Download Δk/k

DISAPPEARANCE OF STATE | 1 :
rd
4 lecture: Summary 3 lecture: Properties of NH3
NH3 IN VARIOUS E FIELDS
0.4
0.2
 1  ( At / ) 2 →
~
the energy uncertainty E  /t is much larger than the energy splitting /t 0  2 A 2  p 2 E 2 .
0
General rule: For very short times, a quantum system
does not take notice of external perturbations, the system is quasi-free.



 E0    ψ1 ( z) H ( z)ψ1 ( z)dz   ψ 2 ( z) H ( z)ψ 2 ( z)dz  E0 .
With tunneling, the energy splitting is
E   E  E  
1
2



1
1.5
time A
2
t
2.5
3
1
c22
0.8
NH3 IN VARIOUS E FIELDS
 ( At / ) 2 →
0.6
0.4
0.2
With no tunneling, the mean energy of the system is

0.5
APPEARANCE OF STATE | 2 :
Alternative derivation from the ordinary Schrödinger equation with solutions ψ1(z), ψ2(z):
The NH3 system has mirror symmetry S with respect to the x-y plane of the 3 H-atoms.
The eigenstates of S are the symmetric and antisymmetric states ψ   (ψ1  ψ 2 ) / 2 .
Symmetry operator S and the Hamiltonian H commute,
which therefore the ψ± are also the energy eigenstates, with eigenvalues E±.
pE A
2
0.6
pE A
Short times: c2 (t )  ( A 2 / 2 ) t 2 : transition probability rises quadratically with time,
~
and is independent of the field strength E , because for times t << dwell time t0,
0.8
c12
Transition probability: Starting in one of the mirror symmetric states, the other state develops as
A2
2
2
~
2
c2 (t )  2
sin 2 ( A 2  p 2 E 2 t/) , c1 (t )  1  c 2 (t ) .
2 ~2
A p E
0, 1, 2 , 4
1
0, 1, 2 , 4
th

0
0.5
1
1.5
time A
2
t
2.5
3
2(ψ1 Hψ 2  ψ 2 Hψ1 )dz  2 ψ1 Hψ 2 dz  2 A .

that is, the coupling constant A is given by the overlap intergral

A   ψ1 ( z ) H ( z )ψ 2 (z)d z .

Ammonia MASER
1. active medium: beam of NH3
2. population inversion: E+ state selection in a quadrupole filter
3. resonator: microwave Ramsey double-cavity at 24 GHz
Dubbers Quantum Physics in a nutshell WS 2008/09 4.1
2.3 The Stern-Gerlach effect
B
μ
a) The experiment and its interpretation
Expectation: The potential energy of a magnetic moment μ
under angle θ to a magnetic field B along axis z is EM   μ  B   μBz cos θ .
θ
s
Therefore, the force on a magnetic moment in an inhomogeneous magnetic field Bz(z)
E
B
is Fz   M   μz z cos θ .
z
z
Classically all angles θ are permitted, so one expects a broad distribution of deflections.
Observation: splitting into two beams
B=0
B≠0
Original data from Stern and Gerlach
Interpretation: only two angular spin orientations are allowed:
z
spin up:
sz = +½ħ
uncertain phase
s
s  s( s  1) 
1
2
3
spin down:
sz = −½ħ
Dubbers Quantum Physics in a nutshell WS 2008/09 4.2
b) Angular momentum and uncertainty relation
The uncertainty relation between momentum p and position r of a particle makes it impossible
to measure simultaneously component px and position x, or py and y, …
This uncertainty in p and x is carried over to the angular momentum, l  p  r ,
 lx 
 y  z  z  y 

 


with p  i , or:  l y   i z  x  x  z  , and (without proof) makes it impossible
l 
 x  y  y  z 
 z


