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PHYS541000 Quantum Mechanics(I)(II) Chung-Yu Mou Unique subjects in -- quantum mechanics and physics cosmology Cosmology: closest subject to God、can understand how God creates the w Quantum Mechanics: Describing the world by using the w that electrons can pass two slits at the same time 1 http://www.phys.nthu.edu.tw/~mou/teach.htm l http://www.phys.nthu.edu.tw/~mou/teach/16Fall_QuantumMech.ht ml s s Quantum mechanics – The new way that was developed at the beginning of the 20th century to interpret & predict behaviors of microscopic objects such as atoms, electrons, .. Purpose of this course Learn to calculate and think in the quantum mechanical way Conceptual problem will not be fully discussed in this course. In fact, as Richard Feynman remarked: s I think that it is fair to say that no one understands the quantum theory… Two tracks of discoveries h Spectroscopy (discrete) Photoelectric Effect (Einstein) Theory of H Atom (Bohr) Matter wave (De Broglie) Matrix Mechanics (Heisenberg) Wave Mechanics (Schrodinger) Equivalent (shown by Schrodinger) (based on the general formalism of Dirac) Third Formulation Richard Feynman Path Integral Equivalent to the above two formulations but may be more general Track 2: particle-wave controversy Huygens: light is wave Newton: light is particle Hook: wave Maxwell: wave Young: wave Planck, Einstein, and Compton: particle .. The Maxwell’s theory of light E Light intensity (I) ~ |E|2 B Wave-like properties of light Hygen’s principle Interference Diffractio n Particle-like properties of light Pierre Gassendi and Newton Reflection: elastic collision of particles Refraction: 1 If q1 > q 2 1y u y = u x cot q u 1y < u 2y \u 1 < u 2 2 y2 However, using n1 sinq1 = n2 sinq 2 n1 < n2 (u = c / n) Radiation and black-body radiation: all objects radiate electromagnetic waves Thermodynamics of Radiation El = Energy of emitted EM wave with l per unit area ×sec Al = energy absorption rate for wavelength l , depending on properties of the object. Il = El = Al I l Energy of incident wave with l = energy flux for wavelength l per unit area ´ sec El =I l is independent of properties of the object Al Black Body Radiation El is universal, independent of the composition Al and geometry of the object. If Aλ=1, Eλ= f(λ,T) is universal => black body radiation Ideal black body = cavity Universe » cavity: cosmic microwave background T = 2.715K Max Planck For a given n , there are about n 2 waves with different directions Classically, the averaged energy for each wave is kBT Expt & Planck Planck’s consideration is based on statistical mechanics not dynamics In an equilibrium system The probability of finding a subsystem with the energy E is e (e = 2.71828..) - E k BT Plank: energy of light is not continuous basic unit is hn , average of energy = hn e Þ e= hn e hn /k BT -1 - hn /k BT + 2hn e -2hn /k BT + 3hn e -3hn /k BT + ××× h = Planck constant =6.626×10-34 joule-sec Solid State Version : C versus T 3 3 U =N k BT (kinetic energy) N k BT (potential energy)=3Nk BT 2 2 dU C 3Nk B dT Quantum region Photoelectric effect: metal Wave-like interpretation of radiatio electron λ x Electric field experienced by electrons Expectation from wave properties electron E x Electric field experienced by electrons • The larger E is, the higher the intensity is. It is easier to shake off electrons with larger current. This is independent of wavelength/frequency. •Any frequency of light can yield photoelectrons. •Need sufficient time to accumulate energy to Photoelectric effect experiment Einstein: E=nhν is dynamical! Light is composed of light quanta, the energy of each light quantum is hn h W W = work function 1 m 2 h - 2 1 2 eV0 h - W mmax min f 0 W / h 2 Compton effect: for waves: l =l ¢ Compton:experimental results are consequences of particle’s collision Light quanta collides with electron elastically: After collision, the change of momentum for light quanta changes its E = hn E = p 2 c 2 + (mc 2 )2 = pc hn h \p= = c l Viewed as collision of particles h h = cosq + pe cosf li l f h sinq = pe sinf lf hni =hn f + Ee ® l i > l f Ee = p 2 c 2 + (me c 2 )2 Is light particle or wave? 1924 de Broglie joined the debate Is the fundamental particle –electron wave or particle? Ph.D. thesis of de Broglie: electrons are also wave-like h l= p What is particle? Wave? particle: (1) One grain by one grain, discontinous (2) trajectory Wave: (1)interference (2)diffraction Both satisfy momentum and energy conservation Need more clear and direct Expectation for electrons being particles: expectation for