5 th lecture: Summary 4 th lecture: Effective spin ½ systems
Classical spinning-top eq.
Spin s , magnetic moment magnetic Hamiltonian
For s = ½: s

1
2
 σ
:
Pauli matrices d s /d t
 ω
0
 s , holds also for gyromagnetic case with ω
0
 γ B .
μ
H


 γ s
 μ 
, with gyromagnetic ratio
B
 γ s

B .
γ 
H

1
2

 σ 
B

1
2

 (
σ x
B x
 σ y
B y
 g e / 2 m ,
σ z
B z
) ,
σ x



0
1
1
0

 ,
σ y



0 i

0 i

 ,
σ z



1
0

0
1

 .
ω
0 d s
s
μ s
Any 2-dim hermitean with
α 
H
( h
12
, h
12
' , 1
2
( h
11


 h
12
 h
22
)) h
11
 ih
12
' h
12
, i.e. as an
 h
22 ih
12
'

 can be expressed as effective spin interaction.
H

1
2
( h
11
 h
22
) I

1
2
σ  α ,
In general : any quantum system of N identical states behaves like a spin s system, where N = 2 s + 1.
Zeeman effect : B -field lifts symmetry, 2 s + 1 splitting

B : E m
Example : B

( B x
, 0 , B z
) : H

1
2
 (
ω
0
σ z
 ω
1
σ x
)

1
2



ω
0
ω
1
 m  ω
0

ω
1
ω
0


, Larmor frequency
ω 
0
,
γB
.
E
E
0 eigenvalues E

   ω 2
0
 ω
1
2
, with respect to the basis states s z
are on hyperbola.
For spin initially along z , spin-flip probability given by Rabi's formula : c
2
( t )
2 
ω
0
2
ω
1
2
 ω
1
2 sin
2
(
ω
0
2  ω
1
2 t ) .
All measureable quantities are either eigenvalues A n
or expectation values

A

With hermitean density operator
ρ   n
| n
 p n
 n | , the expectation value is

of some operator A .
A
   n p n
A n

Tr(
ρA
) .
In the 2-dim case, with superposition ψ
1
' ( t )
 c
1
( t ) ψ
1
 c
2
( t ) ψ
2
, ψ
2
'
  c
1
 ψ
1
 c
2
 ψ
2
, c
1
2  c
2
2 
1 , density matrix for a pure state with p
1
= 1 is
ρ' 
U
 ρU 

 c c
1


2
 c
1 c
2




1
0
0
0




 c
1 c

2 c
2 c
1



 c
1 c
1 c
2

2 c
1
 c
2
2 c
2
. s = ½:
E
+1/2
P z
 c
1
( t )
2  c
2
( t )
2
:
SPIN ROTATION IN VARIOUS B FIELDS
1
0.75
0.5
0.25
0
0.25
0.5
0.75
0
ħω
0
2
m = + ½
E
−1/2
m = − ½
B z
4 time
1 t
6
The 2-dim density operator can always be written in terms of Pauli matrices as ρ 
1
2
( I
  σ   σ ) .
Dubbers: Quantum physics in a nutshell WS 2008/09 5.1
8

e) Spin-polarization
The vector polarization is defined as the expectation value of the spin vector s :
P

 s s 

, normalized such that 0

P

1 . For spin s = ½, P
  σ 
.
1. Mixed spin state: s = ½: two substates with magnetic quantum number m = −½ and m = +½,
P z
P z




1

1

μB
2 kT exp(
 μB
/ kT ) exp(
 μB / kT )
for

μB 
2 kT exp(
 μB
/ 2 kT ) exp(
 μB / 2 kT )
.

 exp(
 μB
/ 2 kT ) exp(
 μB / 2 kT )
,
1.75
1.5
1.25
1/ T
M
(defined with respect to some quantization axis z ), with populations N m
and probabilities p


N

/( N


N

) and p


N

/( N


N

) , have longitudinal polarization
P z
  σ z
 
Tr (
ρσ z
)

Tr

 p
0
 p
0





1
0

0
1



Tr

 p
0


0 p



 p

 p

, or P z

N
N




N
N


.
χ
2
CURIE GESETZ
B s > ½: 2 s + 1 sublevels with magnetic quantum numbers m = − s
, − s
+1, …, + s , with population probabilities p m

