Chapter 40: Quantum Physics

Transcription

1 Chapter 40: Quantum Physics Quantum mechanics deals with the Physics of microscopic systems which are beyond our ordinary physical senses (and conventional understanding). In quantum mechanics, particles (e.g., electrons) can behave like waves, and radiation (electromagnetic waves) can behave like a particle. In classical Physics every aspect of a system (e.g., energy, momentum, position, etc) can be known simultaneously, in quantum mechanics this is not the case! By realizing the particle-wave duality of matter, and by accepting the limited information allowed to the observer, quantum mechanics has become one of the most successful (and thoroughly tested) theories ever. 1

2 1927 Solvay Conference End of 19th century-reductionism: classical mechanics can explain everything? Wrong: new experiments produce startling, paradoxical results that cannot be explained classically Quantum mechanics provides mathematical framework (Schroedinger s wave mechanics, Heisenberg s operator formalism) But what does it mean? Bohr-Einstein debates 2

3 Particle nature of radiation: Blackbody radiation Energy density T u( f, T) Rayleigh Jeans: 8πf u( f, T ) = 3 c 2 kt u f data Planck: assumed that light is quantized in packets u( perfectly! Electromagnetic radiation consists of indivisible quanta with energy E = f, T ) hf and momentum E hf p = = = c c 8πh = 3 c e h λ f hf kt photon 3 1 fits blackbody radiation curve λ = h p 3

4 Example 40-3: Bright star I=1.6x10-9 W/m 2 at Earth s surface and λ = 560nm. Rate of photons entering a night-adapted eye? 4

5 Photoelectric Effect E 1. EM radiation hits metal plate photoelectrons (PE) 2. PE emitted only if f > f 0 ( f 0 depends on metal) 3. If f held constant, number of PE depends linearly on intensity I 4. Max kinetic energy of PE independent of I, but varies linearly with f 5. No measurable time delay between arrival of photon and emission of PE 5

6 Energy and momentum are always conserved Photon (E γ, p γ ) Electron (E e, p e ) electron W (work function) Energy delivered by photon: For a single photon: For N photons: E =hf γ E total = NE γ γ I Energy delivered to exiting electron: 1 E kin 2 = mv= hf W 2 6

7 Max kinetic energy of photoemitted electron 1 2 mv = hf W 2 e of the photon (Hz) What about the number N e of photoemitted electrons? N e α 7

8 Compton effect Photon in λ Photon out λ Electron initially at rest e - 8

9 Compton effect λ λ' Warning in dealing with high energy particles! No particle can travel faster than the speed of light c. For particles that are travelling at relativistic speeds (approaching c) one cannot use the classical expressions for energy and momentum (see Ch. 40). Instead on must use relativistic expression that insure that v c p 2 E = p c + m c! pc when pc>> " mc 2m for kinetic energy pc much greater than rest mass 2 energy mc (homework 1d, 27, and 50) 29d, 55, 22 9

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11 Momentum conservation: x-direction: p= p'cosθ + p cosφ y-direction: 0 = p'sinθ p sinφ p p'cosθ = p p pp θ + p θ + p θ = cosφ p'sinθ = p sin φ square and add p e cos e 2 'cos ' cos ' sin φ + p sin p pp θ + p = p 2 'cos ' e e φ e e e (A) (41-21) ( ) E = p c + m c Energy conservation : + = + 2 E mc E' Ee ' e E = pc, E' = pc, E = p c + m c e e pc p ' c + mc = p c + m c square both sides = e + p p' m c 2 pp' 2mpc 2 mp' c p m c (41-22) (B) (A) subtract Eq from Eq pp '(1 cos θ ) 2mpc = 2 mp ' c = = (1-cos θ ) λ ' λ = # (1-cos θ ) p ' p mc mc (B) 11

