* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Document
Canonical quantization wikipedia , lookup
Hidden variable theory wikipedia , lookup
EPR paradox wikipedia , lookup
Bremsstrahlung wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Renormalization wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Tight binding wikipedia , lookup
Particle in a box wikipedia , lookup
Geiger–Marsden experiment wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Atomic orbital wikipedia , lookup
Elementary particle wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Electron configuration wikipedia , lookup
Hydrogen atom wikipedia , lookup
Double-slit experiment wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Wave–particle duality wikipedia , lookup
Modern Physics (I) Chap 3: The Quantum Theory of Light Blackbody radiation, photoelectric effect, Compton effect Chap 4: The Particle Nature of Matter Rutherford’s model of the nucleus, the Bohr atom Chap 5: Matter Waves de Broglie’s matter waves, Heisenberg uncertainty principle Chap 6: Quantum Mechanics in One Dimension The Born interpretation, the Schrodinger equation, potential wells Chap 7: Tunneling Phenomena (potential barriers) Chap 8: Quantum Mechanics in Three Dimensions Hydrogen atoms, quantization of angular momentums Chapter 3: The Quantum Theory of Light Emission of electromagnetic radiation by solids – continuous spectra (Cf. emission and absorption spectra of atoms – discrete spectra) 科學發展月刊 2005/11 Wavelength of radiation near peak of emission spectrum determines color of object 500oC 700oC 1000oC Radiation at wavelength longer than optical 2500oC Increase fraction in optical wavelengths hc maxT constant 5k B The Problem to answer: The problem is to predict the radiation intensity at a given wavelength emitted by a hot glowing “solid” at a specific temperature T (in thermal equilibrium) To calculate the energy per unit volume per unit frequency of the radiation within the blackbody cavity u = u (f, T ) Energy density in frequency range from f to f+df 1 u(f ,T)df N ( f )df V : average energy per mode P , T A(T )e P d / k BT 0 e / k BT k BT Boltzmann distribution Rayleigh-Jeans formula P 0 P 0 n 0 nhf Ae nhf / kBT Ae nhf / k BT hf ehf / kBT 1 0 nhf (n = 1, 2, 3, ......) Planck’s blackbody radiation theory Energy density in frequency range from f to f+df 2 1 8 f hf N ( ) d u(f )d ,Tdf N ( f )df 3 hf / k BT df V V c e 1 5 1 e hc / k BT 1 d To fit the data by Coblentz (1916) Planck obtained h = 6.5710-34 J · s uT() u ,T d 8 hc (h = 6.626 10-34 J s) h: a very small number which plays significant roles in microscopic worlds Planck thought that his concept of energy quantization was merely a desperate calculational device and moreover a device that applied only in the case of blackbody radiation Einstein elevated quantization to the level of a universal phenomenon by showing that light itself was quantized Photoelectric effect Photoelectric Effect: Einstein’s quantization theory of light Radiant energy is quantized and is localized in a small volume of space, and that it remains localized as it move away from source with a speed of c wave package Radiant EM waves seems like many packages (grains) of energy. Each has energy hf Total radiant energy E = nhf cutoff VS KEmax eVs hf When a light quantum hit an electron, it can either be absorbed completely or no reaction 0 f 1/e 2/e 3/e metals 1, 2, 3 1 < 2 < 3 Compton Effect (1922) Between 1919–23, Compton showed that x-rays collide elastically with electrons, in the same way that two particles would elastically collide What does this tell us ? • Light “particles” (photons) carry momentum ! P E photon E is the photon energy c c is the speed of light • Earlier result that Ephoton= hf = hc/ P h h is Planck’s constant is the wavelength of light graphite (carbon) Experimental details A beam of x-ray of wavelength o is scattered through an angle by a metallic foil, the scattered radiation contains a well-defined wavelength which is longer than o o=0.0709nm Photon can scatter off matter When 0, one more peak with > o appears. depends on o h 1 cos mec 0.00243 1 cos [nm] Collision of particles ŷ after collision before collision x-ray E o hf o Po hf o c x-ray x̂ e- e- E e m ec 2 E hf E'e me2c4 Pe2 c2 Pe 0 P hf Pe 0 Conservation of momentum c hf o hf cos Pe cos c c Conservation of energy hf sin Pe sin c hf o mec2 hf me2c4 Pe2 c2 Summary: Planck: energy quantization of oscillators in the walls of a perfect radiator Einstein: extension of energy quantization