Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation...

38
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

Transcript of Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation...

Page 1: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

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

Page 2: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Lecture 1 Perturbation Theory

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2( ) H = H0 + ˆ ′H

3( ) H = H0 + λ ˆ ′H

1( ) H0ψ 0

0( ) = E00( )ψ 0

0( )

4( ) H ψ 0 = E0ψ 0

5( ) ψ 0 =ψ 0 s,λ( ) E0 = E0 λ( )

6( ) ψ 0 =ψ 0

0( ) + λψ 01( ) + λ 2ψ 0

2( ) + ...

7( ) E0 = E0

0( ) + λE01( ) + λ 2E0

2( ) + ...

8( ) ψ 0

0( ) ψ 0 = 1

9( ) ψ 00( ) ψ 0 = 1= ψ 0

0( ) ψ 00( )

1! "# $#

+ λ ψ 00( ) ψ 0

1( )

0! "# $#

+ λ 2 ψ 00( ) ψ 0

2( )

0! "# $#

+ ...

Lecture 1, p.1

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3( ) H = H0 + λ ˆ ′H

4( ) H ψ 0 = E0ψ 0

6( ) ψ 0 =ψ 0

0( ) + λψ 01( ) + λ 2ψ 0

2( ) + ...

7( ) E0 = E0

0( ) + λE01( ) + λ 2E0

2( ) + ...

10( ) ψ 0 =ψ 00( ) −

ψ k0( ) ˆ ′H ψ 0

0( )

Ek0( ) −E0

0( )k≠0∑ ψ k

0( )

ψ 01( )

! "#### $####

+ ...

11( ) E0 = E00( ) + ψ 0

0( ) ˆ ′H ψ 00( )

E01( )

! "## $##−

ψ k0( ) ˆ ′H ψ 0

0( )

Ek0( ) −E0

0( )k≠0∑ ψ 0

0( ) ˆ ′H ψ k0( )

E02( )

! "###### $######

+ ...

Lecture 1, p.2

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Consider H atom in an uniform electric field in the z direction Represent a polarization function by a 2pz STO

Optimize the 2pz ζ-exponent

J φ⎡⎣ ⎤⎦ = φ H0 −E0

0( ) φ + φ ˆ ′H −E01( ) ψ 0

0( ) + ψ 00( ) ˆ ′H −E0

1( ) φ ≥ E02( )

φ = 2pz ζ( ) ∴ J φ⎡⎣ ⎤⎦ = J 2pz ζ( )⎡⎣ ⎤⎦ = J ζ( )

Lecture 1, p.3

To  work  at  home  a,er  the  EMTCCM-­‐2016:  

δ-δ-δ-

δ+

δ+δ+

a)   b)  

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J φ⎡⎣ ⎤⎦ = φ H0 −E0

0( ) φ + φ ˆ ′H −E01( ) ψ 0

0( ) + ψ 00( ) ˆ ′H −E0

1( ) φ ≥ E02( )

H0 = − ∇

2

2− 1

r∇2 = ∂2

∂r 2 +2r

∂∂r

− 1r 2 L2 L2Yℓ,m = ℓ ℓ +1( )Yℓ,m

ˆ ′H = −z = −r cosθ

1s = R1sY0,0 = 2 e−r( ) 1

2 π

⎛⎝⎜

⎞⎠⎟= 1

πe−r

2pz STO( ) = R2ζY1,0 =

2ζ 5 2

3r e−ζ r⎛

⎝⎜⎞⎠⎟

12

cosθ⎛

⎝⎜

⎠⎟ =

1πζ 5 2r e−ζ r cosθ

E0

1( ) = 1s − z 1s = 0

Lecture 1, p.4

J 2pz⎡⎣ ⎤⎦ = 2pz H0 2pz −E0

0( ) + 2 2pzˆ ′H 1s

E0

0( ) = 1s − ∇2

2− 1

r1s = −0.5

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1s − z 2pz = −ζ5 2

πe− ζ +1( )r

0

∫ r 4dr cos2θ0

π

∫ sinθ dθ dφ = − 430

∫ ζ 5 2 e− ζ +1( )r

0

∫ r 4dr = −32 ζ 5

1+ζ( )5

R2ζ − ∇2

2− 1

rR2ζ = − 2ζ 5

3e−2ζ r

0

∫ ζ 2r 2 − 4ζ r + 2r( )dr = − 16

2− 3ζ( )ζ 3

J 2pz⎡⎣ ⎤⎦ = R2ζ H0 R2ζ − 1s H0 1s + 2 1s − z 2pz ≥ E0

2( )

