PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101

Transcription

1 PHY 114 A General Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lecture 3 (Chapter 40-4): Some topics in Quantum Theory 1. Particle behaviors of electromagnetic waves. Wave behaviors of particles 3. Quantized energies 4/19/01 PHY 114 A Spring Lecture 3 1

2 4/19/01 PHY 114 A Spring Lecture 3

3 Part of SPS zone 5 conference April 0-1, 01 Offer 1 point extra credit for attendance* *After the lecture, me that you attended. In the following exchange you will be asked to answer one question about the lecture. 4/19/01 PHY 114 A Spring Lecture 3 3

4 Webassign hint: d sinθ mλ 1 For N 4160 grooves/cm, d N 4/19/01 PHY 114 A Spring Lecture 3 4

5 Webassign hint: Condition for bright spot : d sinθ mλ λ 4/19/01 PHY 114 A Spring Lecture 3 5

6 If you have not already done so please reply to my concerning your intentions regarding Exam 4. The material you have learned up to now in PHY 113 & 114 was known in 1900 and is basically still true. Some details (such as at high energy, short times, etc. ) have been modified with Einstein s theory of relativity, and with the development of quantum theory. 4/19/01 PHY 114 A Spring Lecture 3 6

7 Which of the following technologies do not need quantum mechanics. A. X-ray diffraction B. Neutron diffraction C. Electron microscope D. MRI (Magnetic Resonance Imaging) E. Lasers Which of the following technologies do not need quantum mechanics. A. Scanning tunneling microscopy B. Atomic force microscopy C. Data storage devices D. Microwave ovens E. LED lighting 4/19/01 PHY 114 A Spring Lecture 3 7

8 Image of Si atoms on a nearly perfect surface at T7 K. Image made using atomic force microscopy. From Physical Review Letters March 0, Volume 84, Issue 1, pp /19/01 PHY 114 A Spring Lecture 3 8

9 Quantum physics Electromagnetic waves sometimes behave like particles one photon has a quantum of energy Ehf Particles sometimes behave like waves momentum ph/λhf/c wavelength of particle related to momentum: λh/p quantum particles can tunnel to places classically forbidden Stationary quantum states have quantized energies 4/19/01 PHY 114 A Spring Lecture 3 9

10 Classical physics Wave equation for electric field in Maxwell s equations (plane wave boundary conditions): E t c E x for example: E( x, t) E max ˆsin j ( k( x ct) ) Equation for particle trajectory r(t) in conservative potential U(r) and total energy E 1 dr m dt + U for example: ( r) r( t) E r 4/19/01 PHY 114 A Spring Lecture v 0 t 1 gkˆ t

11 Particle wave properties in classical physics Particle properties Wave properties Position as a function of time is known -- r(t) Phonomenon is spread out over many positions at an instant of time. Particle is spatially confined when E U(r). Particles are independent. Notion of spatial confinement non-trivial. Interference effects. 4/19/01 PHY 114 A Spring Lecture 3 11

12 Mathematical representation of particle and wave behaviors. Consider a superposition of periodic waves at t0: E( x, t) E max i sin ( k i x ) single wave (one value of k) superposed wave (many values of k) 4/19/01 PHY 114 A Spring Lecture 3 1

13 [ ] ( ) E ( x,0) E sin k i x i max k 1 x k π x k 10 x x smaller more particle like k smaller more wave like 4/19/01 PHY 114 A Spring Lecture 3 13

14 x k π Heisenberg s uncertainty principle De Broglie s particle moment wavelength relation: h h / π p k λ λ/π Heisenberg s hypotheses: x p t E h Js evs 4/19/01 PHY 114 A Spring Lecture 3 14

15 4/19/01 PHY 114 A Spring Lecture 3 15 Wave equations Electromagnetic waves: Matter waves: (Schrödinger equation) x c t E E ( ) ( ) t x x U x m t x t i, ) (, Ψ + Ψ

16 Comparison of different wave equations Electromagnetic waves Vector E or B fields Second order t dependence Examples: E B y z ( x, t) ( x, t) E E c max max sin sin ( kx ωt) ( kx ωt) Matter waves Scalar probability amplitude First order t dependence Examples: Ψ( x, t) Ψ 1 Ψ r, t) πa E sin( kx) 0 e iet / r / a0 ie0t / ( e e 3 0 e 8πε a 4/19/01 PHY 114 A Spring Lecture 3 16

