Physical Chemistry -The ATOMS of Niels Bohr and Wave Particles Duality of Light and Matter, Lecture notes of Physics

Download Physical Chemistry -The ATOMS of Niels Bohr and Wave Particles Duality of Light and Matter and more Lecture notes Physics in PDF only on Docsity! L! ! ! ! ! 5.61 Fall 2007 Lecture #4 page 1 The ATOM of NIELS BOHR Niels Bohr, a Danish physicist who established the Copenhagen school. (a) Assumptions underlying the Bohr atom (1) Atoms can exist in stable “states” without radiating. The states have discrete energies En, n = 1, 2, 3,..., where n = 1 is the lowest energy state (the most negative, relative to the dissociated atom at zero energy), n = 2 is the next lowest energy state, etc. The number “n” is an integer, a quantum number, that labels the state. (2) Transitions between states can be made with the absorption or ΔE emission of a photon of frequency ν where ν = . h En1 hν hν or Absorption Emission En2 These two assumptions “explain” the discrete spectrum of atomic vapor emission. Each line in the spectrum corresponds to a transition between two particular levels. This is the birth of modern spectroscopy. h (3) Angular momentum is quantized: ! = n" where " = 2π Angular momentum L = ! r × p "= L ! r LFor circular motion: L ! is constant if ! r and p ! are constant l = mrv is a constant of the motion ! ! p = mv Other useful properties 1 ! ! ! 5.61 Fall 2007 Lecture #4 page 2 v! = (2π r ) ⋅ ν = r ωrot rot velocity (m/s) circumference frequency angular (m/cycle) (cycles/s) frequency (rad/s) ⇒ " = mvr = mr 2ω rot Recall the moment of inertia I = ∑miri 2 i ∴ For our system I = mr2 ⇒ ! = Iω rot Note: Linear motion vs. Circular motion mass m ↔ I moment of inertia velocity v ↔ ω rot angular velocity momentum p = mv ↔ ! = Iω angular momentum Kinetic energy is often written in terms of momentum: 1 2 p2 1 m2r 2 v2 !2 K.E. = =K.E. = mv = 2 2m 2 mr 2 2I Introduce Bohr’s quantization into the Rutherford’s planetary model. For a 1-electron atom with a nucleus of charge +Ze +Ze r e- Ze2 n2 (4πε0 ) ! 2 r = 4πε0 mv2 Z me2 The radius is quantized!! ⇒ r = (4πε0 ) ! 2 ≡ a0 the Bohr radius 2me 2 5.61 Fall 2007 Lecture #5 page 2 sin(kx + π) (out of phase by λ/2) This leads to many interference phenomena Young’s 2-slit experiment D l λ = s λl ⇒ D = s Light as a particle Compton exp’t If just a wave, expect light to scatter off electron Experimentally: e- x-rays (λ ∼ 1Å) e- λ λ e- λ′ λ Destructive + = 0 interference sin(kx) light, λ s l D Interference pattern screen observed Ehν= att ce 5.61 Fall 2007 Lecture #5 page 3 The backscattered wave is red-shifted (λ′ > λ ), i.e. less energy/photon. hc hcE ′ = < = E λ′ λ Energy (and momentum) transferred to the electron. Need relativistic mechanics to solve hν ⎛ h ⎞ p = c ⎝⎜ = λ ⎠⎟ for the light Light is a particle with energy hν momentum p = c Light can behave both as a wave and as a particle!! Which aspect is observed depends on what is measured. (B) Matter Matter as particles ⇒ obvious from everyday experience Matter as waves (deBroglie, 1929, Nobel Prize for his Ph.D. thesis!) Same relationship between momentum and wavelength for light and for matter