to measure simultaneously more than one angular momentum component lx, ly, lz.
Only the length of vector l and one component lz along an arbitrary quantization axis z
can be measured simultaneously with arbitrarily high accuracy,
and these are quantized to l  l (l  1) , l  0,1,2,... and l z  ml  , ml  l ,...  l .
m = 3:
In the Bohr model, this quantization may pictorially be linked to the requirement
that the rotating particle must form a standing matter wave:
Moreover, as the Stern-Gerlach effect shows (and as the Dirac equation describes),
particles may carry spin angular momentum s, quantized to s  s(s  1) , s  0, 12 ,1, 3 2 ,2,...
and s z  ms  , ms  s,...  s . For the electron s = ½ and ms = ± ½.
The momentum p - position r uncertainty relation leads to
the strange phenomena seen for instance in the double slit experiment.
[Electron makes self-interference, therefore it must go through both slits simultaneously.
When the electron is localized such that one knows which slit it passes,
then the electron is so much disturbed that interference is lost.
In this way knowledge what is going on under the cover of uncertainty is principally prevented.]
These strange phenomena reappear for angular momentum (both orbital l  p  r and spin s)
in the form of new puzzling behaviour that can be seen for instance in the Stern Gerlach effect
[more generally: in 'entangled' systems, to be discussed later].
Dubbers Quantum Physics in a nutshell WS 2008/09 4.3
c) SG effect with unpolarized beams
1. SG apparatus along axis z:
Observation: Two outgoing beams, each 100% polarized along z.
Each atom counted with spin up in detector ↑, or with spin down in detector ↓
Interpretation: directional quantization of spin s:
z
spin up:
ms = +½
s
spin down:
ms = −½
2. SG apparatus rotated through angle θ:
Observation: both outgoing beams and their polarizations are also rotated by θ,
each beam 100% polarized along the new axis z'. As, for an unpolarized beam,
no spatial direction is preferred, this is what we really expect to happen.
However, the
Interpretation is puzzling:
'spin up' and 'spin down' is not a property of the incoming atoms, because the
atoms cannot know in advance what measurement is going to be done downstream.
The property 'spin up' or 'spin down' imposed onto the beam
by the measurement in the SG apparatus.
Dubbers Quantum Physics in a nutshell WS 2008/09 4.4
d) SG effect with polarized beams
Incoming beam is 100% vertically polarized along z,
using for instance the outgoing beam of configuration 1:
3. SG apparatus along z.
Observation: all outgoing atoms counted in detector ↑, as expected.
4. SG apparatus rotated through angle θ.
Observation: each outgoing atom is counted in detector ↑, with probability cos 2
or is counted in detector ↓, with probability sin 2 12 θ (cf. section 3.2a) below).
Examples:
  00 :
  45 0 :
  90 0 :
  180 0 :
e)
cos 2 (0 0 )  1
cos 2 (22.5 0 )  0.85
cos 2 (45 0 )  ½
cos 2 (90 0 )  0
1
2
θ,
sin 2 (0 0 )  0
( case 3.)
2
0
sin (22.5 )  0.15
sin 2 (45 0 )  ½
( case 5.)
2
0
sin (90 )  1.
Dubbers Quantum Physics in a nutshell WS 2008/09 4.5
e) Two successive SG operations
Incoming beam is 100% vertically polarized along z,
5. SG apparatus rotated through angle 90o.
Observation: the outgoing atoms are counted in detector ↑ or in detector ↓,
each with probability ½.
One outgoing beam goes into second SG apparatus:
6. Incoming beam is horizontally polarized along y,
SG apparatus rotated through angle 180o.
Observation: one of the outgoing beams now is polarized 'spin down',
although the original incoming beam of 5 was 100% 'spin up'.
Interpretation: There is an apparent violation of angular momentum conservation,
which in some way is covered by the uncertainty relation.
3.
Dubbers Quantum Physics in a nutshell WS 2008/09 4.6
3. The description of effective spin ½-systems
3.1 Gyromagnetic effects
a) The classical gyromagnetic effect
A rotating charge possesses angular momentum L and magnetic moment μ.
When a magnetic field B is applied, both L and μ will precess about B.
In detail: A rotating particle on a circular orbit, with
constant velocity υ, mass m, charge q = −e, area A  πr 2 , has:
angular momentum L  mυ  r ,
dq
eυ
I