12 h λ' λ = (1-cos θ) mc θ = 0 λ λ' θ θ = 45 θ = 90 θ =

13 Wave nature of matter λ λ = h p debroglie wavelength h = J s Planck s constant 2π p = # k = # = λ mv # h 2π 13

14 Example 40-5: de Broglie wavelength of a neutron: mass=1.6x10-27 kg, v=1500 m/s (35 K gas) 14

15 Experimental evidence for wavelike behavior of matter Davison-Germer electron diffraction experiment d Bragg Peak: 2d sin θ= nλ λ= sin θ n d = nm first diffraction peak (n=1) at 65 2(0.091nm) sin 65 $ λ= = nm 1 h What about λ=? p 2-19 E=54 ev= J = 2m p 54 ev = = = p 2mE 2( kg)( J) kg λ= Ji s s kgim s = nm kg m / s $ kgi m s 15 s

16 Example 40-6: angles for diffraction peaks for electrons with E= 120 ev incident on crystal whose scattering planes are 0.12 nm apart? 16

17 Neutron double slit experiment h h λ = = p 2mE slow neutrons p small λ large 17

18 Tunneling E = U + K Classical particle trapped by potential 18

19 Waves can leak out of trap barrier Total internal reflection Not-so-total internal reflection! 19

20 Tunneling fraction (or probability) of particles getting through! e 2 a 2m U E # ( ) 2 a # e a = barrier width U = average height of potential barrier m = mass of particle E= energy of particle 2m ( U E ) a energy position E <U> 20

21 Heisenberg Uncertainty Relations Time or position Due to wave nature of matter position-momentum uncertainty relation x p > # note x> x and p > p x x x h p = λ 2 p E = or E γ =hf 2m time-energy uncertainty relation ( see HW ) E t > # note E> E and t > t only important on atomic scales kg m/s For a dust particle, x=10 m 10 kgim/s For an electron, x=10 m (atom) 10 kgim/s which is 10 times larger than the electron's momentum in its classical atomic orbit (also v=0.1c)! kg m/s 21

22 Double-slit dilemma check Interference experiments on Physics Trek 2000 web site (Classical and quantum mechanical twoslit experiments demo) 22

23 Electron double-slit experiment 23

24 Resolution of double-slit dilemma Note: both slits open! p # 2 > > # y y d d sinθn = nλ Distance between two adjacent maxima on screen ymax = Dsinθ n+ 1 Dsinθ n ( n + 1) Dλ ndλ = = d d Dλ d Location of peak on screen changes by 2# D D p y d 2D / d Dλ > = = p p k πd D p Displacement due to measurement is comparable to separation between maxima p y 24

25 Double slit analysis exactly the same as for waves, but λ is very short 25

26 Ground state: Lowest energy state for a parabolic potential (simple harmonic oscillator/spring) What is lowest energy for a classical SHO E=? x=? p=? p mω x E = + 2m x = b p> (# ) 2 # b 2 2 b mω b E> + = f(b) 2m 2 df # # m b 0 b 2 2 = + ω = = ω 3 min db mb m 2 # 1 2 # Emin = + mω = # ω 0 2m m 2 mω (# ω) zero point motion or zero point energy 26

27 Example 40-8: Use position-momentum uncertainty relation to estimate the lowest energy of a particle with mass m in a 1-D box of width L. 27

28 Example: use position-momentum uncertainty relation to estimate the lowest energy of an electron (mass m) moving in an attractive Coulomb potential U(r)= -e 2 /4πε o r 28

29 Classical vs. Quantum Position: (x,y,z) P(x,y,z) well defined probability function of wave that time interferes with itself Momentum: (p,p,p ) x y z x p y p z p z x y > # > # > # Energy: E E t > # 29

30 Polarizer experiment Classical EM wave (polarized vertically) I 0 N huge Polarizer at 45 o to vertical I 0 / 2 Single photon (polarized vertically) N=1 Polarizer at 45 o to vertical? Probability=1/2 30

31 Radioactive decay A e - B lower energy Time t later N 0 at t=0 N 0 (t) Nt () = Ne 0 t τ Note that N( τ ) = N e = 0.37N are the older A atoms more likely to decay into B atoms? 31

32 Coin tossing analogy demo: Start with all heads, for each round of tossing, flip each coin that is still heads up, do not flip coins that are already tails (P=1/2 throughout). N(t) is number of remaining heads. H H T H H H H H H T H N 0 at t=0 all heads Time t (or a number of tosses) later H N(t) 32

33 T 1/ Problem The half life 2 is the time in which half of the original set of nuclei have decayed (the other half are still in the original form). Express half life in terms of the lifetime τ. 33