Time-Dependent Perturbation Theory. Absorption and Emission of Radiation. Band Shapes and Convolution

Transcription

1 Lecture 1 Perturbation Theory Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering Workshop Band Shapes and Convolution

2 Lecture 1 Perturbation Theory

3 ( 1) Ĥ ψ = E ψ Lecture 1, p.1 ( 2 ) Ĥ = Ĥ + H ˆ ( 3) Ĥ = Ĥ + λh ˆ ( 4) Ĥ ψ = E ψ ( 5) ψ = ψ ( s,λ) E = E ( λ) ( 6) ψ = ψ + λψ ( 1) + λ 2 ( 2) ψ +... ( 7) E = E + ( 1) λe + λ 2 ( 2) E +... ( 8) ψ ψ = 1 ( 9) ψ ψ = 1= ψ ψ!# "# $ + λ ψ ψ ( 1)!# "# $ + λ 2 ψ ψ ( 2)!# "# $

4 Lecture 1, p.2 ( 3 ) Ĥ = Ĥ + λh ˆ ( 6) ψ = ψ + λψ ( 1) + λ 2 ( 2) ψ +... ( 7) E = E + ( 1) λe + λ 2 ( 2) E +... ( 4) Ĥ ψ = E ψ ( 1) ψ = ψ ψ k H ˆ ψ E k ψ k k E!#### "#### $ ψ ( 11) E = E + ψ ˆ ψ ˆ H ψ!## "## $ k H ψ E k ψ H ˆ ψ k k E!###### "###### $ E 1 E

5 Lecture 1, p.3 J φ = φ Ĥ E φ + φ H ˆ ( 1) E ψ + ψ ˆ H ( 1) E φ ( 2) E To work at home a,er the EMTCCM- 216: Consider H atom in an uniform electric field in the z direction Represent a polarization function by a 2p z STO Optimize the 2p z ζ-exponent δ- δ- δ- δ+ δ+ δ+ φ = 2p z a) b) ( ζ ) J φ = J 2p z ( ζ ) = J ζ

6 J φ = φ Ĥ E φ + φ H ˆ ( 1) E ψ + ψ ˆ H ( 1) E φ ( 2) E Lecture 1, p.4 J 2p z = 2p z Ĥ 2p z E + 2 2pz H ˆ 1s Ĥ = r H ˆ = z = r cosθ 2 = 2 r r r 1 ˆL2 ˆL2 Y r 2 l,m = l( l +1)Y l,m 1s = R 1s Y, = 2 e r 1 2 π = 1 π e r 2p z ( STO) = R 2ζ Y 1, = 2ζ r e ζ r π cosθ = 1 π ζ 5 2 r e ζ r cosθ 2 E = 1s 2 1 r 1s =.5 ( 1) E = 1s z 1s =

7 Lecture 1, p.5 1s z 2p z = ζ 5 2 π e ( ζ +1 )r π 2π r 4 dr cos 2 θ sinθ dθ dφ = 4 ζ 5 2 e ( ζ +1 )r 3 r 4 dr = 32 ζ 5 1+ζ 5 R 2ζ r R = 2ζ 5 2ζ 3 e 2ζ r ( ζ 2 r 2 4ζ r + 2r )dr = 1 ( 6 2 3ζ )ζ 3 J 2p z = R 2ζ Ĥ R 2ζ 1s Ĥ ( 2) 1s + 2 1s z 2p z E J ( ζ ) = ζ ζ ζ 5 ( 1+ζ ) 5 minj ( ζ =.844)

8 Mathematica code and results: Lecture 1, p.6 h11 = ".5; h22 = "(2 $ 3) ζ^5 Integrate[Exp["2 ζ r] (ζ^2 r^2 " 4 ζ r + 2 r), {r,, Infinity}, Assumptions, ζ > ] h12 = "(4 $ 3) Sqrt[ζ^5] Integrate[r^4 Exp["(1 + ζ) r], {r,, Infinity}, Assumptions, ζ > ] ζ $. SetPrecision[ FindMinimum[h22 " h h12, {ζ,.6}][[2]], 3] Plot[h22 " h h12, {ζ,.2, 1.5}, PlotStyle, Black] " 1 6 (2 " 3 ζ) ζ3 32 ζ5 " (1 + ζ)

