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Lecture 31:The Hydrogen Atom 2:
Dipole Moments
Phy851 Fall 2009
• The interaction between a hydrogen atomand an electric field is given to leading orderby the Electric Dipole approximation:
• The dipole moment of a pure dipole:– Vector quantity– Points from - to +.– Magnitude is charge _ distance
• For Hydrogen atom this gives:
Electric Dipole Approximation
( )−+ −= rrqdrrr
)( CME rEDVrr⋅−=
`Semi-Classical’ Approx:
•Electric field is classical
•COM motion is classical
-
+rr
rr −+
−
ReDrr
−=
-Rr
Dipole Moment Operator
• The electric dipole moment is an operator inH(R), which means that its value depends onthe state of the relative motion:
• Choosing the z-axis along the electric fielddirection gives:
• Expanding onto energy eigenstates gives:
)(|| CME rEZeV =
( ) '''''';1 1'
1
0
1'
0'
'
''
mnVmnV mnmnEn n
n n
m mE ll
lll l
l
l
l
l∑∑∑∑ ∑ ∑∞
=
∞
=
−
=
−
= −= −=
=
ReDrr
−=
)( CME rEDVrr⋅−= )( CME rEReV
rr⋅=
( ) )(''';'''; CMmnmnmnmnE rEdV llll=
Dipole-Moment Matrix Elements
• Separate radial and angular Hilbert spaces:
• SELECTION RULES:– Arfken, 3rd ed., 12.213
• The important thing to remember is that
• Electric Dipole Forbidden Transitions
( )
−
−+
−+
−+=Θ −+
Ω
1',2
22
1',2
22
',
)(
14114
)1(''cos llll
l
l
l
lll δδδ
mmmm mm
'''cos'''; mnRmnZ mnmn llll Θ=
)()(
'''; ''cos''Ω
Θ= mmnRnedR
mnmn llllll
1',','''; ±∝ llll δδ mmmnmnd
1
2
34
s p d f
Examples:
Charged particle in a Magnetic Field
• EM fields are described by both a scalarpotential, _ and vector potential, A
• To include such EM fields, we can make thetransformation:
• Here q is the charge and A(R) is the vectorpotential
• The Hamiltonian of an electron thenbecomes:– Units of B are Gauss (G):
• This is known as the ‘minimal couplingHamiltonian’
€
v P →
v P − q
v A (
v R )
€
H =12me
r P − e
r A (
r R )[ ]
2+ eΦ(
r R )
Vector potential of a uniform B-field
• For a uniform B-field, we have:
• Proof:
02
1)( BrrA
rrrr×−=
)()( rArBrrrrr
×∇=
€
r B
r r ( ) = −
12
r ∇ ×
r r ×
r B 0( )
€
r P + e
2r R ×
r B 0
2
= P 2 +e2
r P ⋅
r R ×
r B 0 +
r R ×
r B 0 ⋅
r P [ ] +
e2
4r R ×
r B 0( )
2
€
= P 2 − er L ⋅
r B 0 +
e2
4R2B0
2 −r R ⋅
r B 0( )
2
0)( BrBrrr
=
€
= −12
r r
r ∇ ⋅
r B 0( ) −
r B 0
r ∇ ⋅
r r ( ) − r
r ⋅r ∇ ( )
r B 0 +
r B 0 ⋅
r ∇ ( )r r [ ]
€
= −120 − 3
r B 0 − 0 +
r B 0[ ]
€
=r B 0
An electron in a uniform B-field
• Putting this in the Hamiltonian gives:
• Choosing B along the z-axis gives:
€
H =P 2
2me
−e2me
r L ⋅
r B 0 +
e2
8me
B02R⊥
2 + eΦ(r R )
€
H =P 2
2me
−eB0
2me
Lz +e2B0
2
8me
X 2 + Y 2( ) + eΦ(r R )
€
−e2me
LzB0
€
e2B02
8me
X 2 +Y 2( )
“Paramagnetic term”•Generates linear Zeeman effect
“Diamagnetic term”•Generates quadratic Zeeman effect
Paramagnetic Term: Magnetic DipoleInteraction
• A loop of current, I, and area, a, creates amagnetic dipole:
• The orbital motion of a single electronconstitutes a current– For a circular orbit we have
• An electron therefore has a magnetic dipolemoment associated with its orbital motion
• The paramagnetic term is therefore theenergy of the orbital dipole moment in theuniform field:
€
r µ =
e2me
r L
0BVBrr⋅−= µ
€
µ = I a
€
I = −ev2π r
,
€
a = π r2
€
I a = −ev r2
€
−ev r2
= −e2me
mevr = −e2me
r p ×
r r = e
2me
r r ×
r p
ze
B Lm
eBV
20−=
Dipole Energy scale
• The energy shift between different m statesis very small compared to Hydrogen levelspacing
• Order of magnitude:
• Strongest man-made B-fields ~40 T
T
J
T
J
m
e
B
V
e
B 23303419
0
1010~ −+−− ==h
€
VB ≤10−22J << E1 2.18 ×10−18J( )
mm
eBmnVmn
eB hll
20−=
Energy scale ofbare levels
Diamagnetic Term
• An electron in a uniform field will naturallyundergo circular motion in the planeperpendicular to the field– Cyclotron motion
• Thus the B-field induces a current
• This leads to an induced magnetic moment,which must be proportional to B0
• The energy of this magnetic moment in theuniform B field therefore scales as B2
• Order of magnitude:
– The diamagnetic term can be neglected unlessthe B-field is very strong
0Binduced
rr∝µ
200 BBE induced ∝⋅−=
rrµ
228
2302038
20
2
20
10108
~2
T
J
T
J
m
ae
B
V
e
B −+−− ==
)10()10(10 181
22262 JEJVJV BB
−−− <<<<≤
( )2220
2
82 YX
m
BeV
eB
+=
Zeeman Effect
• The Hamiltonian of a Hydrogen atom in a uniformB-field is– Can neglect diamagnetic term
• Eigenstates are unchanged
• Energy eigenvalues now depend on m:
• The additional term is called the Zeeman shift– We already know that it will be no larger than 10-22
J~10-4eV– E.g. 100 G field:
• EZeeman~10-25 J• EZeeman/EI ~ 10-25+18 ~ 10-7
• To get the correct Zeeman shift, we will also needto include spin.– We will do this next semester using perturbation
theory and the Wigner-Ekert Theorem
zLeB
HHµ20 −=
mnEmnH ,,,, ll =
meB
naE mn µµ 2
1
2 220
2
, −−=h
mnEmnH n ll =0
‘Bare’ Hamiltonian