J ζ( ) = − 16

2− 3ζ( )ζ 3 + 0.5 − 232 ζ 5

1+ζ( )5

minJ ζ = 0.844( )

Lecture 1, p.5

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-���

-���

Mathematica code and results: Lecture 1, p.6 Hpluselectricfield.pdf

h11 = "0.5;h22 ="(2$3) ζ^5 Integrate[Exp["2 ζ r] (ζ^2 r^2 " 4 ζ r + 2 r) ,

{r, 0, Infinity}, Assumptions , ζ > 0]h12 = "(4$3) Sqrt[ζ^5]

Integrate[r^4 Exp["(1 + ζ) r] , {r, 0, Infinity},Assumptions , ζ > 0]

ζ $. SetPrecision[FindMinimum[h22 " h11 + 2 h12, {ζ, 0.6}][[2]], 3]

Plot[h22 " h11 + 2 h12, {ζ, 0.2, 1.5}, PlotStyle , Black]

" 16 (2 " 3 ζ) ζ3

H_plus_electric_field_new.nb 3

Hpluselectricfield.pdf

h11 = "0.5;h22 ="(2$3) ζ^5 Integrate[Exp["2 ζ r] (ζ^2 r^2 " 4 ζ r + 2 r) ,

{r, 0, Infinity}, Assumptions , ζ > 0]h12 = "(4$3) Sqrt[ζ^5]

Integrate[r^4 Exp["(1 + ζ) r] , {r, 0, Infinity},Assumptions , ζ > 0]

ζ $. SetPrecision[FindMinimum[h22 " h11 + 2 h12, {ζ, 0.6}][[2]], 3]

Plot[h22 " h11 + 2 h12, {ζ, 0.2, 1.5}, PlotStyle , Black]

" 16 (2 " 3 ζ) ζ3

H_plus_electric_field_new.nb 3

"32 ζ5

(1 + ζ)5

0.844

0.4 0.6 0.8 1.0 1.2 1.4

!1.0

!0.5

4 H_plus_electric_field_new.nb

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Lecture 2 Time-Dependent Perturbation Theory

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Lecture 2, p.1

1( ) − !

i∂Ψ0

∂t= H0Ψ0

2( ) H0Ψ0 = E0Ψ0

3( ) − !

i∂Ψ0

∂t= E0Ψ0

4( ) Ψ0 s,t( ) = exp −iE0t !( ) ψ 0 s( )

5( ) Ψ0 s,0( ) =ψ 0 s( )

6( ) − !

i∂∂t

− H0⎛⎝⎜

⎞⎠⎟Ψ0 = 0

7( ) Ψ0 s,t( ) = ck

k∑ exp −iEk

0t !( ) ψ k0 s( )

9( ) Ψ s,t( ) = bk t( )

k∑ exp −iEk

0t !( ) ψ k0 s( )

8( ) − !

i∂Ψ∂t

= H0 + ˆ ′H t( )⎡⎣ ⎤⎦Ψ

H0ψ k

0 = Ek0ψ k

0

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Lecture 2, p.2

10( ) − !

i∂Ψ s,t( )

∂t= H0 + ˆ ′H t( )⎡⎣ ⎤⎦Ψ s,t( ) Ψ s,t( ) = bk t( )

k∑ exp −iEk

0t !( ) ψ k0 s( )

11( ) − !i

∂ bk t( )k∑ exp −iEk

0t !( ) ψ k0⎡

⎣⎢

⎦⎥

Ψ s,t( )" #$$$$$ %$$$$$

∂t= H0 + ˆ ′H( ) bk t( )

k∑ exp −iEk

0t !( ) ψ k0

Ψ s,t( )" #$$$$$ %$$$$$⎡

⎢⎢⎢⎢

⎥⎥⎥⎥

12( ) − !

idbk t( )

dtk∑ exp −iEk

0t !( ) ψ k0 = bk t( ) exp −iEk

0t !( ) ˆ ′H ψ k0

k∑

13( ) dbn t( )

dt= − i!

bk t( ) exp iωnkt( ) ψ n0 λ ˆ ′H ψ k

0

k∑ ωnk =

En0 −Ek

0

!