9 Lecture 2 Time-Dependent Perturbation Theory

10 ( 1)! i Ψ t = H Ψ Lecture 2, p.1 ( 2) Ĥ Ψ = E Ψ Ψ ( 3)! = E Ψ i t ( 4) Ψ ( s,t) = exp ie t! ( 5) Ψ ( s,) = ψ ( s) ( 6)! i t Ĥ Ψ = ( 7) Ψ ( s,t) = c k ( 8)! i Ĥ ψ k = E k ψ k Ψ t ( 9) Ψ( s,t) = b k ( t) k ψ s exp ie k t! = Ĥ + H ˆ ( t) Ψ k ψ k s exp ie k t! ψ k s

11 ( 1)! i ( 11)! i ( 12)! i ( 13) db n Ψ( s,t) t = Ĥ + H ˆ ( t) Ψ s,t Ψ s,t " $$$$$ # ( $$$$$ ) % b k ( t) exp( ie k t!) ψ k k t db k t dt exp ie t! k k ( t) dt = i! b k ( 14) b k ( t) = b k + λ ( 1) bk ( 15) ( 16) ( 1) db n dt ( 2) db n dt ( t) ( t) = i! = i! k k k Ψ( s,t) = b k ( t) b k t = Ĥ + H ˆ ψ k = b k t ( t) exp( iω nk t) ψ n λ ˆ H ψ k ( t) + λ 2 ( 2) b k ( t) +... b k exp iω ( nk t) ψ ˆ n H ψ k ( 1) b k ( t) exp( iω nk t) ψ ˆ n H ψ k k exp ie k t! k Lecture 2, p.2 ψ k s Ψ s,t " $$$$$ # ( $$$$$ ) % exp( ie k t!) ψ k k ˆ exp ie k t! H ψ k ω nk = E E n k!

12 ( 17) ( 1) db n dt ( t) = i! k ψ i ( 18) Ψ i = exp ie i t! b k exp iω ( nk t) ψ ˆ n H ψ k ( 19) H ˆ on : < t < τ t τ bk ( t > τ ) = b k ( τ ) ( 2) b k = δ ki ( 1) ( 21) b f ( τ ) b f ( τ ) = i! ( 1) db f dt τ ( t) = i! exp iω t ( fi ) ψ ˆ f H ψ i ψ ˆ f H ψ i exp( iω fi t) dt Lecture 2, p.3 ( 22) b f ( τ ) 2 1 ψ ˆ! 2 f τ H ψ i exp( iω fi t) dt 2 ψ 1 b b f (τ) 2 ψ n (τ) 2 n b 1 (τ) 2 ψ f ψ i b i () 2 = 1

13 Lecture 3 Absorption and Emission of Radiation

14 ( 1 ) E x ( z,t ) = E sin( ω t k z) ω = 2π τ = 2πν k = 2π λ = 2π!ν ( 2) H ˆ ( t ) = E x ˆµ x = E sin( ω t k z) ˆµ x ˆµ x = q α x α ( 3) b f ( τ ) i! τ ψ ˆ f H ψ i exp( iω fi t) dt α Lecture 3, p.1 ( 4) b f ( τ ) ie! ψ f ( 5) b f ( τ ) ie! ψ f ( 6) b f ( τ ) E 2! ψ f ( 7) b f ( τ ) E 2!i ψ f ˆµ x ψ i ˆµ x ψ i ˆµ x ψ i ( 8) s = ω nm ± ω lim τ τ τ sin( ω t k z) exp( iω fi t) dt sin( ω t) exp( iω fi t) dt sin( ω t) = { exp i ( ω fi + ω )t exp i ( ω } ω fi )t dt exp( iωt ) exp iωt τ 2i exp( at ) dt = exp i ( ω fi + ω )τ ˆµ x ψ 1 i exp i ( ω ω fi )τ 1 ω fi + ω ω fi ω e isτ 1 = d ( eisτ 1) ds exp i( ω fi ± ω )τ = i τ lim 1 = i τ s s ds ds ω fi ±ω ω fi ± ω exp( aτ ) 1 a ( 9) b f ( τ ) 2 E 2 4! 2 ψ f ˆµ x ψ i 2 exp i ( ω fi + ω )τ 1 exp i ( ω ω fi )τ 1 ω fi + ω ω fi ω 2 ( 1) ψ f x ψ i ψ f y ψ i ψ f z ψ i

