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