PHYS 571 Radiation Physics

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1 PHYS 571 Radiation Physics Prof. Gocha Khelashvili login

2 Bohr s Theory of Hydrogen Atom

3 Bohr s Theory of Hydrogen Atom

4 Bohr s Theory of Hydrogen Atom Electrons can move on certain (stationary) orbits without radiating. Atom radiates when electron makes transition from one stationary orbit to another: hf = E E i f f - emission frequency and NOT the frequency of circular motion. Correspondence Principle: In the limit of large orbits quantum model classical model

5 Bohr s Theory of Hydrogen Atom E n = mk 2 Z 2 e 4 = E Z n n 2 E0 =13. 6eV

6 Hydrogen Energy Levels

7 X-Ray Spectra

8 X-Ray Spectra

9 Mosley Plot E n mk Z e = = hf n ( ) 1/2 f = An Z b K - series b = 1 L - series b = 7.4 n f 2 n 2 4 cmk e = ( Z 1) 4π 1 n f = CR ( Z 1) 1 A = CR 1 1 n 2 1 n 2

10 Critique of Bohr s Theory and the Old Quantum Mechanics Inconsistent theory based on various ad hoc quantum assumptions The theory is silent about atomic transition rates Little success in applying the theory to the optical spectra of more complex atoms No reason why Coulomb s law would work but the laws of radiation would not No reason why Newton s law could be used even though only certain values of angular momentum allowed.

11 New Quantum Mechanics The de Broglie Waves Wave Packets The Uncertainty Principle Wave Particle Duality Elements of Quantum Mechanics The Schrödinger Equation in 1D Infinite Well Barrier Reflection and Transmission

13 The de Broglie Waves Wave Property λ = h p Particle Property

14 The Davisson-Germer Experiment Bragg condition for constructive interference: nλ = 2dsinθ = 2dcosα d = Dsinα nλ = Dsin 2α = Dsinφ

The Davisson-Germer Experiment D = 0 The spacing 0.215 nm (for Ni), The peak observed at =50. λ 0 = = 2 ( 2 ) 0.215sin 50 0.

15 The Davisson-Germer Experiment D = 0 The spacing nm (for Ni), The peak observed at =50. λ 0 = = 2 ( 2 ) 0.215sin nm From de Broglie relation for 54 ev electrons: h hc hc λ = = = = 1/2 p pc mc ev φ ev nm = nm = = nm 6 1/2 1/2 1/2 ( ev)( ev ) ( ev ) ( 54)

$Diffraction of Other Particles - Neutrons Diffraction pattern$

16 Diffraction of Other Particles - Neutrons Diffraction pattern produced by ev neutrons and a target of polycrystalline copper.

17 Wave Packets

18 Wave Packets

19 Wave Packets

20 General Wave Packet

21 The Probability and Wave Function

22 Uncertainty Principle

23 Uncertainty Principle

24 Quantum Mechanics - History

25 Quantum Mechanics - History Erwin Schrodinger Wave Mechanics Werner Heisenberg Matrix Mechanics Schrodinger Equation Energy, position Momentum - measurable Wave Function Represented by matrices, with measurable Not Measurable quantities as a diaginal elements Ψ 2 * =ΨΨ f = f ΨdV - Probability Expectation Values Measurable Paul Dirac Quantum Mechanics Schrodinger Both theories are equiavelent Each can be derived from another WM and MM are two formulations of theory that can be presented in very general terms

26 Quantum Mechanics vs. Classical Mechanics Find - Position, Velocity, Energy, Momentum of the object 2 2 Ψ ( xt, ) Ψ ( xt, ) + V 2 ( xt, ) Ψ ( xt, ) = i F = ma 2m x t Find -, Ψ ( xt) Find - a( xt, ) Find Expectation Values of: x = xψdx - position E = EΨdx - Energy p = pψdx - Momentun L = LΨdx - Angular Momentum Kinematics x - Position 2 d x a = = 2 dt υ - Velocity E - Energy p - Momentum L - Angular Momentum dυ dt

27 TIME-DEPENDENT SCHRDINGER EQUATION Ε xt, Ε xt, = c t x ( ) ( ) ( ) 2 2 Ψ xt, Ψ ( xt, ) i = + V xt, Ψ xt, 2 t 2m x ( ) ( )

28 The Schrödinger Equation in 1D 2 Ψ ( x) 2 + V x Ψ x = EΨ x 2 2m x ( ) ( ) ( ) Ψ ( x) - must exist and satisfy Schrodinger Equation ( x) and ( x) Ψ Ψ ( x) and ( x) Ψ Ψ ( x) and ( x) Ψ Ψ - must be continuous. - must be finite. - must be single valued. ( x) Ψ 0 as x ± - normalization integral remains bounded.

29 The Infinite Square Well ( ) V x 0, 0 < x< L =, x < L and x > L Potential is clearly artificial Exact Solution of Schrodinger Equation Closely related to vibrating string problem in classical physics Illustrates important features of all QM problems 1D potential is Relatively good approximation to some real situations - free electron in a metal. 3D potential is good approximation to some nuclear physics problems

32 The Infinite Square Well 2 Ψ n ( x) = sin L n = 1,2,3,... nπ x L

33 The Infinite Square Well

34 Comparison with Classical Results

35 ELECTRON IN A BOX

37 PROBABILITY FOR FINDING ELECTRON IN A REGION (a < x < b)

38 Wave Reflection and Transmission ( ) ( ) 0 V x = 0 for x< 0 V x = V for x> 0

39 E < V 0 Classical Analogy

40 E > V 0 Classical Analogy

41 Wave Reflection and Transmission E > V 0

42 Particle Incident on a Step Potential

43 Wave Reflection and Transmission E < V 0

44 Tunnel Effect α = 2m V ( E) sinh αa E E T = 1 + or for αa 1 T 16 1 e E E V0 V 4 1 V0 V 2αa

de Broglie Waves h p de Broglie argued Light exhibits both wave and particle properties

de Broglie Waves h p de Broglie argued Light exhibits both wave and particle properties de Broglie argued de Broglie Waves Light exhibits both wave and particle properties Wave interference, diffraction Particle photoelectric effect, Compton effect Then matter (particles) should exhibit both