1) Normal and inverse distribution of population N i Ce Ei / kT N1 / N 2 e N1 E1 ( E1 E2 ) / kT N 2 E2 known E2 E1 N1 N 2 shows that the electron population at lower energy level greater than that at higher level, this is normal distribution Chap 15 Quantum Physics 167 Physics N 2 N1 is instead inverse distribution of population, or simply population inversion Population normal distribution and inversion E2 E1 N2 E2 ............. .. 。 。 。 。 。 。 。 。 。N1 。 。 。 。 E1 。 。。 。。 N1 .. .. . E2 E1 Normal Chap 15 E2 E1 N2 Inversion Quantum Physics 168 Physics T. H. Maiman (U.S. physicist) made the first functional ruby laser in sept., 1960 E3 。 Excited state . E Metastable state 2 . 。 Ground E1 state Energy level of ion Cr in Ruby laser Chap 15 Quantum Physics 169 Physics (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 condition l k 2 Chap 15 Quantum Physics 170 Physics Optical resonator . Laser beam l HRM PTM Demonstration of O.R. Chap 15 Quantum Physics 171 Physics 3. Laser (1) Helium-Neon Gas Laser PTM: partially transmissive mirror HRM: highly reflectance mirror A K PTM HRM He-Ne Laser Chap 15 He 1 2 Metastable Ne 632.8 nm 3 Ground state Energy levels of He and Ne Quantum Physics 172 Physics HELIUM-NEON GAS LASER Chap 15 Quantum Physics 173 Physics (2) Ruby (CrAlO3) laser Its active medium is ruby crystal rod, generates pulse laser with wavelength 694.3 nm. Pulse High reflectance mirror 。。 U 0 Ruby rod Partialy transmissiv e mirror 。 U。 Demo. of Ruby Laser Chap 15 Quantum Physics 174 Physics NEODYMIUM YAG LASER Rear Mirror Adjustment Knobs Safety Shutter Polarizer Assembly (optional) Coolant Beam Tube Adjustment Knob Output Mirror Beam Q-switch (optional) Beam Tube Nd:YAG Laser Rod Flashlamps Pump Cavity Laser Cavity Harmonic Generator (optional) Courtesy of Los Alamos National Laboratory Chap 15 Quantum Physics 175 Physics 4. Characteristics and Applications of Laser (1) highly-directional, a laser collimator can reach accuracy of 16 nm/2.5 km. (2) highly-monochromatic, 1010 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 Chap 15 Quantum Physics 176 Physics Incandescent vs. Laser Light 1. Many wavelengths 1. Monochromatic 2. Multidirectional 2. Directional 3. Incoherent 3. Coherent Chap 15 Quantum Physics 177 Physics 1. Energy Gap of Solids Fully Separated Energy Levels of Two H-atom 2p 2s 2p 2s 1s 1s e A + e Chap 15 e + B e Quantum Physics 178 Physics Six closed H-atom’s energy level split Two closed H-atom’s energy level split E E O 2p 2s 2s 1s 1s r r O E Energy Band of Solids 2s r O Chap 15 Quantum Physics 179 Physics 2(2l 1) quantum states per energy level Electron distribution of different energy bands in Na 3p 2(2l 1) electrons per N 3s 6N 2p 2N 2s 2N 1s energy level 2(2l 1) N electrons per energy band Chap 15 Quantum Physics 180 Physics Experiments show that: The interval between the highest and the lowest energy level in a energy band is less than the 2 10 eV , the number of N atoms is of order of 19 3 order 10 mm , hence the distance of the neighboring energy levels is about 10 2 eV/1019 10 17 eV Chap 15 Quantum Physics 181 Physics Energy band of crystals E Conduction Forbi band Empty band Eg CondForbi uction -dden band band Eg Valence band (not full) -dden band Valence band (full) Chap 15 Quantum Physics 182 Physics Comparison between Conductor, Semi-conductor and Insulator Conductor Semiconductor Insulator Resistance 10 8 ~ 10 4 (Ω m) Temp. Coeff. Pos. + F-band V-band Not full Chap 15 10 4 ~ 108 108 ~ 10 20 Neg. - Neg. - Small Large Full Quantum Physics Full 183 Physics Typical Semiconductors Silicon GaAs Diamond Cubic Structure ZnS (Zinc Blende) Structure 4 atoms at (0,0,0)+ FCC translations 4 atoms at (¼,¼,¼)+FCC translations Bonding: covalent 4 Ga atoms at (0,0,0)+ FCC translations 4 As atoms at (¼,¼,¼)+FCC translations Bonding: covalent, partially ionic Chap 15 Quantum Physics Physics 2. Intrinsic and Extrinsic semi-conductor (1) Intrinsic: pure, no dopants Normal Bond in Ge electron e Eg C-band Ge Ge Ge Ge Ge Ge Ge Ge Ge F-band Electrons are excited, Holes appear e Full band Ge Ge G e e G e Ge e Ge Ge Ge Ge hole Chap 15 Quantum Physics 185 Physics (2) Extrinsic semiconductor) Electron type (n-type) Phosphorus atom are dopant Si atoms are hosts,As Si Si Si Si Si Si eS i Si As Si Si Si Si C-band Si Si Si Si Si Si Si Si Donor Level Si Donor level Si Si Chap 15 V-band Si Quantum Physics 186 Physics p-type semiconductor Boron atom doping into Ge atom lattice Acceptor level Hole C band Ge Ge Ge Ge Ge B Ge Ge Acceptor level Ge Ge Chap 15 Ge Quantum Physics V band 187 Physics 3. PN Junction Current-Volt Characteristics of pn Junction p n I U I U p n U I Chap 15 Quantum Physics 188 Physics p e e Hole n p e e -- --- --- n + + ++ ++ ++ ++ x0 Electron Voltage variation between p-layer and n-layer U0 x x0 Chap 15 Quantum Physics 189 Physics 4. Photovoltaic effect e e n e P e Light γ e e Photovoltaic effect is the creation of voltage or electric current in pn upon exposure to light Chap 15 Quantum Physics 190 Physics 1. The transition temperature of superconductor R ( ) around T=4.20K risistance is ZERO 0.150 ** * 0.100 0.050 Tc : the critical temperature 0.000 4.00 Chap 15 4.20 4.40 Quantum Physics T /K 191 Physics 2. Major Properties of Superconductors (1) Null resistance When T Tc , I I c (critical electric flow) resistance 0 conductance (2) Critical magnetic field The critical point of applied magnetic fields that breaks the superconducting states Chap 15 Quantum Physics 192 Physics T 2 H c H 0 1 ( ) Tc H H c (T ) T 0 K , Hc H0 Normal Super- conductor (3) Meissner effect o TC T dΦ d( B S ) E dl dt dt Chap 15 Quantum Physics 193 Physics in superconductor E 0 when H applied H c dB / dt 0 H in 0 H H H in 0 S N I Chap 15 Quantum Physics 194 Physics 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. Chap 15 Quantum Physics 195 Physics BCS=Bardeen, Cooper, Schrieffer Chap 15 Quantum Physics 196 Physics 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 e Chap 15 deformation of lattice Quantum Physics 197 Physics Chap 15 Quantum Physics 198 Physics 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 10 6 m their spins and momenta are opposite, the total momenta is zero. Chap 15 Quantum Physics 199 Physics 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 Chap 15 Quantum Physics 200