14( ) bk t( ) = bk

0( ) + λ bk1( ) t( ) + λ 2bk

2( ) t( ) + ...

15( ) dbn

1( ) t( )dt

= − i!

bk0( ) exp iωnkt( ) ψ n

0 ˆ ′H ψ k0

k∑

16( ) dbn

2( ) t( )dt

= − i!

bk1( ) t( ) exp iωnkt( ) ψ n

0 ˆ ′H ψ k0

k∑

Page 12: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

ψi0   |bi(0)|2 = 1

|bf(τ)|2

|b1(τ)|2

|bn(τ)|2 ψn

0  

ψf0  

ψ10   …   …  

Lecture 2, p.3

17( ) dbn

1( ) t( )dt

= − i!

bk0( ) exp iωnkt( ) ψ n

0 ˆ ′H ψ k0

k∑

18( ) Ψ i

0 = exp −iEi0t !( ) ψ i

0

19( ) ˆ ′H on : 0 < t < τ t ≥ τ ⇒ bk t > τ( ) = bk τ( )

20( ) bk

0( ) = δ ki( ) dbf1( ) t( )dt

= − i!

exp iω fit( ) ψ f0 ˆ ′H ψ i

0

21( ) bf τ( ) ≈ bf

1( ) τ( ) = − i!

ψ f0 ˆ ′H ψ i

0

0

τ

∫ exp iω fit( ) dt

22( ) bf τ( ) 2≈ 1!2 ψ f

0 ˆ ′H ψ i0

0

τ

∫ exp iω fit( ) dt2

Page 13: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Lecture 3 Absorption and Emission of Radiation

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1( ) Ex z,t( ) = E0 sin ω t − k z( ) ω = 2π

τ= 2πν k = 2π

λ= 2π !ν

2( ) ˆ ′H t( ) = −Exµx = −E0 sin ω t − k z( ) µx µx = qαxα

α∑

3( ) bf τ( ) ≈ − i

!ψ f

0 ˆ ′H ψ i0

0

τ

∫ exp iω fit( ) dt

4( ) bf τ( ) ≈ iE0

!ψ f

0 µx ψ i0 sin ω t − k z( ) exp iω fit( ) dt

0

τ

5( ) bf τ( ) ≈ iE0

!ψ f

0 µx ψ i0 sin ω t( ) exp iω fit( ) dt

0

τ

∫ sin ω t( ) = exp iωt( )− exp −iωt( )2i

6( ) bf τ( ) ≈ E0

2!ψ f

0 µx ψ i0 exp i ω fi +ω( )t⎡⎣ ⎤⎦ − exp i ω fi −ω( )t⎡⎣ ⎤⎦{ }dt

0

τ

∫ exp at( ) dt =exp aτ( )−1

a0

τ

7( ) bf τ( ) ≈ E0

2!iψ f

0 µx ψ i0

exp i ω fi +ω( )τ⎡⎣ ⎤⎦ −1

ω fi +ω−

exp i ω fi −ω( )τ⎡⎣ ⎤⎦ −1

ω fi −ω

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

8( ) s =ωnm ±ω lim

s→0

eisτ −1s

=d eisτ −1( ) ds

ds ds= i τ lim

ω fi ±ω→0

exp i ω fi ±ω( )τ⎡⎣ ⎤⎦ −1

ω fi ±ω= i τ

9( ) bf τ( ) 2≈

E02

4!2 ψ f0 µx ψ i

02 exp i ω fi +ω( )τ⎡⎣ ⎤⎦ −1

ω fi +ω−

exp i ω fi −ω( )τ⎡⎣ ⎤⎦ −1

ω fi −ω

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2

10( ) ψ f

0 x ψ i0 ψ f

0 y ψ i0 ψ f

0 z ψ i0

Lecture 3, p.1

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-110 -100 -90 -80

0.2

0.4

0.6

0.8

1.0

90 100 110 120

0.2

0.4

0.6

0.8

1.0

s = 250;a = 1. × 10^2;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, (0.8 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, 0.8 a, 1.2 a }, ImageSize , {s, s}, PlotStyle , Black];

graph = Row[{emission, absorption}, " "]Export["absorption_emission.pdf", graph]

