N
rd
3 lecture:
Summary 2nd lecture: The ammonia molecule NH3.
c 
State vector ψ   1  with respect to the basis | 1 , | 2 , with:
 c2 
 A  c1 
 c   E
  .
iψ  Hψ , with energy matrix H, or: i 1    0


A
E
c
0  c 2 
 2 
V(z)
V(z)
0.38 Å
Energy eigenvalues E  E0  A .
z
E+
E−
0.2 eV
| 1
In an energy eigenstate of the molecule, the amplitude to find the nitrogen atom in state | 1 is
| 2
c1  exp( iE 0 t / ) cos( At / ) , that is the probability oscillates back and forth like:
c1
2
 cos ( At /  ) , c 2
2
2
 1  c1
2
| 2
1.75
 sin ( At / ) , with frequency ω  2A /  .
E
1.25

2
2 ~2
~  , with energy eigenvalues E   E 0  A  p E .
E 0  pE 
1

1 
, sin 2 φ 


1 
2



1

1 




2 


| 1
~
p2E
~
2 p2E

~
p eff

p   p for small field E .
, or p eff  
~
E
1  2
A2  p 2 E 2
That is, in zero field ξ = 0, the NH3 molecule has no EDM, i.e. it only has an induced EDM.
| E  
(| 1  | 2 ) / 2
0.25
| 2
4
2
0
2
4
ξ = pE A
for level E+:
EFFECTIVE ELECTRIC DIPOLE MOMENT
1
~
, ξ  pE/A .
~
~
The effective EDM is defined as the slope of the function E (E ) : peff  E/E .
~
A2  p 2 E 2  
2A
0.5
0.5
peff
mixing coefficients cosφ, sinφ: cos 2 φ 




2 


1
0.75
A


1 
2



| 1
| E  
(| 1  | 2 ) / 2
1.5
 cos φ sin φ 
 containing the eigenvectors of H,
Diagonalization: U†HU = E, with U  

sin
φ
cos
φ



 ~
E
ENERGIES OF NH3 MOLECULE IN E FIELD
2
2
~ ~
With an electric dipole moment p  p z in an external electric field E  E z ,
~
the energies of states | 1 , | 2 changes from E0 to E  E0  pE ,
~
 E 0  pE

H

Energy matrix is
 A

z
E0
2A
←peff = 0
0
0.5
1
4
2
0
ξ = pE A
2
4
3.1
CLASSICAL COUPLED OSCILLATOR
1
x12
the other state was found to develop like (p. 1.3):
0.5
0.5
0
0
x 2 (t )  sin 2 ω t sin 2 ω t , and x1 (t )  1  x 2 (t ) , fast ω oscill. + slow Δω-beats.
2
The energy flows back and forth between 1 and 2 with the slow beat frequency Δω.
c2 (t )  exp( iE 0 t/) sin( At/) , with E0 
1
2
( E  E ), A 
1
2
30 40
time t
50
60
0.2 E0
c12
 c (0)   1 
ψ (t  0)   1     , the other state is found to develop like (p. 2.5):
 c2 (0)   0 
20
NH3 MOLECULE
1
0.5
10
c22
For the qu.-mech. NH3 molecule, starting in one state
10
( E  E ) ,
0.5
A
2
k'
10
x2 (t )   sin ω t sin ω t , with ω  ½(ω  ω ) , ω  ½(ω  ω ) ;
2
0.2 k
 x (0)   1 
For a classical coupled oscillator starting in one state x (t  0)   1     ,
 x2 (0)   0 
x22
i) The time development of an NH3 molecular states
0
0
10 20 30 40 50 60
time t
the fast E0/ħ oscillation is complex, that is it disappears in the transition probability:
2
CLASS. ASYM. COUPLED OSCILLATOR
1
2
0.2 k
2
x12

