PHYS541000 Quantum Mechanics(I)(II) The first hour: p1-p12 The second hour: p13-p24 The third hour: p25-p32 and more Chung-Yu Mou

Unique subjects in physics -- quantum mechanics and cosmology Cosmology： closest subject to God、can understand how God creates the world Quantum Mechanics： Describing the world by using the way that electrons can pass two slits at the same time

http://www.phys.nthu.edu.tw/~mou/teach.html

http://www.phys.nthu.edu.tw/~mou/teach/16Fall_QuantumMech.html

Quantum mechanics – Purpose of this course 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: I think that it is fair to say that no one understands the quantum theory…

Two tracks of discoveries Spectroscopy (discrete) Theory of H Atom (Bohr) Matrix Mechanics (Heisenberg) Photoelectric Effect (Einstein) Matter wave (De Broglie) 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 Light intensity (I) ~ |E|2 B

Wave-like properties of light Hygen’s principle

Interference Diffraction

Particle-like properties of light Pierre Gassendi and Newton Reflection: elastic collision of particles Refraction：

Radiation and black-body radiation: all objects radiate electromagnetic waves

Thermodynamics of Radiation

Black Body Radiation If Aλ=1, Eλ= f(λ,T) is universal => black body radiation Ideal black body ＝ cavity

Max Planck Expt & Planck

h = Planck constant =6.626×10-34 joule-sec Planck’s consideration is based on statistical mechanics not dynamics h = Planck constant =6.626×10-34 joule-sec

Solid State Version : C versus T Quantum region

Photoelectric effect: metal

Wave-like interpretation of radiation λ electron Electric field experienced by electrons λ x

Expectation from wave properties Electric field experienced by electrons electron x 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 shake off electrons.

Photoelectric effect experiment

Einstein: E=nhν is dynamical!

Compton effect: λ

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 wavelength.

Viewed as collision of particles

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

Need more clear and direct evidence ! What is particle? Wave? particle： (1) One grain by one grain, discontinous (2) trajectory Both satisfy momentum and energy conservation Wave： (1)interference (2)diffraction Need more clear and direct evidence !

Expectation for electrons being particles: expectation for particles electron source

Expectation for electrons being waves:

Electronic version of Young’s experiment electron source

Interference of waves Destructive interference Constructive interfrence

Quantitative analysis dsinθ = (m+1/2)λ destructive inteference dsinθ= mλ constructive interference dsinθ = m θ d

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)

Tonomura et al. American Journal of Physics 57, 117(1989) λ = 0.054Å (50kV), Va = 10V a = 0.5μm, b = 5mm

Matter waves and mechanical properties h = Planck constant (6.626×10-34 joule-sec) DeBroglie: λ=h/p Einstein: E=hν=p2/2m

Bulk size Nano size

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.

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+ …

How do we describe the results ? Experience from electro-magentism: (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 r1 r2

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 The electron does not pass 1 or 2 “It can pass 1 and 2 at one time.” electron source 1 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 behavior 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 ? focal plane

Zeilinger, Rev. Mod. Phys. S288, (1999) 焦平面上Detector之距離 Dopfer, B., 1998 Zeilinger, Rev. Mod. Phys. S288, (1999)

Einstein: Spooky action at a distance! “ghost-like behavior” Due to the nature of probability, the propagation of information does not exceed speed of light.

I think that it is fair to say that no one understands the quantum theory…

Description of plane waves x λ t T phase

Sinusoidal wave in higher dimension: being directional phase:

Counting number of waves Number of possible directions for a given ν