Out[38]=

!110 !100 !90 !80

0.2

0.4

0.6

0.8

1.0

90 100 110 120

0.2

0.4

0.6

0.8

1.0

Out[39]= absorption_emission.pdf

Mathematica code and results: Lecture 3, p.2

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a  

b  

ρa  

ρb  

abs.   st.  em.   sp.  em.  

Lecture 3, p.3

1( ) B d ρa

abs.( )!"#−B d ρb

st .em.( )!"$ #$−A ρb

sp.em.( )!"#= 0

2( ) ρb = ρa exp −!ωba kT( )

3( ) A = B d exp −!ωab kT( )−1⎡⎣ ⎤⎦

4( ) d ∝ω 3 exp −!ω kT( )−1⎡⎣ ⎤⎦

−1

5( ) A ∝B ω 3 exp −!ωab kT( )−1

exp −!ω kT( )−1

6( ) A ∝B ω 3

Page 17: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Lab report on the electronic absorption spectrum of iodine: www.art-xy.com/2009/10/lab-report-on-electronic-absorption.html Electronic absorption spectrum of iodine: http://www.uni-ulm.de/physchem-praktikum/media/literatur/1987_The_Iodine_Spectrum_Revisited.pdf ww2.chemistry.gatech.edu/class/pchem/expt6m.pdf  

Lecture 3, p.4

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Lecture 4 Raman Scattering

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incident laser beam

scattered light

molecule

ν0

(ν0, ν0 - Δνmolecule)

νRaman = ν0 − Δνmolecule

Δνmolecule = ν0 −νRaman

Lecture 4, p.1

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Inte

nsity

Wavenumber shift Δν/cm-1

790

762

459

314

218

-790

-7

62

-459

-314

-218

Stokes anti-Stokes Rayleigh

Lecture 4, p.2

!νRaman = !ν0 − Δ !νvib.trans.

Δ!νvib.trans. = !ν0 − !νRaman

Page 21: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

1( ) µind = α E

2( ) E = E0 cos 2πν0t( )

3( ) α = α0 +

∂α∂Q

⎛⎝⎜

⎞⎠⎟ 0

Q + ...

4( ) Q = A cos 2πνt( )

5( ) µind = α0E0 cos 2πν0t( ) + ∂α

∂Q⎛⎝⎜

⎞⎠⎟ 0

A E0 cos 2πν0t( ) cos 2πνt( )

6( ) µind = α0E0 cos 2π ν0

Rayleigh!t

⎝⎜⎜

⎠⎟⎟+ 1

2∂α∂Q

⎛⎝⎜

⎞⎠⎟ 0

A E0 cos 2π ν0 −ν( )Stokes"#$ %$

t⎡

⎢⎢

⎥⎥+ cos 2π ν0 +ν( )

anti−Stokes"#$ %$t

⎢⎢

⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

7( ) ∂α

∂Q⎛⎝⎜

⎞⎠⎟ 0

≠ 0

Lecture 4, p.3

cos A+B( ) = cos A( )cos B( )− sin A( )sin B( )cos A−B( ) = cos A( )cos B( ) + sin A( )sin B( )12

cos A+B( ) + cos A−B( )⎡⎣ ⎤⎦ = cos A( )cos B( )