N m
  s m
  s
N m
, have longitudinal polarization
P z

 s z


1 mp m
, with

1

P z
 
1 .
 s s
  m s
  s
Example : N independent s = ½ atoms at temperature T in a magnetic field B :
Example
1 /( 2 kT )
: Atom has

μ 
10

4 eV/Tesla
40 / 2

300 / 70 , P z

, at T
LN
2
85

10

4 
= 70 Kelvin:
0 .
85 % .
In thermal equilibrium , they occupy the two Zeeman levels E
±
, with
N

/ N

 exp(
 μB
/ kT ) , for E
+
> E
−
, and N = N
+
+ N
−
. Their polarization is:
N


N

N
 tanh

N
μB

2 kT
The magnetization curve is M
The magnetic susceptibility is χ

Nμ P z


M

B


Nμ tanh
Nμ
2
2 kT
μB
2 kT

N
μ 2
B
2 kT
.
, which is Curies Law .
1
0.75
0.5
∂
M /∂ B
0.25
Dubbers: Quantum physics in a nutshell WS 2008/09 5.2
0 2 4 6 8
TEMPERATUR
T

2. Pure state, given for instance by:
ρ 


1
0
0
0

 , has
 σ z
 
Tr (
ρσ z
)

Tr


1
0
0
0




1
0

0
1



Tr


1
0
0
0



1 , or P z
= 100%.
Any coherent superposition ψ
1
' ( t )
 c
1
( t )
ψ
1
 c
2
( t )
ψ
2
then also is a pure state, with:
 σ z
 
Tr ( ρσ z
)

Tr c
1 c
1 c

2
2 c
1
 c
2
2 c
2


1
0

0
1



Tr
2
 c
1 c
1 c
2
  c
1
 c
2 c
2
2
 c
1
2  c
2
2
.
Example : Spin rotation of 100% polarized atoms in a transverse magnetic field, see above.
3. A coherent superposition made from a mixed ensemble:
 σ z
 
Tr (
ρσ z
)

Tr p
1 c
1 c
1 c

2
2 
( p
1 p
2
 c
2 p
2
)
2 p c
1
 c
2
( p
1
1 c
2
2 
 p
2 p
2
)
2 c
1


1
0

0
1

 ,
 σ z
 
( c
1
2  c
2
2
)( p
1
 p
2
) , i.e. the same happens as for a pure states, but at a lower degree of polarization
 σ
( t )
 z
can be measured in a pick-up coil outside the atomic vapor vessel.
4. The mean energy for the mixed B x
, B z
-atomic spin rotation case is

E
 
Tr (
ρE
)
  Tr

 c
1 p c
1
2 c
1
(
2 p

1 p

2 c
2 p
2
2
) c
1 c p
1
2 c
2
2
( p

1
 p
2 p c
2
1
2
)




ω
ω
0
1

ω
1
ω
0

 , or

E
 
[( c
1
2  c
2
2 )  ω
0

2 c
1 c
2
 ω
1
]( p
1
 p
2
) . p
1
 p
2
.
Dubbers: Quantum physics in a nutshell WS 2008/09 5.3

f) Other notations for the density matrix
H in energy representation can be written as H
  n
| n

E n
 n | , where n is a set of 'good' quantum numbers (for instance | n , l , m l
, s , m s
, for the H-atom).
The matrix elements of H in another representation | m

, where m stands for another set of quantum numbers, then read:
 m | H | m '
   n
 m | n

E n
 n | m '
 or H m , m '
  n c mn
E n c
 nm '
or H

UEU

, with diagonal matrix E

U

HU
In the same way, the density operator can be expressed in diagonal form as
ρ   n
| n
 p n
 n | , whatever the choice of base states | n

.
| and can be transformed to any other base states U
 ρU
(Examples: | n
 n

= eigenstates of energy for the NH
3
in a similar way.
molecule,
= eigenstates of spin in the Stern-Gerlach experiment, ...).
.
Dubbers: Quantum physics in a nutshell WS 2008/09 5.4