2 2
c 2 (t )  c c  sin ( At/) , and c1 (t )  cos ( At/) , and only slow A/ħ beats remain.
2
0.5
2
A2
2
2
~
sin 2 ( A 2  p 2 E 2 t/) , and c1 (t )  1  c 2 (t ) .
2
2 ~2
A p E
k'
k
20
30 40
time t
50
60
NH3 MOLECULE IN E FIELD
0.2 E0
1
10
0.5
A
10
0.5
pE
c 2 (t ) 
0
c12
~
Similarly, for the the qu.-mech. NH3 molecule, with coupling strength A, in electric field E :
~
c2 (t )  cos φ sin φ (exp( iE  t/)  exp( iE  t/)) , with 12 ( E   E  )  A 2  p 2 E 2
A
~
c2 (t ) 
exp( iE 0 t/) sin( A 2  p 2 E 2 t/) ,
~
A2  p 2 E 2
0.5
0
c22
For the classical asymmetric coupled oscillator, with coupling strength k', asymmetry Δk:
k'
x2 (t )  cos φ sin φ (cos ω t  cos ω t ) 
cos ω t cos ω t (p. 1.8).
k ' 2 k 2
x22
10
0
0
10
20
30 40
time t
50
60
3.2
k) The short-time behaviour of a quantum state
Starting in one state, the other state develops with probability
A2
~
2
c 2 (t )  2
sin 2 ( A 2  p 2 E 2 t/) :
2 ~2
A p E
~
For large external electric field p 2 E 2  A 2 :
A2
~
( A 2  p 2 E 2 ) t 2 / 2  ( A 2 / 2 ) t 2 ,
2
2 ~2
A p E
the transition probability rises quadratically with time,
and becomes independent of the field strength:
c 2 (t ) 
2
0, 1, 2
0.6
0.4
pE A
~
For short times ( A 2  p 2 E 2 /  ) t  1 :
 t2
0.8
c22
~
the probability decreases strongly with 1 / E 2 ,
~
while the oscillation frequency | 1 | 2 increases  E .
NH3 IN VARIOUS E FIELDS
1
A2
~
c 2 (t )  2 ~ 2 sin 2 ( pEt/) ,
p E
2
0.2
0
2
4
~
t   / A2  p 2 E 2
6
8
10
time t
Explanation:
1


~2 .
2
ω 2 A  p2E
For short times t << τ, the energy uncertainty:
~
E  /t  /τ  2 A 2  p 2 E 2 is much larger than the energy splitting ħω of the system.
The dwell time in state | 1 is τ 
General rule:
For very short times, a quantum system does not take notice of external perturbations, the system is quasi-free.
(Even the exponential decay law, for instance for radioactivity, should become invalid at very short times.)
3.3
l) The connection with the ordinary Schrödinger equation
The probability amplitude ψ (z ) of the N-atom in the double-well potential V(z) obeys the ordinary Schrödinger equation:
 2 d 2ψ
Hψ  
 V ( z )ψ  Eψ , with reduced mass m.
2m dz 2
NH3-molecule:
For infinite height of the barrier V0   , one can distinguish states ψ1 ( z ) which are completely
within the potential well 1 and states ψ 2 ( z) which are completely within the potential well 2.
infinite V0
position eigenfunctions
ψ1
ψ
ψ2
finite V0
position eigenfunctions
energy eigenfunctions
ψ
ψE
ψ+
ψ1
ψ2
V(z)
V0
ψ(z)
E0
| 1
| 2
ψ−
The actual barrier height V0 = 0.2 eV is much larger than the coupling A  0.5  10 4 eV through the barrier,
therefore, when the wavefunction ψ is primarily located in well 1, then there is very little spill-over to well 2,
and we can still use the states ψ1 ( z ) and ψ 2 ( z) as base states (or in the notation | 1  ψ1 ( z) and | 2  ψ 2 ( z) ).
A real gas then is a mixture of equally populated states ψ1 ( z ) and ψ 2 ( z) .
3.4
The NH3 system has mirror symmetry S with respect to the x-y plane of the 3 H-atoms.
Applying the mirror operation twice must give back the original system:
S 2 ψ  ψ , that is S has the eigenvalues ±1: Sψ  ψ
The symmetry operator S and the Hamiltonian H commute:
H ( Sψ )   Hψ   Eψ and S ( Hψ )  S ( Eψ )  E  Sψ   Eψ , with eigenenergy E,
therefore they have the same set of eigenstates.
N
| 1
H
| 2
For a linear superposition of states ψ1 and ψ 2 : ψ( z)  c1ψ1 ( z)  c2ψ 2 ( z) , the eigenstates of S are:
the symmetric state
ψ   (ψ1  ψ 2 ) / 2 with Sψ   (ψ 2  ψ1 ) / 2  ψ  , and the
antisymmetric state
ψ   (ψ1  ψ 2 ) / 2 with Sψ   (ψ 2  ψ1 ) / 2  ψ  .
Therefore, the states ψ   (ψ1  ψ 2 ) / 2 are also the energy eigenstates, with definite energy eigenvalues E±.
With no tunneling, the mean energy of the system is




 E0    ψ1 ( z ) H ( z )ψ1 ( z )dz   ψ 2 ( z ) H ( z )ψ 2 ( z )dz  E0 .
With tunneling, the mean energies are