δ-δ-δ-

δ+

δ+δ+

Page 22: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Lecture 4, p.4

1( ) µ ind = α E

2( )µx

µy

µz

⎜⎜⎜⎜

⎟⎟⎟⎟

=

α xx α xy α xz

α yx α yy α yz

α zx α zy α zz

⎜⎜⎜⎜

⎟⎟⎟⎟

Ex

Ey

Ez

⎜⎜⎜⎜

⎟⎟⎟⎟

µx

µy

µz

⎜⎜⎜⎜

⎟⎟⎟⎟

=

α xxEx +α xyEy +α xzEz

α yxEx +α yyEy +α yzEz

α zxEx +α zyEy +α zzEz

⎜⎜⎜⎜

⎟⎟⎟⎟

3( ) Ψ i =ψ i exp −iω it( ) Ψ f =ψ f exp −iω f t( )

4( ) Ψr =ψ r exp −i ω r − iΓ r( )t⎡⎣ ⎤⎦

5( ) ΔE Δt ∼ " ⇒ E = "ω( ) ⇒ Δω Δt ∼1 ⇒ 2Γ rτ r ∼1 ⇒ Γ r ∼

12τ r

6( ) ω ri =ω r −ω i

7( ) Eσ = Eσ 0 exp −i ω0 −ω fi( )⎡⎣ ⎤⎦ ω0 = E0 ! σ = x, y, z

8( ) αρσ( )fi= ψ f

0 αρσ ψ i0 = 1!

ψ f0 µρ ψ r

0 ψ r0 µσ ψ i

0

ω ri −ω0 − iΓ r

+ψ f

0 µσ ψ r0 ψ r

0 µρ ψ i0

ω rf +ω0 + iΓ r

⎝⎜⎜

⎠⎟⎟r≠i,f

Page 23: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Normal Pre-Resonance Discrete Resonance

αρσ( )fi= ψ f

0 αρσ ψ i0 = 1!

ψ f0 µρ ψ r

0 ψ r0 µσ ψ i

0

ω ri −ω0 − iΓ r

+ψ f

0 µσ ψ r0 ψ r

0 µρ ψ i0

ω rf +ω0 + iΓ r

⎝⎜⎜

⎠⎟⎟r≠i,f

Lecture 4, p.5

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,  3519-­‐3537  (1976).  

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“Tryptophan  UV  Resonance  Raman  ExcitaWon  Profiles”,  J.A.  Sweeney,  S.A.  Asher,  J.  Phys.  Chem.  1990,  94,  4784-­‐4791  

Lecture 4, p.6

tryptophan  

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Available on 22 Nov 2016 (see amazon.co.uk)

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Workshop Introduction to Mathematica®

Mathema'ca  (www.wolfram.com)  Author  of  Mathema<ca:  Wolfram  Research  Inc.  Champaign,  Illinois,  2010.    

Page 27: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

 Basic  Rules  

 4.  Shi,+Enter  to  evaluate  input  

 1.  Capital  leNers  in  all  command  names  

 2.  [  ]  surround  funcWon  arguments  

3.  {  }  are  used  for  lists  and  ranges  

Page 28: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Basic Rules1. Capital letters in all command names2. [ ] surround function arguments3. { } are used for lists and ranges4. Shift+Enter to evaluate inputExamplesIntegrate[Cos[2 " x], x]1

2Sin[2 x]

Plot[Si n[2 " x] $ 2, {x, &1 0, 1 0}]

!10 !5 5 10

!0.4

!0.2

0.2

0.4

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

2 introduction.nb

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Page 29: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Plot[Sin[2 " x] $ 2, {x, &1 0, 1 0}]

!10 !5 5 10

!0.4

!0.2

0.2

0.4

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

2 introduction.nb

Page 30: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Manipulate[Plot[Sin[frequency " x + phase], {x, &10, 10}],{frequency, 1, 5}, {phase, 1, 10}]

frequency

phase

!10 !5 5 10

!1.0

!0.5

0.5

1.0

introduction.nb 3

Manipulate[Plot[Sin[frequency " x + phase], {x, &10, 10}],{frequency, 1, 5}, {phase, 1, 10}]

frequency

phase

!10 !5 5 10

!1.0

!0.5

0.5

1.0

introduction.nb 3

Page 31: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Manipulate[Plot[Sin[frequency " x + phase], {x, &10, 10}],{frequency, 1, 5}, {phase, 1, 10}]

frequency

phase

!10 !5 5 10

!1.0

!0.5

0.5

1.0

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}

introduction.nb 3

myListPlot = ListPlot[myPrimes]