partic electro n source 32 Expectation for electrons being waves: 33 Electronic version of Young’s experiment electr on source Interference of waves 1 2 Destructive Constructiveinterfer interfr 1 /2 2 Quantitative analysis L d θ dsinθ (m+1/2)λ dsinθ==mλ m constructive destructive interferenceinteference y sin tan L L y d y d L L 1m l 700nm l 0.17nm d 101 mm 104 m L y d 7mm y 1.7 m 37 The Feynman Lectures on Physics (III) p. 1-4~1-5 …This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossible small scale … We are doing a “thought experiment”… Reference: Davisson and Germer: diffraction of electrons wavelength: 0.165nm(1.65 Å, 50eV) 38 Tonomura et al. American Journal of Physics 57, 117(1989) λ = 0.054Å (50kV), Va = 10V a = 0.5μm, b = 5mm 40 41 Matter waves and mechanical properties h = Planck constant (6.626×10-34 joule-sec) DeBroglie: λ=h/p Einstein: E=hν=p2/2m p 3 h = k BT Þ lth = 2m 2 3mk BT 2 free electron: lth (300K ) = 6.2nm atom: l (300K ) £ 0.2nm Bulk size L l L Nano size L Neutron Reviews of Modern Physics 60, 1067 (1988) Helium atoms Physics Review Letters 66, 2689 (1991) Macromolecules C60 http://www.quantum.univie.ac.at/research/c60/index.html Nature 409, 304(2001) Nature 401,680 (1999) (1) Diffration grating is SiNx grating (period 100 nm) with width 0.1 m. (2) C60 is thermal ionized by a laser. The ions are then accelerated and directed towards a conversion electrode. The ejected electrons are subsequently counted by a Channeltron electron multiplier. 48 Other atoms: Na, Phys. Rev. 66, 2693 (1991) Bio-molecules 3D structure of tetraphenylporphyrin C44H30N4(TPP) 3D structure of the fluorofullerene C60F48 L Hackermuller et al. Phys. Rev. Lett. 91 090408, (2003) Interference of 106 Na atoms Science 275, 637 (1997) Conclusions (i) Number of counting ~ probability of finding the particle (ii) When there are many exclusive choices, 1,2,3,.. P (total probability) ≠ P1+P2+P3+ … Particle-like behavior: P = P1+P2+P3+ … 53 How do we describe the results ? Experience from electromagentism: (i) Light intensity (I) ~ |E|2 (ii) For multiple choices: I (total intensity) ~ |E1+E2|2 ≠ |E1|2+|E2|2 A way out to escape the classical sum rule of probability P (total probability) = P1+P2 Max Born’s probability interpretation Ψ(x,y,z,t) = wave function of matter wave (complex) E(x,y,z,t) = Electric field (i) Occurrence of events (particle’s appearance) = Probability density ~ |Ψ |2 (ii) For many alternative routes (choices), each alternative route is represented by Ψi total probability ~ |Ψ1+ Ψ2+ …. |2 (principle of superposition) Difference between classical and quantum: |Ψ1+ Ψ2|2 – (|Ψ1|2+ |Ψ2|2)= Ψ1Ψ2 *+ Ψ1*Ψ2 = interference term 1 eikr 1 r1 2 eikr 2 r2 | 1 2 | |1 e 2 ik ( r1 r2 ) 2 2 2 cos k (r1 r2 ) k(r1 - r2 ) = 2p l (r1 - r2 ) = 2p path difference l | Particle-wave duality • When the material is detected, it is a whole particle that is being detected. It is particle-like. • The matter wave commands the particle where to go and arrive in different positions with the associated probability. pilot wave New concepts classical: |Ψ1|2+ |Ψ2|2 quantum: |Ψ1+ Ψ2|2 electron source The electron does not pass 1 or 2 1 “It can pass 1 and 2 at one time.” 2 Ref: The ghost in atom Which-way experiment Electron source Nature 395, 33(1998) PRL70,2359(1993) Once which-way is known... Particle-like behav Electron source Which-way experiment 198Hg+ polarized Phys. Rev. Lett. 70,2359(1993) Ground state: 6s2S1/2, Excited state: 6p2P1/2 degenerate: mJ Photons: σ & π polarized π : ΔmJ=0 two atoms are in the same state σ: |Δ mJ|=1 two atoms are not in the same state π polarized σ polarized Five quantum effects (to be covered) • • • • • Interference Quantization Tunneling effect Spin Entanglement Entanglement/糾纏 (Quantum telepathy) Entangled photon-pair experiment photon 1 photon 2 ? focal plane 焦平面上Detector之距離 Dopfer, B., 1998 Zeilinger, Rev. Mod. Phys. S288, (1999) Einstein: Spooky action at a distance! Spooky action: “ghost-like behavior” Due to the nature of probabi the propagation of informati does not exceed speed of lig 67 s I think that it is fair to say that no one understands the quantum theory… Description of plane waves x λ n = 1/ T c = ln x t T t sin(2 2 ) sin(kx t ) T ei ( kx t ) cos(kx t ) i sin(kx t ) phase Sinusoidal wave in higher dimension: being directional sin(kx - w t) Þ sin(k × r - w t) k y (x, y) l y sinq = l l x cosq = l k sinq = l k = kx2 + k y2 + kz2 x 2p k= 2p = ky ly 2p k cosq = = kx lx phase:kx x + k y y = k × r = 2p x lx + 2p y ly Counting number of waves c k z Number of possible directions for a given ν x » 4pn µ n 2 y 2