Download Δk/k

5th lecture: Summary 4th lecture: Effective spin ½ systems
ds
s
μ   γ s , with gyromagnetic ratio γ  g e/ 2m ,
H  μ  B  γ s  B .
H  1 2   σ  B  1 2   (σ x B x  σ y B y  σ z B z ) ,
0 1
0  i
1 0 
μ
 , σ y  
 , σ z  
 .
σ x  
1
0
i
0
0

1






h12  ih12 ' 
 h11
 can be expressed as H  12 (h11  h22 ) I  12 σ  α ,
Any 2-dim hermitean H  
h22 
 h12  ih12 '
s
Pauli matrices
E
E+1/2
with α  (h12 , h12 ' , 12 (h11  h22 )) , i.e. as an effective spin interaction.
In general: any quantum system of N identical states behaves like a spin s system, where N = 2s + 1.
ħω0
E0
Zeeman effect: B-field lifts symmetry, 2s + 1 splitting  B : Em  m ω0 , Larmor frequency ω0  γB .
ω1 
,
 ω0 
eigenvalues
E   ω02  ω12 , with respect to the basis states sz are on hyperbola.
(ω0 σ z  ω1σ x ) 
1
2
For spin initially along z, spin-flip probability given by Rabi's formula: c 2 (t ) 
2
All measureable quantities are either eigenvalues An or expectation values  A of some operator A.
With hermitean density operator ρ  n | n p n  n | , the expectation value is  A  n pn An  Tr( ρA) .
In the 2-dim case, with superposition ψ1' (t )  c1 (t )ψ1  c2 (t )ψ 2 , ψ 2'  c1ψ1  c2ψ 2 , c1  c 2
2
 c1
density matrix for a pure state with p1 = 1 is ρ'  U ρU   
 c2

2
 1,
2
c2   c1
 
c1   c1c 2
 c2  1 0  c1

 
c1  0 0   c2
The 2-dim density operator can always be written in terms of Pauli matrices as ρ 
2
1
2
c1 c2 
2 .
c 2 
2
Pz  c1 (t )  c 2 (t ) :
ω12
sin 2 ( ω02  ω12 t ) .
2
2
ω0  ω1
SPIN ROTATION IN VARIOUS B FIELDS
1
0.75
0.5
0.25
0
0.25
0.5
0.75
0,1,2,4
B  ( Bx ,0, Bz ) : H 
2
m = −½
Bz
E−1/2
Pz
ω
 0
 ω1
Example:
1
m = +½
s = ½:
0
2
4
time
6
1
8
t
( I  σ   σ ) .
Dubbers: Quantum physics in a nutshell WS 2008/09
5.1
1
Spin s, magnetic moment
magnetic Hamiltonian
For s = ½: s  12 σ :
ω0
ds/dt  ω0  s , holds also for gyromagnetic case with ω0  γB .
0
Classical spinning-top eq.
e) Spin-polarization
The vector polarization is defined as the expectation value of the spin vector s:
 s
P
, normalized such that 0  P  1 . For spin s = ½, P   σ  .
s
1. Mixed spin state:
s = ½: two substates with magnetic quantum number m = −½ and m = +½,
(defined with respect to some quantization axis z), with populations Nm and probabilities
p  N  /( N   N  ) and p  N  /( N   N  ) , have longitudinal polarization
p
Pz  σ z   Tr ( ρσ z )  Tr  
 0
0  1 0 
p

  Tr  
p  0  1
 0
M
0 
N  N
  p  p , or Pz  
.
 p 
N  N
B
s > ½: 2s + 1 sublevels with magnetic quantum numbers m = −s, −s+1, …, +s,
s
with population probabilities pm  N m m s N m , have longitudinal polarization
s z  1 s
 m s mpm , with  1  Pz  1 .
s
s
Example: N independent s = ½ atoms at temperature T in a magnetic field B:
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  1  exp(  μB/kT ) exp(  μB/ 2kT )  exp(  μB/ 2kT )
,


N   N  1  exp(  μB/kT ) exp(  μB/ 2kT )  exp(  μB/ 2kT )
μB
μB
Pz  tanh

for μB  2kT .
2kT 2kT
μB
μ2B
N
The magnetization curve is M  Nμ Pz  Nμ tanh
.
2kT
2kT
M Nμ 2
The magnetic susceptibility is χ 
, which is Curies Law.

