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
Lecture 1 Perturbation Theory
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
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
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)
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
3π
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
1s − z 2pz = −ζ5 2
πe− ζ +1( )r
0
∞
∫ r 4dr cos2θ0
π
∫ sinθ dθ dφ = − 430
2π
∫ 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
��� ��� ��� ��� ��� ���
-���
-���
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
Lecture 2 Time-Dependent Perturbation Theory
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
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∑
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
Lecture 3 Absorption and Emission of Radiation
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
-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
a
b
ρa
ρb
abs. st. em. sp. em.
Lecture 3, p.3
1( ) B d ρa
abs.( )!"#−B d ρb
st .em.( )!"$ #$−A ρb
sp.em.( )!"#= 0
2( ) ρb = ρa exp −!ωba kT( )
3( ) A = B d exp −!ωab kT( )−1⎡⎣ ⎤⎦
4( ) d ∝ω 3 exp −!ω kT( )−1⎡⎣ ⎤⎦
−1
5( ) A ∝B ω 3 exp −!ωab kT( )−1
exp −!ω kT( )−1
6( ) A ∝B ω 3
Lab report on the electronic absorption spectrum of iodine: www.art-xy.com/2009/10/lab-report-on-electronic-absorption.html Electronic absorption spectrum of iodine: http://www.uni-ulm.de/physchem-praktikum/media/literatur/1987_The_Iodine_Spectrum_Revisited.pdf ww2.chemistry.gatech.edu/class/pchem/expt6m.pdf
Lecture 3, p.4
Lecture 4 Raman Scattering
incident laser beam
scattered light
molecule
ν0
(ν0, ν0 - Δνmolecule)
νRaman = ν0 − Δνmolecule
Δνmolecule = ν0 −νRaman
Lecture 4, p.1
��� ��� ��� ��� � -��� -��� -��� -���
Inte
nsity
Wavenumber shift Δν/cm-1
790
762
459
314
218
-790
-7
62
-459
-314
-218
Stokes anti-Stokes Rayleigh
Lecture 4, p.2
!νRaman = !ν0 − Δ !νvib.trans.
Δ!νvib.trans. = !ν0 − !νRaman
1( ) µind = α E
2( ) E = E0 cos 2πν0t( )
3( ) α = α0 +
∂α∂Q
⎛⎝⎜
⎞⎠⎟ 0
Q + ...
4( ) Q = A cos 2πνt( )
5( ) µind = α0E0 cos 2πν0t( ) + ∂α
∂Q⎛⎝⎜
⎞⎠⎟ 0
A E0 cos 2πν0t( ) cos 2πνt( )
6( ) µind = α0E0 cos 2π ν0
Rayleigh!t
⎛
⎝⎜⎜
⎞
⎠⎟⎟+ 1
2∂α∂Q
⎛⎝⎜
⎞⎠⎟ 0
A E0 cos 2π ν0 −ν( )Stokes"#$ %$
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+
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
∑
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).
“Tryptophan UV Resonance Raman ExcitaWon Profiles”, J.A. Sweeney, S.A. Asher, J. Phys. Chem. 1990, 94, 4784-‐4791
Lecture 4, p.6
tryptophan
Available on 22 Nov 2016 (see amazon.co.uk)
Workshop Introduction to Mathematica®
Mathema'ca (www.wolfram.com) Author of Mathema<ca: Wolfram Research Inc. Champaign, Illinois, 2010.
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
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
-�� -� � ��
-���
-���
���
���
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
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
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
� �� �� �� ��
��
��
��
��
���
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
� �� �� �� ��
��
��
��
��
���
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
� �� �� �� ��
��
��
��
��
���
Workshop Band Shapes and Convolution
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
-� -� � �
���
���
���
���
���
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
-� -� � �
����
����
����
����
����
����
Workshop, p.2
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
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]
!10 !5 5 10
0.2
0.4
0.6
0.8
1.0
-�� -� � ��
���
���
���
���
���
Workshop, p.4