3.4
Time dependence of the density matrix a) The Liouville equation
Back to p. 2.4: i  ψ 
Hψ , with Hermitean H . In a given representation for s = ½, i 

 c  c 
2
1





H
H
11

12
H
12
H
22



 c c
2
1

 with
ψ 

 c c
2
1

 .
The pure-case density matrix is ρ 

 c
1 c
1
 c
1 c

2
More general:
ρ ij and i  c  i

 c i c
 j k

,
H ik c k
, c.c:
 i  c  i
 c
1
 c
2 c
2 c
2


 .
  k
H ik
 c
 k
  k
H ki c
 k
.
The equation of motion of
ρ
then is and that is i  i 
 ij
 ij

 i 

( c  i k c
H
 j ik

ρ kj c i c 

 j
)
 k

 ij

 k
ρ ik
H c  i
H kj c ik
 j

 c k
( c c i
 j c 

Hρ )
 j ij
, c i

 k
(
ρH
H
) ki ij c
,
 k
  k
[ H ik
( c k c
 j
)

H ki
( c i c
 k
)] , which is the Liouville equation for the density operator: i  ρ 
Hρ
 ρH 
[ H ,
ρ
] .
For the mixed case , at t = 0: ρ mix
( 0 )
  n p n
| n
  n |
  n p n
ρ n
( 0 ) .
As the various parts in the mixture are assumed to be completely independent, the statistical weight p n
of the pure ρ n

| n
  n | in the mixture will stay the same in the course of time, that is
ρ mix
( t )
  n p n
ρ n
( t ) .
Therefore when the pure state density operator obeys the Liouville equation, then so does the mixed state density operator i  ρ mix
( t )
 i 
 n p n
 ρ n
( t )
  n p n
[ H , ρ n
]

[ H ,
 n p n
ρ n
( t )]

[ H , ρ mix
] , that is the Liouville equation holds for all density operators.
Dubbers: Quantum physics in a nutshell WS 2008/09 5.5

b) The Bloch equation
For s = ½, the density operator in terms of the polarization vector
 σ 
is:
ρ 
1
2
( I
  σ   σ )

1
2
( I
  σ x
  σ x
  σ y
  σ y
  σ z
  σ z
) , or
ρ 


ρ
11
ρ 
12
ρ
ρ
12
22



1
2


 σ
1 x



 σ z i
 σ
 y

 σ x

1

 i
 σ z
 σ
 y
 
 .
Example : atomic beam 100% polarized along z : ρ 


1
0
Conversely,
 σ x
  ρ
12
 ρ
21
,
 σ y
  i (
ρ
12
 ρ
21
) ,
 σ z
 
0
0

 ; along x : ρ
ρ
11
 ρ
22
.



0
1
For the magnetic Hamiltonian with the relations H
11

H
22
H



H
11

H
12
  γB z
, H
11

H
12
H
22


H
22

  γ


B
0 , H
12 x
B z
 iB y

H
21
B x

 iB
B z y

 ,
  i  γB y
, H
12
1
0

 , etc.

H
21
  γB x
. the Liouville equation i i  ρ
12
 ρ
21 i  ρ
11



H
11
H
21
ρ
12
ρ
11
H
12
ρ
21


 i   ij

ρ
11
H
12
ρ
21
H
11
ρ
12
H
21
 k
( H ik
ρ kj
 ρ ik
H kj
) reads

H
12
ρ
22

, i
H

22

22
ρ
21



H
ρ
12
H
22
ρ
21
22
ρ
H
12
21
 ρ
21
H
12
  i  ρ
11 i  ( i  ( i  (

12

12

11
 
21
 ρ
21
 
22
)
)
)

( H
11


( i
H
 ρ
11
11



H
H
22
22
)(
)(
2 ( H
12
ρ
ρ
12
12
ρ
21
 ρ
21
)


ρ
21
)
H
21

( H
12

ρ
12
( H
12
)

(


H
H
H
12
21
)(
ρ
11
21

)(
H
ρ
11
21
)(
 ρ
22
)
 ρ
ρ
21
22
)
 ρ
12
)