h h p = ⇒ λ = ≡ de Broglie wavelength λ p Amazing notion! But wavelength only observable for microscopic momentum Diffraction pattern e- crystal l i 5.61 Fall 2007 Lecture #5 page 4 Consequences (I) (1) on Bohr atom + If e- wave does not close on itself, eventually destructive interference will kill it! + If e- wave does close on itself, then constructive interference preserves it. Criterion for stability: nh nh2πr = nλ = = p mv nh or mvr = = n! 2π ⇒ " = n! As Bohr had assumed angular momentum is quantized! 5.61 Fall 2007 Lecture #6 page 3 ! ! 1 Δpx ≥ ⇒ Δv ≥ ≈ x10−6 m s 2Δx x 2mΔx 2 Basically, if we know the e- is in the atom, then we can’t know its velocity at all! Bohr had assumed the electron was a particle with a known position and velocity. To complete the picture of atomic structure, the wavelike properties of the electron had to be included. So how do we properly represent where the particle is?? Schrödinger (1933 Nobel Prize) A particle in a “stable” or time-independent state can be represented mathematically as a wave, by a “wavefunction” ψ(x) (in 1-D) which is a solution to the differential equation Time-independent Schrödinger equation − !2 2m ∂2ψ ∂x2 + V x x x( )ψ ( ) = Eψ ( ) potential energy total energy We cannot prove the Schrödinger equation. But we can motivate why it might be reasonable. φ1 (x,t) = Asin (kx −ωt) is a right-traveling wave. A v 2πk = ω = 2πν λν = v ω Similarly, a left-traveling wave can be represented as φ2 (x,t) = Asin (kx + ωt) . Both are solutions to the wave equation 5.61 Fall 2007 Lecture #6 page 4 ∂2φ (x,t) 1 ∂2φ (x,t) = ∂x2 v2 ∂t2 Further, the sum Ψ (x,t) = φ1 (x,t) + φ2 (x,t) of left and right traveling waves is also a solution. Ψ (x,t) = A⎡⎣sin (kx −ωt) + sin (kx + ωt) ( )cos (ωt)⎤⎦ = 2 Asin kx This is a stationary wave or standing wave. Its peaks and nulls remain stationary. At various times during a full cycle (2π/ω): ) = 2 Asin kxΨ (x,0 ( ) t = 0 2 2 ⎞⎛ 1 2π ⎞ ⎛ Ψ ⎝⎜ x, 8 ⋅ ω ⎠⎟ = 2 A ⎝ ⎜ ⎟ sin kx( ) ⎠ t = π/4ω ⎛ 1 2π ⎞ t = π/2ω nodes Ψ x, ⎝⎜ 4 ⋅ ω ⎠⎟ = 0 − 2 2 ⎞ t = 3π/4ω ⎛ 3 2π ⎞ ⎛ Ψ ⎝⎜ x, 8 ω ⎠⎟ = 2 A ⎝ ⎜⋅ ⎟ sin kx( ) ⎠ t = π/ω ⎛ 1 2π ⎞ Ψ x, ⎠⎟ = 2 A −1 ( ) ⎝⎜ 2 ⋅ ω ( )sin kx As in a vibrating violin string, the node positions are independent of time. Only the amplitude of the fixed waveform oscillates with time. More generally, we can write wave equation solutions in the form Ψ x,t) = ψ ( )cos (ωt)( x In the particular case above, ψ x ( ) .( ) = 2 Asin kx 5.61 Fall 2007 Lecture #6 page 5 For the general case, ∂2 Ψ ∂x ( 2 x,t) v 1 2 ∂2Ψ ∂t ( 2 x,t) − v ω 2 2 Ψ (x,t) = −k 2Ψ (x,t) (ω k = v)= = Plugging in Ψ x,t) = ψ ( )cos (ωt)( x ∂2ψ x ⎛ 2π ⎞ 2( ) ( ) = −⇒ ∂x2 = −k 2ψ x ⎝⎜ λ ⎠⎟ ψ ( ) x ∂2ψ x 2 de Broglie relation λ = h p ⇒ ∂x ( ) = − ⎝⎜ ⎛ ! p ⎠⎟ ⎞ ψ x( ) 2 ∂2ψ x( ) −!2 = p2ψ x( ) ∂x2 2But p ) = 2m ⎡⎣E − V x ⎤⎦ (assuming t-independent potential) = 2m(K.E. ( ) !2 ∂2ψ x + V x x x∴ − ( ) ( )ψ ( ) = Eψ ( ) 2m ∂x2 time-independent Schrödinger equation in one dimension We now have the outline of: • a physical picture involving wave and particle duality of light and matter ! • a quantitative theory allowing calculations of stable states and their properties !