17 What is the meaning of the matter wave function Ψ(x,t)? Ψ(x,t) is not directly measurable Ψ(x,t) is measurable represents the density of particles at position x at time t. For a single particle system represents the probability of measuring particle at position x at time t. For many systems of interest, the wave function can be written in the form Ψ(x,t) ψ(x)e -iet/ Ψ(x,t) ψ(x) Ψ( x, t) dx 1 4/19/01 PHY 114 A Spring Lecture 3 17

18 4/19/01 PHY 114 A Spring Lecture 3 18 Wave-like properties of particles Louis de Broglie suggested that a wavelength could be associated with a particle s momentum x i x h i h p π λ Wave equation for particles Schrödinger equation ( ) ( ) t x t h i t x x U x m, π, ) ( Ψ Ψ + ( ) ( ) ( ) ( ),, ) ( ψ, Stationary - state wavefunctions : / t x E t x x U x m e t iet Ψ Ψ + Ψ r r

19 Example - - free particle - - m x Ψ ( x, t) EΨ( x, t) Ψ E k ( x,t) Ψ k m π λ 0 U ( r) sin(kx)e λ me ( r, t) ψ( r) h mλ 4/19/01 PHY 114 A Spring Lecture : -iet/ h Ψ or E e iet / Example: Suppose we want to create a beam of electrons (m9.1x10-31 kg) for diffraction with λ1x10-10 m. What is the energy E of the beam? E h mλ 91. ( 34 ) J ( 10 kg 10 m) J 150eV

20 Electron microscope Typically E10,000-00,000 ev for high resolution EM From Microscopy Today article May 009 4/19/01 PHY 114 A Spring Lecture 3 0

21 Electrons in an infinite box: E ψ ψ ( x) ψ( x) nπx L for 0 x ( x) ψ sin n 1,,3 0 m x L π n E n m 4/19/01 PHY 114 A Spring Lecture 3 1

22 Electrons in a finite box: finite probability of electron existing outside of classical region 4/19/01 PHY 114 A Spring Lecture 3

23 Why would it be interesting to study electrons in a finite box? A. It isn t B. It is the mathematically most simple example of quantum system C. Quantum well systems can be manufactured to design new devices 4/19/01 PHY 114 A Spring Lecture 3 3

24 Tunneling of electrons through a barrier surface region tip vacuum 4/19/01 PHY 114 A Spring Lecture 3 4

25 How a scanning tunneling microscope works: Developed at IBM Zurich by Gerd Binnig and Heinrick Rohrer who received Nobel prize in /19/01 PHY 114 A Spring Lecture 3 5

26 Visualization of ψ(x) A surface if a nearly perfect Si crystal Physical Review Letters -- March 0, Volume 84, Issue 1, pp /19/01 PHY 114 A Spring Lecture 3 6

27 The physics of atoms Features are described by solutions to the matter wave equation Schrödinger equation: reduced mass of electron and proton i, t, Ψ m r Stationary - state wavefunctions : ( r t) + U ( r) Ψ( r t) Solutions : Ψ E a 0 ( r, t) ψ( r) n Z e 8πε a 4πε0 me 0 0 e Ze 4πε iet / 1 n 0 r nm Z n ev 4/19/01 PHY 114 A Spring Lecture 3 7

28 Form of probability density for ground state (n 1) ψ( ) 4πr r 4/19/01 PHY 114 A Spring Lecture 3 8

29 Angular degrees of freedom -- since the force between the electron and nucleus depends only on distance and not on angle, angular momentum L r x p is conserved. Quantum numbers associated with angular momentum: Notation: L ( + 1) 0,1,,... ( n 1) Lz m m total 0 s, 1 p, d of + 1 states s p 4/19/01 PHY 114 A Spring Lecture 3 9

30 Summary of results for H-atom: E n Z 13.6 n ev n 4 n 3 n Balmer series spectra degeneracy associated with each n: n n 1 4/19/01 PHY 114 A Spring Lecture 3 30

31 Atomic states of atoms throughout periodic table: E m r ( r) + U ( r) ψ( r) ψ effective potential for an electron in atom 4/19/01 PHY 114 A Spring Lecture 3 31

32 Example: Cu (Z9) 1s s p 6 3s 3p 6 3d 10 4s 1 s p d 4/19/01 PHY 114 A Spring Lecture 3 3

33 ψ( ) 4πr r 4/19/01 PHY 114 A Spring Lecture 3 33