5 10 15 20 25

20

40

60

80

100

myFittedCurve = Fit[myPrimes, {1, x, x^2}, x]&2.89565 + 2.5392 x + 0.0555927 x2

4 introduction.nb

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Page 32: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

myListPlot = ListPlot[myPrimes]

5 10 15 20 25

20

40

60

80

100

myFittedCurve = Fit[myPrimes, {1, x, x^2}, x]&2.89565 + 2.5392 x + 0.0555927 x2

4 introduction.nb

myFittedPlot = Plot[myFittedCurve, {x, 0, 25}]

5 10 15 20 25

20

40

60

80

100

Show[myFittedPlot, myListPlot]

5 10 15 20 25

20

40

60

80

100

introduction.nb 5

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Page 33: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

myFittedPlot = Plot[myFittedCurve, {x, 0, 25}]

5 10 15 20 25

20

40

60

80

100

Show[myFittedPlot, myListPlot]

5 10 15 20 25

20

40

60

80

100

introduction.nb 5

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Page 34: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Workshop Band Shapes and Convolution

Page 35: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Workshop, p.1

Gaussian Band Shapeg[x_, x0_, a_] := Exp[%(a (x % x0))^2]norm = Integrate[g[x, x0, a], {x, %Infinity, Infinity}, Assumptions %> {x0 > 0, a > 0}]Plot[g[x, 0, 1.] + (norm +. a %> 1), {x, %4, 4}]

π

a

!4 !2 2 4

0.1

0.2

0.3

0.4

0.5

Lorentzian Band Shapef[x_, x0_, a_] := 1 + (a^2 + (x % x0)^2)norm = Integrate[f[x, x0, a], {x, %Infinity, Infinity}, Assumptions %> {x0 > 0, a > 0}]Plot[f[x, 0, 1.] + (norm +. a %> 1), {x, %4, 4}]

π

a

!4 !2 2 4

0.05

0.10

0.15

0.20

0.25

0.30

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Page 36: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Gaussian Band Shapeg[x_, x0_, a_] := Exp[%(a (x % x0))^2]norm = Integrate[g[x, x0, a], {x, %Infinity, Infinity}, Assumptions %> {x0 > 0, a > 0}]Plot[g[x, 0, 1.] + (norm +. a %> 1), {x, %4, 4}]

π

a

!4 !2 2 4

0.1

0.2

0.3

0.4

0.5

Lorentzian Band Shapef[x_, x0_, a_] := 1 + (a^2 + (x % x0)^2)norm = Integrate[f[x, x0, a], {x, %Infinity, Infinity}, Assumptions %> {x0 > 0, a > 0}]Plot[f[x, 0, 1.] + (norm +. a %> 1), {x, %4, 4}]

π

a

!4 !2 2 4

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0.10

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Workshop, p.2

Page 37: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

ideal spectrum S0(ω)

ideal resolution function R(ω’-ω)

spectrum scanning

ideal observed spectrum S(ω’)

resolution function R(ω’-ω)

ideal spectrum S0(ω)

observed spectrum S(ω’)

spectrum scanning

S ω '( ) = S0 ∗R( ) ω '( ) ≡ S0 ω( )

−∞

∫ R ω '−ω( )dω

Workshop, p.3

Page 38: Lecture 1 Perturbation Theory Lecture 2 Time-Dependent ... · Lecture 2 Time-Dependent Perturbation Theory Lecture 3 Absorption and Emission of Radiation Lecture 4 Raman Scattering

Convolutionf[x_,a_]:=a%Pi%(a^2+x^2)aa=Plot[{f[x,1.]%f[0,1.],f[x,2.]%f[0,2.]},{x,+10,10},PlotRange+>All,PlotStyle+>{Red,Blue}];conv=Convolve[f[x,1.],f[x,2.],x,y];conv%.y+>0;bb=Plot[conv%%,{y,+10,10},PlotRange+>All,PlotStyle+>Black];Show[aa,bb]

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Workshop, p.4