( H
12
 σ  x
  γB y
 σ z
  γB z
 σ y

,
 σ  y
  γB z
 σ x
  γB z
 σ z

,
 σ  z


H
21
)(
ρ
21
 γB x
 σ y
  γB y
 σ x

,
 ρ
12
)
NB: The Bloch equatio is a 3×3 real representation which is the Bloch equation
    ω
0
  σ 
, with Larmor frequency ω
0
 γ B .
of a 2×2 complex problem: SO(3)↔SU(2).
It holds for arbitrary spin s :
 s    ω
0
  s

, polarization P
  s

/  s :
 ω
0

P , or magnetization
It is the quantum version of the equation of motion of the spinning top, and holds for
 s

,
 μ 
M

, but not for
 μ  s ,
μ

.
 γ  s

:
 ω
0

M .
For N atoms,
 μ 
is well defined when its direction varies by an angle Δ
φ
<< 1, with Δ
φ
from the uncertainty relation

N
  φ 
N
  φ 
1 .
Dubbers: Quantum physics in a nutshell WS 2008/09 5.6

c) Outlook to higher spins: generalized Bloch equations
For atom (system) with an s > ½ it is not sufficient to give the polarization vector P to characterize the population of its 2 s + 1 sublevels and their coherences.
Example : the angular distribution of light emitted by unpolarized atoms is isotropic.
But this distribution does not depend on the components of the polarization vector, for instance P z
  s z

/ s   s
1   m
 s
 s mp m
, but depends on quantities quadratic in magn. qu. number m , s = ½: for instance the zz -component of alignment A zz

( 3
 s
2 z
  s ( s

1 )) / 3 s
2  2
.
Example s = 1: a linear population of the 3 sublevels: N
+
= 2, N
0
= 1, N
−
= 0 gives
P z

N

N



N
0
N


N


1
3
, but A zz

N
N




N
N

0


2 N
N

0

0 , so emission is isotropic.
Quadratic population N
+
= 1, N
0
= 0, N
−
= 1, on the other hand, gives P z
= 0, but A zz
= 1 , so emission is anisotropic.
P z is one of 3 components of a vector. A zz
is one of 5 components of a tensor.
Altogether one needs (2 s + 1)
2
− 1 polarization components to characterize an ensemble of atoms with spin s and all its coherences
(i.e. 4 − 1 = 3 components for s = ½, 9 − 1 = 8 for s = 1, 16 − 1 = 15 for s = 3/2, etc.).
E m
E m
p m p m
Dubbers: Quantum physics in a nutshell WS 2008/09 5.7

The Bloch equation gives the time evolution of the polarization vector P of an atomic system under a magnetic dipole interaction μ·B , with an external field B :

P
 γ

B


P , with
γ
proportional to the magnetic dipole moment
μ
.
This equation holds for any spin s > 0.
Also the alignment tensor A

 γ
'

B

,
follows a very similar precession equation where the rank 2-cross product is between a tensor of rank 1 and a tensor of rank 2, valid for any s > 1.
For s > ½, in general there is also a quadrupole interaction between an atomic electric quadrupole moment Q and the tensor V
ˆ of external electric field gradients.
(While B involves the first spatial derivatives of the magnetic vector potential A ,
V
ˆ involves the second spatial derivatives of the electric potential
V .
One can extend the Bloch equation to include the quadrupole interaction:

P
ˆ

A


γ
γ '

B

B



P


η
V
ˆ
η ' V
ˆ



P
,
, with
η
and
η
' proportional to the electric quadrupole moment Q .
For atomic spin s , up to 2 2 s -pole interactions, described by tensors of rank 2 s , are allowed:
s = ½: magnetic dipole interaction and rank 1 tensors = vectors;
s = 1: electric quadrupole interaction and rank 2 tensors; s

3
2
: magnetic octupole interaction and rank 3 tensors;
s = 2: electric hexadecupole interaction and rank 4 tensors, etc.
NB: The polarization tensors P , A , etc. are the irreducible representation of the density matrix ρ .
Dubbers: Quantum physics in a nutshell WS 2008/09 5.8

c) Coherence
Example double slit experiment: amplitude through slit 1:
ψ
1
 amplitude through slit 2: ψ
2