B 2kT
Pz 
Example: Atom has μ  10 4 eV/Tesla , at TLN 2 = 70 Kelvin:
1 /( 2kT )  40 / 2  300 / 70 , Pz  85  10 4  0.85 % .
χ
SUSZEPTIBILITAET
Pz 
CURIE GESETZ
2
1.75
1.5
1.25
1/T
1
0.75
0.5
0.25
∂M/∂B
Dubbers: Quantum physics in a nutshell WS 2008/09
0
2
4
6
TEMPERATUR T
8
5.2
2. Pure state, given for instance by:
1 0
 1 0  1 0 
1 0
 , has σ z   Tr ( ρσ z )  Tr 

  Tr 
  1 , or Pz = 100%.
ρ  
 0 0
 0 0  0  1
 0 0
Any coherent superposition ψ1' (t )  c1 (t )ψ1  c2 (t )ψ 2 then also is a pure state, with:
 c1 2
σ z   Tr ( ρσ z )  Tr  
c c
 1 2
 c1 2
c1 c2  1 0 

  Tr 
2 

  c c
0

1
c 2 

 1 2
c1 c2 
2
2
 c2 .
2  c1
 c2 
Example: Spin rotation of 100% polarized atoms in a transverse magnetic field, see above.
3. A coherent superposition made from a mixed ensemble:
 p1 c1 2  p 2 c 2 2
σ z   Tr ( ρσ z )  Tr 
 c c ( p  p )
2
 1 2 1
2
2
σ z   ( c1  c 2 )( p1  p 2 ) ,
c1 c 2 ( p1  p 2 )  1 0 

 ,
2
2 
p1 c 2  p 2 c1  0  1
i.e. the same happens as for a pure states, but at a lower degree of polarization p1  p2 .
σ z (t ) can be measured in a pick-up coil outside the atomic vapor vessel.
4. The mean energy for the mixed Bx, Bz-atomic spin rotation case is
 p1c12  p2 c22 c1c2 ( p1  p2 )  ω0 ω1 

 , or
 E  Tr ( ρE)  Tr 
2
2 
ω  ω0 
c
c
(
p

p
)
p
c

p
c
2
1 2
2 1  1
 1 2 1
 E   [(c12  c 22 )ω0  2c1c 2 ω1 ]( p1  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 En  n | ,
where n is a set of 'good' quantum numbers (for instance | n, l , ml , s, ms , 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 En  n |m' 



or H m,m'  n cmn En cnm
' 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 in a similar way.
(Examples: | n = eigenstates of energy for the NH3 molecule,
| n = 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
H 12  c1 
 c   H
c 
  with ψ   1  .
for s = ½,
i 1    11

 c2   H 12 H 22  c2 
 c2 
 c1c1
The pure-case density matrix is ρ   
 c1c2
More general: ρij  ci c j ,
and
c1 c2 
.
c2 c2 
ici  k H ik ck , c.c:  ici  k H ik ck  k H ki ck .
The equation of motion of ρ then is ρ ij  ci c j  ci c j ,
and
that is
iρ ij  i(ci c j  ci c j )  k H ik ck c j  ci k H ki ck  k [ H ik (ck c j )  H ki (ci ck )] ,
iρ ij  k H ik ρkj  k ρik H kj  ( Hρ) ij  ( ρH ) ij ,
which is the Liouville equation for the density operator:
iρ  Hρ  ρH  [ H , ρ] .
For the mixed case, at t = 0: ρmix (0)  n pn | n  n |  n pn ρn (0) .
As the various parts in the mixture are assumed to be completely independent,
the statistical weight pn of the pure ρn | n  n | in the mixture
will stay the same in the course of time, that is ρmix (t )  n p n ρn (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 pn ρ n (t )  n pn [ H , ρn ]  [ H , n pn ρn (t )]  [ H , ρmix ] ,
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  12 ( I  σ   σ )  12 ( I  σ x   σ x  σ y   σ y  σ z   σ z ) ,
σ x   iσ y  
 1  σ z 

.
 σ   iσ 

1


σ

y
z
 x

1 0
0 1
 ; along x: ρ  
 , etc.
Example: atomic beam 100% polarized along z: ρ  
 0 0
1 0
Conversely, σ x   ρ12  ρ21 , σ y   i ( ρ12  ρ21 ) , σ z   ρ11  ρ22 .
or
ρ
ρ   11

