Download Document

Interaction with electromagnetic radiation
Electric Dipole approximation:
dipole
r r
E (r , t )
r
r
d = er
r r
interacts with field E (r , t )
a0 = 1 Å
r
k
r r
B (r , t )
Interaction hamiltonian:
r r r
H = er ⋅ E (r , t )
r r
r −iωt ikr ⋅rr
E (r , t ) = E0e e
r r 1 r r2
r − iωt ⎡
n ⎤
(
)
(
)
1
...
= E0e
+
i
k
⋅
r
+
k
⋅
r
+
+
O
r
⎢⎣
⎥⎦
2
r − iωt
≈ E0e
( )
Note:
λ = 5000 Å
⎛ 2πa ⎞
⎟
⎝ λ ⎠
Higher order terms in O⎜
r r
2πa
<< 1
k ⋅ r ≈ ka =
λ
E-field does not vary over
size of the atom, hence:
H EM
Electric quadrupole, octupole
r r r
H
=
μ
⋅ B (r , t )
Magnetic interactions:
M
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
r r
= er ⋅ E (t )
Quantum analog of electromagnetic radiation
Classical electric dipole radiation
Classical oscillator
I rad
Transition dipole moment
Quantum jump
r2
∝ er
2
* r
ψ 1 er ψ 2
2
I rad ∝ μ fi =
The atom does not radiate when it is in a stationary state !
The atom has no dipole moment
μii = ∫
*r
ψ 1 r ψ 1dτ
=0
Intensity of spectral lines linked
to Einstein coefficient for absorption:
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Bif =
μ fi
2
6ε 0h 2
Selection rules
depend on angular behavior of the wave functions
Parity operator
r
r
θ
φ
r
−r
r
r
Pr = −r
( x , y , z ) → ( − x, − y , − z )
(r ,θ , φ ) → (r , π − θ , φ + π )
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
All quantum mechanical wave functions
have a definite parity
r
r
Ψ (− r ) = ± Ψ (r )
r
Ψ f r Ψi ≠ 0
If
Ψf
and
Rule about the
Ψi
Ylm
have opposite parity
functions
PYlm (θ , φ ) = (− )l Ylm (θ , φ )
Selection rules
Mathematical background: function of odd parity gives 0 when integrated over space
In one dimension:
∞
Ψ f x Ψi =
∞
∫
Ψ *f xΨi dx
−∞
=
with
∫ f ( x)dx
f ( x) = Ψ *f xΨi
−∞
∞
0
∞
0
∞
∞
∞
−∞
−∞
0
∞
0
0
0
∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx = ∫ f (− x)d (− x ) + ∫ f ( x)dx = ∫ f (− x)dx + ∫ f ( x)dx
∞
= 2 ∫ f ( x)dx ≠ 0
if
Ψi
f ( − x) = f ( x)
and Ψ f
opposite parity
0
=0
if
f (− x) = − f ( x)
Ψi
and Ψ f
same parity
Electric dipole radiation connects states of opposite parity !
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Selection rule
r
Ψ f r Ψi
Δl
Transition dipole matrix element
Parity consideration
r
r
P Ψ f r Ψi = P RnlYlm r Rn'l 'Yl 'm'
= (− )l (− )(− )l ' = even
if
l + 1 + l' = even
Δl = ±1,±3,...
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Selection rule Δm
cos φ
x
Ψ f y Ψi = r Ψ f sin φ Ψi = r Ψ f
1
z
[x]:
2π
=
∫e
−im 'φ
0
2π
[y]:
[z]:
= ∫e
−im 'φ
0
2π
=
(e φ + e φ )/ 2
(e φ − e φ )/ 2i Ψ
i
−i
i
−i
i
1
ei (−m'±1+ m )φ
dφ ≠ 0
2
if
m − m'±1 = 0
2π i ( − m ' m1+ m )φ
eiφ − e − iφ imφ
e
e dφ = ∫
dφ ≠ 0
2i
2i
0
if
m − m'±1 = 0
eiφ + e −iφ imφ
e dφ =
2
∫e
−im 'φ imφ
e
dφ ≠ 0
2π
∫
0
if
m − m'= 0
0
Combinations of x, y, z related to the polarization of light
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Polarization vectors
(
)
1
rˆ = xê x + yê y + zê z
r
= sin θ cos φê x + sin θ sin φê y + cos θê z
rˆ ∝ Y1,−1
ê x +iê y
2
ê rad = A
σ−
+ Y10ê z + Y11
ê x −iê y
2
2π
Y1,−1 − Y11
3
2π
sin θ sin φ = i
Y1,−1 + Y11
3
)
(
-ê x +iê y
cos θ =
2
⎛ ê x +iê y ⎞
⎟
⎜−
σ+⎜
2 ⎟⎠
⎝
(
sin θ cos φ =
)
4π
Y10
3
Definition of Yij
+ Aπ ê z + A
definition of polarized light vectors
r̂ ⋅ ê rad = A
Y
+ Aπ Y10
σ − 1,−1
+A
Y
σ + 11
Verify selection rules for
x+iy and x-iy
Circular polarization
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
m=0
m=-1
σ-
σz
m=1
σ+
l=0,m=0
Properties of
Write r-vector in terms of
spherical harmonics
Ylm
Y11 = −
Y1,−1 = −
Y10 =
*
Y
∫ l f m f Y1,mYl imi sin θdθdφ ≠ 0
if
l f = li ±1
m f = mi ± m
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
z
r
− 1 ( x + iy ) 3
3
sin θeiφ =
4π
8π
2 r
1 ( x − iy ) 3
3
sin θe −iφ =
4π
8π
2 r
3
4π
Triangular property of
Spherical harmonics
Verify with “Mathematica”
Photon picture of selection rules
r
J
r
J'
absorption
r
J f =1
photon
r r r
J '= J + 1
so
J ' = J +1
J'= J
J ' = J −1
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
ΔJ = 0,±1
For the total angular momentum
Non-electric dipole selection rules
Higher order transitions
Scale like xy, r2
*
Y
∫ l f m f Y2,mYl imi sin θdθdφ ≠ 0
l f − l i = 0,±2
Magnetic dipole interactions have different selection rules
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Selection rules in Hydrogen atom
Intensity of spectral lines given by
r
r
μ fi = ∫ Ψ*f μΨi = Ψ f − er Ψi
n
1) Quantum number
no restrictions
Balmer series
2) Parity rule for l
Δl = odd
3) Laporte rule for
l
Angular momentum rule:
r
r r
l f = li + 1
From 2. and 3.
so
Δl ≤ 1
Δl = ±1
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Lyman series