current
,
dt
2πr
eυ
e
e
μ  IA  
πr 2  
mυ r  
L,
magnetic moment
2πr
2m
2m
e
μ  γ L , with gyromagnetic ratio γ 
directed along L:
.
2m
A
μ
L
ω0
When a spinning top experiences a torque M  r  F it undergoes precession:
With M  dL / dt and dφ  dL/L  Mdt / L ,
the precession frequency is ω0  dφ/dt  M/L .
The equation of motion of the spinning top then becomes dL/dt  M  0 L ,
dL
 ω0  L .
in vectorial form:
dt
In the gyromagnetic case with μ  γ L in field B, the torque is
M  μ  B   γL  B .
With M  γLB , we find that the precession frequency ω0  M/L becomes
dL
 ω0  L .
linear in field B:
ω0  γB , and again
dt
B
ω0
M
μ
L
Later on we shall rederive this expressions for quantum mechanics: they remain valid for the expectation value  L .
Dubbers Quantum Physics in a nutshell WS 2008/09 4.7
b) The quantum Zeeman effect
Particle with spin s, magnetic moment μ   γ s , in a magnetic field B directed along z.
Like in the classical case, the gyromagnetic ratio is γ  g e/ 2m , but with a 'fudge factor' g.
In the relativistic Dirac equation, electron spin arises naturally and has g = 2.
Non-relativistic ansatz:
Hamiltonian is H tot  H 0  H , with magnetic interaction H   μ z Bz  γ s z Bz .
In the 'matrix representation of the operator H in 2s+1-dimensional spin space'
H is diagonal with respect to the spin eigenfunctions ψm,
with spin quantum number s and magnetic quantum numbers m  s,...,s :
H ψ m  γ s z Bz ψ m  γ m Bz ψ m  Em ψ m .
B≠0
B=0
Hence the ψm are also energy eigenstates with
energy eigenvalues Em  m γB z  m ω0 , m  s,...,s ,
with Larmor frequency ω0  γBz .
E
E+1/2
m = +½
s = ½:
Zeeman effect: The energy E0 is split into 2s+1 levels, splitting is linear in Bz.
E0
For the s = ½ two-state system in 2-dim spin space,
1
 0
with basis vectors ψm, m = ±½, i.e. ψ  12    and ψ  12    ,
 0
1
the Hamiltonian is,
1 0 
1 0 
 , with matrix σ z  2s z  

H  ω0 s z  12 ω0 
0

1
0

1




ħω0
E−1/2
m = −½
Bz
Alternative notations for the matrix elements: H mm'  m ω0  mm' or  m | H | m'   m ω0  mm' .
Dubbers Quantum Physics in a nutshell WS 2008/09 4.8
3.2 Spin Rotation
a) Rotation matrices
What happens if the B-field is not along z but a different axis?
We simply have to rotate the coordinate system to the new axis,
with the help of a rotation operator R, with RR† = 1, which changes the wave function from ψ to R†ψ,
and the Hamiltonian from H to R † Hψ  R † H ( RR † )ψ  ( R † HR ) R †ψ , with the new Hamiltonian H ' R † HR . z
We shall assume that in the Hamiltonian Htot only the magnetic interaction H is affected by the rotation.
Rz:
When the coordinate system is rotated about the z-axis through an azimuth angle φ,
then the wave function changes merely by a phase factor e imφ ,
 e iφ/ 2
c 
that is the spin-½ wave function ψ   1  changes to Rz† ψ, with rotation matrix Rz (φ)  
 c2 
 0
where the successive lines and rows correspond to m = +½ and m = −½.
0 
,
e iφ/ 2  x'
y
x
z
y'
Ry:
θ
θ
- e iφ/ 2 sin θ/ 2 e iχ/ 2 
,
e iφ/ 2 cos θ/ 2 e iχ/ 2 
 12 (1  cos θ )

For spin 1, the rotation matrix about the y-axis through polar angle θ is R y (θ )   12 2 sin θ
 1
 2 (1  cos θ )
φ
z'
When the coordinate system is rotated about the y-axis through polar angle θ,
 cos θ/ 2  sin θ/ 2 
 .
then the wave function ψ changes to Ry†ψ, with rotation matrix R y (θ )  
 sin θ/ 2 cos θ/ 2 
 e iφ/ 2 cos θ/ 2 e iχ/ 2
In the general case, the rotation matrix is R   iφ/ 2
iχ/ 2
 e sin θ/ 2 e
with Euler angles θ, φ, χ.
φ
y
x
x'
 12
1
2
2 sin θ
cos θ
2 sin θ
(1  cos θ ) 

 12 2 sin θ  .