 ρ12
ρ12 

ρ22 
1
2
Bx  iB y 
H 12 
 Bz
H
,
  γ
For the magnetic Hamiltonian H   11


B

iB

B
x
y
z
 H 12 H 22 


with the relations H 11  H 22  γB z , H 11  H 22  0, H 12  H 21  iγB y , H 12  H 21  γB x .
the Liouville equation iρ ij  k ( H ik ρkj  ρik H kj ) reads
iρ12  H11 ρ12  ρ11H12  H12 ρ22  ρ12 H 22
iρ 21  H 21 ρ11  ρ21H11  H 22 ρ21  ρ22 H 21
iρ11  H12 ρ21  ρ12 H 21 , iρ 22  H 21 ρ12  ρ21H12  iρ11
i( ρ12  ρ 21 )  ( H11  H 22 )( ρ12  ρ21 )  ( H12  H 21 )( ρ11  ρ22 )
i( ρ12  ρ 21 )  ( H11  H 22 )( ρ12  ρ21 )  ( H12  H 21 )( ρ11  ρ22 )
i( ρ11  ρ 22 )  iρ11  2( H12 ρ21  H 21 ρ12 )  ( H12  H 21 )( ρ21  ρ12 )  ( H12  H 21 )( ρ21  ρ12 )
σ x   γB y σ z   γB z  σ y  , σ y   γB z σ x   γB z  σ z  , σ z   γB x σ y   γB y σ x  ,
NB: The Bloch equatio is a 3×3 real representation
of a 2×2 complex problem: SO(3)↔SU(2).
which is the Bloch equation σ   ω0  σ  , with Larmor frequency ω0  γB .
 ω M .
It holds for arbitrary spin s:  s  ω0   s , polarization P   s / s : P  ω0  P , or magnetization M   μ  γ s : M
0
It is the quantum version of the equation of motion of the spinning top, and holds for s ,  μ , but not for s, μ.
For N atoms,  μ is well defined when its direction varies by an angle Δφ << 1, with Δφ 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 2s + 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,
1 s
for instance Pz   s z  /s  m   s mpm , but depends on quantities quadratic in magn. qu. number m, s = ½:
s
for instance the zz-component of alignment Azz  (3 s z2   s(s  1)) / 3s 2  2 .
Example s = 1: a linear population of the 3 sublevels: N+ = 2, N0 = 1, N− = 0 gives
N  N   2N 0
N  N
Pz 
 13 , but Azz  
 0 , so emission is isotropic.
N  N0  N
N  N0  N
Quadratic population N+ = 1, N0 = 0, N− = 1, on the other hand,
gives Pz = 0, but Azz = 1 , so emission is anisotropic.
Em
pm
Em
Pz is one of 3 components of a vector. Azz is one of 5 components of a tensor.
Altogether one needs (2s + 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.).
pm
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  γB  P ,
with γ proportional to the magnetic dipole moment μ.
This equation holds for any spin s > 0.
Also the alignment tensor  follows a very similar precession equation


Aˆ  γ' B  Aˆ ,
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 Q̂ and the tensor V̂ of external electric field gradients.
(While B involves the first spatial derivatives of the magnetic vector potential A,
V̂ involves the second spatial derivatives of the electric potential V.
One can extend the Bloch equation to include the quadrupole interaction:

 
P  γB  P  ηVˆ  Aˆ ,



Aˆ  γ' B  Aˆ  η'Vˆ  P ,
with η and η' proportional to the electric quadrupole moment Q.
For atomic spin s, up to 22s-pole interactions, described by tensors of rank 2s, 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, Â , 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:
ψ1  ψ10 e i ( p x  E t ) / ,
amplitude through slit 1:
ψ 2  ψ10 e i ( p x  E t ) / iδ , with relative phase shift iδ,
amplitude through slit 2:
total amplitude
px
x
ψ  ψ1  ψ 2 .
Probabiliy density:
ψ
2
 ψ1  ψ 2  (ψ1  ψ 2 )(ψ1  ψ 2 )
 ψ1ψ1  ψ 2ψ 2  ψ1ψ 2  ψ1ψ 2
2
 ψ10
2
 ψ 20
2
 ψ10 ψ 20 (e iδ  e iδ )
 ψ10
2
 ψ 20
2
 2 ψ10 ψ 20 cos δ.
dN/dθ
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   .
with spin-flip:
incoherent neutron scattering
Then the uncertainty x , p x of the system is so large that we cannot know which path was taken.
Self-interference is impossible when both paths are not within the same V , so we can know
2
2
which path was taken. In this case p  ψ1  ψ 2 , and interference is lost.
dN/dθ
2θ
no spin-flip:
coherent neutron scattering
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.
2θ
Whether the incoming neutron beam is coherent or not depends only on its divergence and spot size.
Dubbers: Quantum physics in a nutshell WS 2008/09
5.9
d) Decoherence
2
In the density matrix, the non interfering probabilities ψ1 and ψ 2
2
are on the diagonal,
while the mixed interference terms ψ1ψ 2  ψ10ψ 20 e  iδ are off-diagonal.
 ψ1 2 ψ1ψ 2 
 interference signals are strongest.
In the pure case: ρ  
ψ ψ  ψ 2 
2
 1 2

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 ρ2 = 1, while in the mixed case, Tr ρ2 < 1.
2-dim example: as the trace is independent of representation, in the pure case,
1 0
  1  0  1 , while in the mixed case, with 0  p1 , p2  1 ,
Tr ρ 2  Tr ρ  Tr 
 0 0
 p2
Tr ρ 2  Tr  1
 0
0
  p12  p22  p1  p2  1 .
2
p2 
Changes of phase δ affect the off diagonal elements ψ iψ j  ψ i 0 ψ j 0 e
while the diagonal elements ψ i
2
 i ( δi δ j )
,
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 pi 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 
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
1
 3.
V 
1
1
n 3
1
is
such
that
ρ


 3 , or n 3  1 ( nλ 3  2.6 for BEC).
3
3 3
3
x
x p


Dubbers: Quantum physics in a nutshell WS 2008/09
5.10