ψ
10 e i ( p x

E t ) / 
,
ψ
10 e i ( p x

E t ) /
  iδ total amplitude ψ  ψ
1
 ψ
2
.
, with relative phase shift iδ ,
Probabiliy density:
ψ 2 

ψ
1

ψ
1
ψ
1

 ψ
10
2
ψ
2

2
ψ
2

ψ 
2
(
ψ
1
 ψ

1
ψ
ψ 
2
2
 ψ
20
2  ψ
10
ψ
)(
ψ
1

 ψ
1


ψ
2
20
(e
 i δ
ψ
2

 e
) i δ
)
 ψ
10
2  ψ
20
2 
2
ψ
10
ψ
20 cos
δ
.
p x x
Self-interference is possible, if both paths are within the same cell volume
 3 x
 3 p x
 
V
  3 of phase space ( x , p ), that is if the distance x between the slits (+ extension of the source and the detector) and divergence
 p x
of the electron beam and are so small that x
 p x
  x
 p x
  .
Then the uncertainty
 x ,
 p x
of the system is so large that we cannot know which path was taken. d N /d
θ with spin-flip: incoherent neutron scattering
Self-interference is impossible when both paths are not within the same

V , so we can know which path was taken. In this case p
 ψ
1
2  ψ
2
2
, and interference is lost. d N /d
θ
2
θ
A process/system is called coherent when interference is possible.
Example : Neutron scattering with and without neutron spin flip in the sample: with spinflip it is certain on which nucleus the neutron was scattered, there are no diffraction peaks, without spinflip it is uncertain. In the first case scattering is incoherent, in the second coherent.
This holds irrespective whether the scattering involves energy transfer.
Hence, for neutrons there are four different scattering processes: coherent elastic, coherent inelastic, incoherent elastic, incoherent inelastic.
Whether the incoming neutron beam is coherent or not depends only on its divergence and spot size. no spin-flip: coherent neutron scattering
2 θ
Dubbers: Quantum physics in a nutshell WS 2008/09 5.9

d) Decoherence
In the density matrix, the non interfering probabilities while the mixed interference terms ψ
1
 ψ
2
ψ
1
2 and ψ
2
2
are on the diagonal,
 ψ
10
ψ
20 e
 iδ
are off-diagonal.
In the pure case:
ρ 
ψ
1
2
ψ
1
ψ
2

ψ
1
 ψ
2
ψ
2
2
interference signals are strongest.
In the mixed case, the beam consists of different populations distinguished, say, by energy, that is wavelength, and the interference signal becomes blurred (e.g. white light interference).
How can we know whether a given density matrix is pure? In both cases, pure and mixed, Tr
ρ
= 1.
But in the pure case, also Tr
ρ 2
= 1, while in the mixed case, Tr
ρ 2
< 1.
2-dim example: as the trace is independent of representation, in the pure case,
Tr ρ 2 
Tr ρ 
Tr


1
0
0
0



1

0

1 , while in the mixed case, with 0
 p
1
, p
2

1 ,
Tr
ρ 2 
Tr

 p
1
0
2
0 p
2
2


 p
1
2  p
2
2
 p
1
 p
2

1 .
Changes of phase
δ
affect the off diagonal elements
ψ i
 ψ j
 ψ i 0
ψ j 0 e
 i ( δ i
 δ j
)
, while the diagonal elements
ψ i
2
remain stable against phase shifts.
Environmental influences (like air turbulence in astronomical observation) first destroy the relative phases of the paths before they affect the intensities. Therefore, for a system in contact with the environment, the off-diagonal elements fade away ( decoherence ), and coherence of the system is lost.
If the environment is a heat bath with a definite temperature, then in the long run the p i
will aquire a Boltzmann distribution.
This goes over into a Fermi ( s

1
2
, 3
2
, ...) or Bose-Einstein ( s

0 , 1 , 2, ...) distribution if phase space density becomes ≈ 1atom/cell: ρ 
1

V
This is the case when the mean distance x between particles and their mean momentum p are p
 x
  , that is when the spatial density n

1 x
3
is such that ρ 
1 x
3 p
3
 n  3
 3

1
 3
, or n  3 
1 ( nλ
3 
2 .
6 for BEC).

1
 3
.
Dubbers: Quantum physics in a nutshell WS 2008/09 5.10