1 (1  cos θ ) 
2

1
2
Dubbers Quantum Physics in a nutshell WS 2008/09 4.9
z
b) Pauli matrices
Let the B-field be along the x-axis. This configuration can be reached from the case with B along z
 cos π/ 4 sin π/ 4 
1  1 1
 

 .
by rotating the system through θ = π/2 about y, by applying R y  
2   1 1
  sin π/ 4 cos π/ 4 
Then the rotated Hamiltonian has, instead of σz, the matrix
0 1
1 1  1 1 0  1 1  0 1 


  
 , that is H '  ω0 s x  12 ω0 
 .
σ x  R †y σ z R y  
1
0
2 1 1  0  1  1 1  1 0 


π/2
y
x
z
Let the B-field be along the y-axis. This configuration can be reached from the case with B along x
 eiπ/ 4
0 
 to σx,
by rotating the system about z through φ = π/2, by applying Rz  
 iπ/ 4 
0
e


 iπ/ 4
iπ/ 4
 iπ/ 2
 0  i
0  0 1  e
0   0 e
1 e

   iπ/ 2



which gives σ y  Rz† σ x Rz  
iπ/ 4 
iπ/ 4 

2 0
e  1 0  0 e
0   i 0 
 e
0  i
 .
that is H "  ω0 s y  12 ω0 
i 0 
π/2
y
x
x
0 1
0  i
1 0 
 , σ y  
 , and σ z  
 are the 3 Pauli matrices.
The matrices σ x  
1 0
i 0 
 0  1
Together with the 2-dim unit matrix I they form a complete set in the sense that
any Hermitean 2 × 2 matrix can be expressed as a linear combination of these 4 matrices:
c  id  1
 a

  2 (a  b) I  12 (a  b)σ z  cσ x  dσ y .
b 
 c  id
The first term only shifts the overall energy, the second term can be expressed as a scalar product α·σ
between a vector α  (c, d , (a  b) / 2) and a vector of matrices σ  (σ x , σ y , σ z ) ,
which gives the operator s = ½ħσ in spin representation.
Hence, any 2-state quantum system behaves like quantum-angular momentum and has all its peculiar properties.
Extension to N identical states: any N-state quantum system corresponds to a spin system with spin quantum number s = ½(N − 1).
Dubbers Quantum Physics in a nutshell WS 2008/094.10
Bz
B
c) Spin rotation in a transverse magnetic field
For magnetic field B  ( Bx ,0, Bz ) , and 0  Bz , 1  B x ,
the Hamiltonian is
H  12 (ω0 σ z  ω1σ x ) ,
H
or
1
2
ω
 0
 ω1
ω0
ω1 
.
 ω0 
ω
ω1
ZEEMAN EFFECT with transverse Bx field
4
Bx
From the characteristic equation ( 12 ω0  E )(  12 ω0  E )  14  2 ω02  0
ω02
E2

1,
 2 ω12 ω12
2
Bx
0
E
and the energy eigenvalues E   ω02  ω12 ,
The curves E  ( B0 ) or E ( B1 ) are the same as for the eigenvalues E (k1  k 2 )
of the classical coupled asymmetric oscillator,
with the asymmetry k1  k 2 of spring constants, cf. summary p. 2.1,
and of the NH3 molecule as a function of the external electric field, p. 3.1.
2
4
4
2
that is the evolution of spin polarization along axis z:
2
2
2
Pz  c1 (t )  c 2 (t )  2 c1 (t )  1 .
Pz
2
2
4
SPIN ROTATION IN VARIOUS B FIELDS
1
0.75
0.5
0.25
0
0.25
0.5
0.75
0,1,2,4
no spin-flip c1 (t )  1  c 2 (t ) ,
0
Bz Bx
0
2
4
time
6
1
The same equations have the same solutions, and all the other features
found earlier carry over, like the temporal development of probabilities
ω2
2
for spin-flip c2 (t )  2 1 2 sin 2 ( ω02  ω12 t ) , and
ω0  ω1
2
0
we find the hyperbolic equation
8
1t
Dubbers Quantum Physics in a nutshell WS 2008/094.11
3.3 The density matrix
a) Pure states and mixed states
In quantum mechanics, event probabilities are determined, for instance by counting particles.
Examples:
NH3 molecules in upper energy state, focussed by the the quadrupole separator;
Ag atoms in the spin-up state, deflected upwards in the Stern-Gerlach apparatus;
monoenergetic electrons elastically scattered on protons into a certain direction.
In these examples, the statistical ensembles of the outgoing particles are in a
pure state:
NH3 all in energy state | E   ;
Ag atoms all in spin state | ;
electrons all in the same momentum state | p .
A pure state can also be a superposition of these states,
the Ag atoms, for instance, can all be in the pure state |  (|  |) / 2 .
Often the incoming particles are in a
mixed state:
NH3 in thermal equilibrium, i.e. Boltzmann distributed over | E   and | E   ;
Ag atoms unpolarized or only partially polarized;
electrons are emitted from a cathode with Maxwell velocity distribution;
the electrons can be transformed into an almost pure | p state by acceleration in an electric field.
Dubbers Quantum Physics in a nutshell WS 2008/094.12
b) The density matrix in diagonal form
E
In the beam, the particles come in completely independent from each other.
When they are in mixed state, then one can only give a probability pn
that they are, for instance, in the eigenstate ψ n | n of energy En.
 pE ,
 p
when the population is normalized to  p  1 .
The mean energy of such a mixed ensemble then is  E  
or  E  n p n En
En
n
n
n
n
n
n
pn
n
Define the density matrix in energy representation as
 p1 0 ... 0 
 E1