to light in the photoelectric effect Compton: further confirmation of the existence of the photon as a particle carrying momentum in x-ray scattering experiments Rutherford’s model of the Nucleus The Bohr Atom Constituents of atoms (known before 1910) There are electrons with measured charge and mass There are positive charge to make the atom electrical neutral The size of atom is known to be about 10-10 m in radius Rutherford’s scattering experiment Projectile: particle with charge +2e Target: Au foil KE 5 MeV ! Rutherford’s -particle Scattering Experiment (1911) To probe the distribution of the positive charge with a suitable projectile How is the mass of the positive charge distributed within the atom? Experimental results: (Geiger and Marsden) 99% of deflected particles have deflection angle 3o However, there are 0.01% of particles have larger angle > 90o Rutherford’s model of the structure of the atom to explain the observed large angle scattering A single encounter of particle with a massive charge confined to a volume much smaller than size of the atom Nucleus -14 r ~ 10 4 m 10 R All positive charges and essentially all its mass are assumed to be concentrated in the small region Rutherford’s scattering model (r, ) (m,v) fast, massive particles r r (t ) (t ) b b: impact parameter Trajectory of particle (r, ) Ze+ 2e Ze rˆ m d 2r r d 2 rˆ Deflection due to Coulomb interaction: 4 o r P( )d 2 2 dt dt # of particles detected by detector at scattering angle d N D ( )d NP(b)db 1 sin 4 / 2 # of particles detected by detector at scattering angle N D ( )d ND 1 sin 4 / 2 sin 4 / 2 In the case when the KE of the particle is so high that the equation begins to fail, this distance of the closest approach is approximately equal to the nuclear radius mv 2 (2e)( Ze) 2 4 o D D 5 1015 m (Rutherford assumed that particles do not penetrate the nucleus) Rutherford Scattering: Rutherford’s calculations and procedures laid the foundation for many of today’s atomic and nuclear scattering experiments By means of scattering experiments similar in concepts to those of Rutherford, scientists have elucidated (1) the electron structure of the atom, (2) the internal structure of the nucleus, and even (3) the internal structure of the nuclear constituents, protons and neutrons Einstein: Splitting the atom by bombardment is like shooting at birds in the dark in a region where there are few birds Schematics of energy levels and radiated spectrum of H atom 1890 Rydberg & Ritz formula 1 nm 1 1 R 2 2 n m R 1.0968 107 m 1 n, m integers with n < m Bohr’s quantum model of the Atom (1913) Four postulates: 1. An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb attraction between the electron and the nucleus 2. The allowed orbit is a stationary orbit with a constant energy E 3. Electron radiates only when it makes a transition from one stationary state to another with frequency f Ei E f h 4. The allowed orbit for the electron: L n n h . The quantum 2 number n labels and characterizes each atomic state n = 1, 2, 3, …… (“quantum number”) Bohr atom e- Consider an atom consists of nucleus with +Ze protons and a single electron –e at radius r r 1 Ze2 mv 2 2 4 o r r Ze Coulomb attraction Orbital angular momentum L n v mr mr Centripetal force Ln r mv n = 1, 2, 3, … n 2 4 o 2 n 2 Radius of allowed orbit: r 2 ao Z e m Z Quantized orbits !! ao Bohr radius = 0.529 Å For n = 1 and Z = 1, r = ao = 0.529 10-10 m Correct prediction for atomic size !! e- Ze Total Energy of the electron 1 Ze2 mv 2 1 ze2 E KE U 8 r 2 o 4 o r Z 2 e2 Z2 En 2 2 Eo n 8 oa0 n Eo 13.6 eV n2 r ao Z (3) f Ei E f h Z 2 Eo 1 1 c 2 2 h n f ni 1 1 1 Z R 2 2 n f ni Allowed transition Rydberg constant 2 Eo R 1.097 107 m 1 hc Good to describe the observed spectra of any Hydrogen-like atom with nucleus charge +Ze and a single orbital eH, He+, Li2+, … The Bohr atom— “Bohr’s original quantum theory of spectra was one of the most revolutionary, I suppose, that was ever given to science, and I do not know of any theory that has been more successful …… I consider the work of Bohr one of the greatest triumphs of the human mind.” (Lord Rutherford) “This is the highest form of musicality in the sphere of though.” (Einstein) 「他不但具有關於細節的全部知識,而且還始終堅定的注視著 基本原理。」