15 Lecture 3, p.2 Mathematica code and results: s = 25; a = 1. 1^2; t = 1.; emission = Plot[ (Abs[(Exp[I t (a + x)] ( 1) ) (a + x) ( (Exp[I t (a ( x)] ( 1) ) (a ( x)])^2, {x, (1.2 a, (.8 a }, ImageSize, {s, s}, PlotStyle, Black]; absorption = Plot[ (Abs[(Exp[I t (a + x)] ( 1) ) (a + x) ( (Exp[I t (a ( x)] ( 1) ) (a ( x)])^2, {x,.8 a, 1.2 a }, ImageSize, {s, s}, PlotStyle, Black]; graph = Row[{emission, absorption}, " "] Export["absorption_emission.pdf", graph]

16 Lecture 3, p.3 ( abs. ) st.em.!"#! $ ( " #$ ) ( 1) B d ρ a B d ρ b sp.em.!"# A ρ b = b ρ b abs. st. em. sp. em. ( 2) ρ b = ρ a exp(!ω ba kt ) a ρ a ( 3) A = B d exp!ω ab kt 1 ( 4) d ω 3 exp!ω kt 1 ( 5) A B ω exp!ω kt 3 ab exp!ω kt ( 6) A B ω 3

17 Lecture 3, p.4 Lab report on the electronic absorption spectrum of iodine: Electronic absorption spectrum of iodine: ww2.chemistry.gatech.edu/class/pchem/expt6m.pdf

18 Lecture 4 Raman Scattering

19 Lecture 4, p.1 incident laser beam molecule ν scattered light (ν, ν - Δν molecule ) ν Raman = ν Δν molecule Δν molecule = ν ν Raman

20 Lecture 4, p.2 Stokes Rayleigh anti-stokes Intensity Wavenumber shift Δν/cm -1!ν Raman =!ν Δ!ν vib.trans. Δ!ν vib.trans. =!ν!ν Raman

21 ( 1 ) µ ind = α E ( 2) E = E cos 2πν t ( 3) α = α + α Q ( 4) Q = A cos( 2πνt ) Q +... δ- δ- δ- Lecture 4, p.3 δ+ δ+ δ+ Q ( 5) µ ind = α E cos( 2πν t) + α ( 6) µ ind = α E cos 2π ν ( 7) α Q Rayleigh! t A E cos( 2πν t) cos( 2πνt ) α Q A E Stokes anti Stokes "# $ %$ "# $ %$ cos 2π ( ν ν ) t + cos 2π ( ν +ν ) t = cos A = cos A cos A + B cos A B 1 2 cos A + B cos B cos B + cos( A B) sin A + sin A sin B sin B = cos ( A )cos B

22 ( 1) µ ind = α E Lecture 4, p.4 ( 2) µ x µ y µ z = α xx α xy α xz α yx α yy α yz α zx α zy α zz E x E y E z µ x µ y µ z = α xx E x + α xy E y + α xz E z α yx E x + α yy E y + α yz E z α zx E x + α zy E y + α zz E z ( 3) Ψ i = ψ i exp( iω i t) Ψ f = ψ f exp( iω f t) ( 4) Ψ r = ψ r exp i( ω r iγ r )t ( 5) ΔE Δt " ( E = "ω ) Δω Δt 1 2Γ r τ r 1 Γ r 1 2τ r ( 6) ω ri = ω r ω i ( 7) E σ = E σ exp i ω ω fi ω = E! σ = x, y, z ( 8) ( α ρσ ) = ψ fi f α ρσ ψ i = 1! r i,f ψ f ˆµ ρ ψ r ψ r ω ri ω iγ r ˆµ σ ψ i + ψ f ˆµ σ ψ r ψ r ω rf + ω + iγ r ˆµ ρ ψ i