0
 0 p 2 ... 0 
,
energy
matrix
being
ρ
H

E

 ...
... ... ... ... 



0
 0 0 ... p 
n


The density operator has unit trace: Tr ρ  n pn  1 ,
0
E2
...
0
... 0 

... 0 
.
... ... 

... E n 
For the Hamilton operator energy conservation requires Tr H  n En  0 , for a interacting closed system,
(when the energy of the non-interacting system is set to E0 = 0).
 p1 E1

 0
The product of both is ρH  
...

 0

0
p2 E2
...
0
...
...
...
...
0 

0 
,
... 

p n E n 
and we find for the mean energy  E  n pn En  Tr( ρH ) .
The density matrix also covers the special case of a pure state.
In this case, for instance p1 = 1, all other pn = 0.
Dubbers Quantum Physics in a nutshell WS 2008/094.13
c) The density matrix for pure states
The result  E   Tr ρH not only holds for H and ρ in their energy represention,
but holds for any set of base states, because the trace of an operator is independent of representation.
With the same arguments, we find for the expectation value of any operator A:  A  Tr ρA .
When we choose a different basis ψ'  Uψ , with UU+ = 1,
then ρH transforms to ρ'H'  U  ρH U  U  ρ (UU  ) HU  (U  ρU )(U  HU ) , but the trace remains the same.
b
 a
2
2
 , with a  b  1 .
2-dim case: The most general unitary 2 × 2 matrix U , with UU+ = 1, is U   

 b a 
 cos φ sin φ 
 , with mixing angle φ.
For real Hamiltonian H, reduces to our earlier U  
  sin φ cos φ 
ψ 
1 0
 , for basis ψ   1  (for instance in energy representation).
The density matrix for a pure state is ρ  
 0 0
ψ2 
For the new base states ψ1'  aψ1  bψ 2 and ψ 2'  b ψ1  a ψ 2 (for instance in spin representation)
the density matrix then is
that is
b   a
 a   b  1 0  a



ρ'  U ρU   
 0 0   b  a     b 
b
a

 


2

a
a b 
2
2
a  b  1.
ρ'   
2 , with
 ab

b 

In terms of the Pauli matrices: ρ' 

1
( I  (a b  ab  )σ x  i (a b  ab  )σ y  ( a  b )σ z ) , cf. p. 4.9.
2
2
 b  a b 

,
a  0 0 
2
Dubbers Quantum Physics in a nutshell WS 2008/094.14
d) The density matrix for mixed states
In the same way, the density matrix for a mixed state can be written as
2
2
b   a p1  b p 2 a b( p1  p 2 ) 
 a   b  p1 0  a




 

,
ρ'  U ρU   
2
2

a  0 p 2   b a   ab  ( p1  p 2 ) b p1  a p 2 
b
in terms of Pauli matrices: ρ' 
1
( I  [( a b  ab  )σ x  i (a b  ab  )σ y  ( a  b )σ z ]( p1  p 2 )) .
2
2
2
The spin polarization along axis z then is σ z   Tr ( ρσ z )  ( a  b )( p1  p 2 ) .
2
2
Similarly, σ x   Tr ( ρσ x )  (a b  ab  )( p1  p 2 ) and σ x   i(a b  ab  )( p1  p 2 )
Then, both for pure and for mixed states, the density matrix can be expressed
in terms of the vector σ of Pauli matrices ρ  12 ( I  σ   σ ) ,
with the vector polarization σ  , for spin operator s = ½ħσ.
When the density matrix is given in spin representation, then σ z   p1  p2 , and σ z   1 for a pure state.
Dubbers Quantum Physics in a nutshell WS 2008/094.15