(Einstein) Franck-Hertz Experiment (1914) To observe current I to collector as a function of accelerating voltage Va (6 V) Accelerating voltage (0–40 V) Retarding voltage (1.5 V) Vdrop n 4.9 V Vo When the tube is filled with low pressure of mercury vapor, there are collisions between some electrons and Hg atoms Peaks in current I with a period of 4.9 V Low-energy electrons ( a few tens of volt) Incoming electron Orbital e- nuclear Scattered electron Inelastic collision, 4.9 eV of KE of incident electron raises Hg electron from the ground state to the first excited state Inelastic collision leaves electron with less than Vs, so the electron cannot contribute to current Energy levels of outer electron of Hg atom E=0 10.4 eV 6.7 eV 2nd excited state 1240 eV nm 253 nm 4.9 eV 1st excited state 4.9 eV Ground state Confirmed by emission of single photons !! Significance of the Franck-Hertz Experiment The Franck-Hertz provided a simpler and more direct experimental proof of the existence of discrete energy levels in atoms The experiment confirmed the universality of energy quantization in atoms, because the quite different physical processes of photon emission (optical line spectra) and electron bombardment yielded the same energy levels Summary: D Rutherford’s scattering of particles from gold atoms Bohr’s model provides the explanation of the motion of electrons within the atom and of the rich and elaborate series of spectral lines emitted by the atom Chapter 5: Matter Waves de Broglie’s intriguing idea of “matter wave” (1924) Extend notation of “wave-particle duality” from light to matter For photons, P E hf h c Suggests for matter, c The wavelength is detectable only for microscopic objects h de Broglie wavelength P P: relativistic momentum E f de Broglie frequency h E: total relativistic energy The Davisson-Germer Experiment (1927) a clear-cut proof of wave nature of electrons 1 h h 2 mev eV 2 mev 2eVme d sin q Applying Ni atoms as a reflecting diffraction grating a constructive peak in excellent agreement with the de Broglie formula !! 50o Kept detector at a fixed angle and varied the accelerating voltage V d sin 1 qq (q = 1, 2, 3, ……) 1 h q q 2meVq Constructive peaks occur at wavelengths: q = 1 /q h Vq q q const. 1 2me Experiment of Davisson and Germer confirmed that low-energy electrons with mass (v << c) do have wave-like properties Wave groups and Dispersion Toward a Wave description of Matter Particle (波群與色散) “wave packet” m v Large probability to be found in a small region of space at a specific time t vg = v Wave representation “Wave group” or summed collection of waves with different wavelengths: amplitudes and relative phases chosen to produce constructive interference in small region Δx the group velocity of the matter wave = the velocity of the particle a k f ( x )e ikx dx 波 數 值 越 密 集 , 波 週包 期在 性空 越間 大的 f ( x) ikx a k e dk Matter waves are represented by wavefunctions: (x,y,z,t ) (a solution for the Schrodinger equation) Matter waves is not measurable; they require no medium for propagation (x,y,z,t ) is a complex number and is used to calculate the probability of finding the particle at a given time in a small volume of space The statistical view (Max Born): the probability of finding a particle is directly proportional to ||2 = The Heisenberg Uncertainty Principle (1927) It is impossible to determine simultaneously with unlimited precision the position and momentum of a particle If a measurement of position x is made with an uncertainty x and a simultaneous measurement of momentum Px is made within an uncertainty Px, then the precision of measurement is inherently limited by Px x /2 (momentum-position uncertainty) Similarly, E t /2 (energy-time uncertainty) Double-slit electron diffraction experiment first minimum: D sin min / 2 wave properties While the electrons are detected as particles at a localized spot at some instant of time, the probability of arrival at that spot is determined by finding the intensity of two interfering matter waves particle properties Accumulated results with each slit closed half the time The experimental result contradicts this sum of probability !! A thought experiment: Measuring through which slit the electron passes py Once one measures unambiguously which slit the electron passes through, the act of measurement disturbs the electron’s path enough to destroy the interference pattern Summary The existence of matter waves (de Broglie) Davisson-Germer experiment (electron diffraction from Ni crystal) Constructing “wave packets” by superposition of matter waves with different frequencies, amplitudes, and phases Uncertainty principles Wave-particle duality; double-slit electron diffraction experiment Need a new mechanics that incorporates both wave and particle natures of subatomic objects