23 Lecture 4, p.5 ( α ) ρσ = ψ fi f α ρσ ψ i = 1! r i,f ψ f ˆµ ρ ψ r ψ r ω ri ω iγ r ˆµ σ ψ i + ψ f ˆµ σ ψ r ψ r ω rf + ω + iγ r ˆµ ρ ψ i Normal Pre-Resonance Discrete Resonance Continuum Resonance Further Reading: D. Long, Raman Spectroscopy and D.L. Rousseau, P.F. Williams, Resonance Raman scanering of light from a diatomic molecule, J. Chem. Phys., 64, (1976).

24 Lecture 4, p.6 Tryptophan UV Resonance Raman ExcitaWon Profiles, J.A. Sweeney, S.A. Asher, J. Phys. Chem. 199, 94, tryptophan

25 Available on 22 Nov 216 (see amazon.co.uk)

26 Workshop Introduction to Mathematica Mathema'ca ( Author of Mathema<ca: Wolfram Research Inc. Champaign, Illinois, 21.

27 Basic Rules 1. Capital leners in all command names 2. [ ] surround funcwon arguments 3. { } are used for lists and ranges 4. Shi,+Enter to evaluate input

28 Examples Integrate[Cos[2 " x], x] 1 Sin[2 x] 2 Plot[Si n[2 " x] $ 2, {x, &1, 1 }]

29 Plot3D[Sin[x +y], {x, &3, 3}, {y, &3, 3}]

30 Manipulate[ Plot[Sin[frequency " x + phase], {x, &1, 1}], {frequency, 1, 5}, {phase, 1, 1}] frequency phase 1..5!1!5 5 1!.5!1.

31 myprimes = Prime[Range[1, 25]] {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} mylistplot = ListPlot[myPrimes] 1

32 myfittedcurve = Fit[myPrimes, {1, x, x^2}, x] & x x 2 myfittedplot = Plot[myFittedCurve, {x,, 25}] 1

33 Show[myFittedPlot, mylistplot]

34 Workshop Band Shapes and Convolution

35 Workshop, p.1 Gaussian Band Shape g[x_, x_, a_] := Exp[%(a (x % x))^2] norm = Integrate[g[x, x, a], {x, %Infinity, Infinity}, Assumptions %> {x >, a > }] Plot[g[x,, 1.] + (norm +. a %> 1), {x, %4, 4}] π a - -

36 Workshop, p.2 Lorentzian Band Shape f[x_, x_, a_] := 1 + (a^2 + (x % x)^2) norm = Integrate[f[x, x, a], {x, %Infinity, Infinity}, Assumptions %> {x >, a > }] Plot[f[x,, 1.] + (norm +. a %> 1), {x, %4, 4}] π a - -

37 Workshop, p.3 ideal resolution function R(ω -ω) ideal spectrum S (ω) ideal observed spectrum S(ω ) spectrum scanning resolution function R(ω -ω) ideal spectrum S (ω) observed spectrum S(ω ) spectrum scanning S( ω ') = ( S R) ( ω ') S ( ω ) R( ω ' ω )dω

38 Workshop, p.4 Convolution f[x_,a_]:=a%pi%(a^2+x^2) aa=plot[{f[x,1.]%f[,1.],f[x,2.]%f[,2.]},{x,+1,1},plotrange+>all, PlotStyle+>{Red,Blue}]; conv=convolve[f[x,1.],f[x,2.],x,y]; conv%.y+>; bb=plot[conv%%,{y,+1,1},plotrange+>all,plotstyle+>black]; Show[aa,bb] - -