 E    ψ  Hψ  dz 
1
 E    ψ  Hψ  dz 
1



2
2






(ψ1  ψ 2 ) H (ψ1  ψ 2 )dz  E and
(ψ1  ψ 2 ) H (ψ1  ψ 2 )dz  E
The energy splitting then is, for Hermitean H,
 E   E  E  
1
2




2(ψ1 Hψ 2  ψ 2 Hψ1 )dz  2 ψ1 Hψ 2 dz  2 A .

that is, our coupling constant A is given by the overlap intergral

A   ψ1 ( z) H ( z )ψ 2 (z)d z

For our treatment so far, we did not need to calculate A and ΔE,
but only had to assume that they are non-zero.
3.5
NH3-molecule:
V(z)
m) The real NH3 molecule
υ=2
The NH3 molecule has many degrees of freedom. At room temperature, mainly
the lowest vibrational state υ = 0 with its zero-point vibrations is populated,
but many rotational states are built on this state and are equally populated.
E2
E1
υ=1
The rotational energies E rot  12 J ( J  1) 2 /θ (angular momentum quantum number J,
moment of inertia θ) are small compared to the height V0 of the barrier,
so in all rotational states the nitrogen atom obeys the same ordinary Schrödinger equation:
E0
υ=0
V0
| 1
| 2
 2 d 2 ( z )

 V ( z ) ( z )  E ( z )
2m dz 2
with essentially the same double-well potential V(z),
and therefore roughly the same tunnel splitting for all rotational states
(the different rotational levels can be resolved, however,
with precision microwave spectroscopy).
Energy levels of ammonia (NH3) in the lowest vibrational state υ = 0.
On the abscissa, K = Jz is the quantum number corresponding to the
z-component of the angular momentum.
Transitions between the two states of the nitrogen atom cause the
line splitting shown and yield emission at frequencies near 24 GHz.
NH3 is a very useful thermometer for molecular clouds.
roomtemperature→
3.6
n) Extension to multiple potential minima
Multiple potential minima occur for electron motion
within a molecule, for instance in a benzene ring C6H6
or through a crystal lattice,
or for the molecular rotation of, for instance, a methyl group CO3 in condensed matter.
For an N-fold minimum, there is an N-fold splitting of the energy levels.
For a crystal, N ~ 1023, and the energy levels form energy bands.
3.7
2.2 The ammonia maser
Maser = Microwave Amplification by Stimulated Emission of Radiation
Proposed by Einstein 1917
Realized by Basov, Prokhorov 1952, independently Townes 1953 (Nobel 1964)
(Laser = Microwave Amplification by Stimulated Emission of Radiation, realized 1960)
Maser (like Laser) needs:
1. active medium
2. population inversion
3. resonator
1. active medium: beam of NH3:
pumps
2. population inversion: E+ state selection in a quadrupole filter:
~
In the x-y plane, on a line through the z-axis of the quadrupole, the direction of the electric field E
~
changes sign when passing the axis, i.e. E is zero on the axis. Near the axis,
~
the absolute size of the field grows linearly with the distance r to the axis: E  ar .
~
~
The EDM vector p follows the local field E adiabatically. So, with E also p changes direction.
~
Therefore, the sign of the interaction energy E   p  E stays the same
~
even when the field components of E change sign.
Therefore, the force F  E / r on the NH3 molecule in the inhomogeneous
~
quadrupole field E is always inward bound, or always outward bound,
~
depending on the orientation of the N-atom, i.e. the sign of p relative to E .
~
attractive
potential E+
Field E z
r
Lines of equal field strength:
or: repulsive
potential E−
r
r
3.8
ENERGIES OF NH3 IN E FIELD
2
~
E
E E
The absolute size of the force is F 
 ~
 apeff ,
r
E r
~
with electric field gradient E/r  a .
1.75
1.5
1.25
E
~
~
~
For a given field value E , the sign of the EDM peff  E / E ( = slope of E  (E ) )
1
in the upper energy state | E   is always opposite to the sign of peff
in the lower energy state | E   .
0.75
Therefore, if one state is focussed, the other is defocused by the quadrupole filter.
0.25
0.5
4
2
0
pE A
2
4
3. resonator: microwave Ramsey double-cavity at 24 GHz:
Einstein: 3 processes
atomic e.s.
A*
A
atomic g.s.
1. photon
absorption
ption
A*
A*
2. spontaneous
emission
A
3. induced
emission
A
3.9
Applications of the maser:
Frequency standards
Radio astronomy: low-noise microwave amplifiers in radio telescopes
Astrophysics: natural masers in interstellar clouds
etc.
3.10