PHYS 571 Radiation Physics Prof. Gocha Khelashvili http://blackboard.iit.edu login
Bohr s Theory of Hydrogen Atom
Bohr s Theory of Hydrogen Atom
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
Bohr s Theory of Hydrogen Atom E n = mk 2 Z 2 e 4 = E Z 2 2 0 2 2 n n 2 E0 =13. 6eV
Hydrogen Energy Levels
X-Ray Spectra
X-Ray Spectra
Mosley Plot E n 2 2 4 mk Z e = = hf 2 2 2 n ( ) 1/2 f = An Z b K - series b = 1 L - series b = 7.4 n f 2 n 2 4 cmk e 2 1 1 = ( Z 1) 4π 1 n 3 2 2 2 f = CR ( Z 1) 1 A = CR 1 1 n 2 1 n 2
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.
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
The de Broglie Waves Wave Property λ = h p Particle Property
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 0.165 nm From de Broglie relation for 54 ev electrons: h hc hc λ = = = = 1/2 p pc mc ev φ 3 1.24 10 ev nm 1.226 1.226 = nm = = 0.167 nm 6 1/2 1/2 1/2 ( 2 0.511 10 ev)( ev ) ( ev ) ( 54)
Diffraction of Other Particles - Neutrons Diffraction pattern produced by 0.0568 ev neutrons and a target of polycrystalline copper.
Wave Packets
Wave Packets
Wave Packets
General Wave Packet
The Probability and Wave Function
Uncertainty Principle
Uncertainty Principle
Quantum Mechanics - History
Quantum Mechanics - History 1926 - Erwin Schrodinger Wave Mechanics 1926 - 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
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
TIME-DEPENDENT SCHRDINGER EQUATION 2 2 1 Ε xt, Ε xt, = 2 2 2 c t x ( ) ( ) ( ) 2 2 Ψ xt, Ψ ( xt, ) i = + V xt, Ψ xt, 2 t 2m x ( ) ( )
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.
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
The Infinite Square Well 2 Ψ n ( x) = sin L n = 1,2,3,... nπ x L
The Infinite Square Well
Comparison with Classical Results
ELECTRON IN A BOX
PROBABILITY FOR FINDING ELECTRON IN A REGION (a < x < b)
Wave Reflection and Transmission ( ) ( ) 0 V x = 0 for x< 0 V x = V for x> 0
E < V 0 Classical Analogy
E > V 0 Classical Analogy
Wave Reflection and Transmission E > V 0
Particle Incident on a Step Potential
Wave Reflection and Transmission E < V 0
Tunnel Effect α = 2m V ( E) 0 1 2 sinh αa E E T = 1 + or for αa 1 T 16 1 e E E V0 V 4 1 V0 V 2αa