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Quantum Dynamics
of a Kicked Harmonic
Oscillator
Laura Ingalls Huntley
Prof. Calvin Stubbins
Franklin & Marshall College
Department of Physics & Astronomy
12 May 2006
Presentation Outline
• Introduction to Coherent States
• The Model and Solving the Schrödinger Equation
– The Probability Distribution and Coherent States
– The Average Energy
• The General Potential
– The Heisenberg Picture Method
– The Infinite Series Method
• The Delta-Kicked Harmonic Oscillator
– The Heisenberg Picture Method
– The Infinite Series Method
• Conclusion
Introduction to Coherent States
• Discovered in 1926 by Erwin
Schrödinger as part of his
description of wave
mechanics.
• Schrödinger described nonstationary states of the
quantum harmonic oscillator
that preserve the shape of
the wave packet and have
classical motion.
• Coherent states have
applications in atomic
interactions, optics, BoseEinstein condensation, lasertrapping of ions, and
quantum electrodynamics.
Minimum-Uncertainty Coherent States
Minimum-uncertainty coherent states occur when the Heisenberg
uncertainty relation is minimized according to
xp  .
2
In traditional coherent states, the uncertainty in the x-operator and
the p-operator are equal. If this is not the case, the state is called a

squeezed coherent state.
Displacement-Operator Coherent States
The displacement-operator definition of coherent states requires that
the wave packet keep its shape and move with some classical
motion. The name refers to a method of generating the states,
whereby the ground state of the unperturbed system is acted upon
by the displacement operator. This has the effect of “displacing” the
system from its equilibrium. In the case of the quantum harmonic
oscillator, this causes the wave packet to oscillate within the potential.
Annihilation-Operator Coherent States
The annihilation operator definition of coherent states states that
they are the eigenfunctions of the annihilation operator such that
a z  z z ,
where, in general, z is complex. A result of this definition is that the
probability of the system
 being in an unperturbed state, m, is given
by
2m
2 z
2
C m  e z
.
m!
Thus, we see that there is always a probability, however small, that the
system remains in an
 unexcited state. This result is tangentially related
to the Mössbauer effect, which is the “recoilless emission” of
a photon from an atom in a solid.
The Model
Previous research has found that the case of a single kicked particle
in a potential well provides an incomplete understanding of the
kicked harmonic oscillator. Thus, a new model is proposed.
• R is the position of the center of
mass of the system.
• The position of the leftmost mass,
m1, is given by x1.
• The position of the rightmost
mass, m2, is given by x2.
• The leftmost mass is kicked by a
time-dependent force, F(t).
• The following relationships hold:
x1  R 

m1
x
x  x1  x2 .
The Potential Energy
The classical potential energy of our model is the sum of the harmonic
potential energy and the energy bestowed by the force, given by
1
V   2 (x 2  x1)2  F(t)x1 .
2
The quantum mechanical Hamiltonian is
 constructed using this expression for the
potential energy and the center of
mass and relative coordinates that we
have defined. It is given by
2
2
2 1 2 2 
H 
 FR 
  x 
Fx.
2
2
2M R
2 x
2
m1
2
Schrödinger’s Equation
The Schrödinger equation is thus written
2
 2
 2 1 2 2



 FR 
  x  
Fx  i
.
2
2
2M R
2 x
2
m1
t
2
Assume the wave function is separable in the following way
(x,R,t)  rel (x,t)CM (R,t).

The separated form of the Schrödinger equation is then obtained
2
1  2CM
i CM
1  2 rel 1 2 2 
i  rel

 FR 

  x  Fx 
.
2
2
2M CM R
CM t
2  rel x
2
m1
 rel t
2
Since the left side is only a function of R and t, and the right side is
only a function of x and t, we may define a time-dependent
separation constant, C(t).
Solving for the Center of Mass Wave Function

The equation for the center of mass wave function is given by
 2CM
CM


FR


i
 CCM .
CM
2
2M R
t
2
In order to further separate the spatial and time dependencies in this
expression (remember, the force depends on time), an extended
transformation must be applied as follows
Galilean
R R  u(t) and t  t
2
2
 

Ý
=
and


u
,
2
2
R R
t  t
R
where u(t) is a function that
describes the motion of the
center of mass of the system.
The Center of Mass Wave Function, Cont.
We also assume that the wave function has the following form
CM (R,t)  ( R,t)e Rg(t ) .
Where g(t) is an arbitrary function, such that

g(t)  i
M
uÝ,
i gÝ(t)  F(t)  0,

where the combination of these functions yield Newton’s Second Law
for the center of mass, given by

Ý.
F  MuÝ
The Center of Mass Wave Function, Cont.
The substitutions and definitions we have made give the following
expression
2
 2 1

2
Ý


M
u


i
 C.
2
2M R 2
t
We apply the separation of variables technique once more,
assuming

(R,t)  R (R)T (t).
We obtain the separated equation, given by
2
1
d
R i dT 1


 MuÝ2  C.
2
2M R dR T dt
2
2
If we set each side equal to a separation constant, E, we eventually
find that



i  1
2
Ý

M
u
C
dt
Et





2ME

 2

R  AeicR 
c 
T e
.
and
The Center of Mass Wave Function, Cont.
It is now possible to write the full expression for CM, given by
CM (R,t)  Ae


i  1
   MuÝ2 C dt Et  iR M uÝ

ic(Ru)
 2

e

e
.
This equation is enlightening in many ways. Inspection shows that
it is very 
similar to the wave function of the free particle case, where A
is a normalization constant and E is the kinetic energy of the particle.
This result is, upon reflection, unsurprising. As the center of mass
coordinate system treats our system as a single particle moving with
some translational motion and having mass equal to the sum of the
two masses, it makes sense that the center of mass wave function is
analogous to the free particle case.

Solving for the Relative Wave Function
We consider the relative portion of our separated Schrödinger
equation
2
1  2 rel 1

i  rel
2 2


x

Fx

 C.
2
2  rel x
2
m1
 rel t
The method for solving this expression is very similar to the method
used to find CM. An extended Galilean transformation such that
x'
x   (t) is needed to make it possible to separate the spatial and
time dependencies, then a series of substitutions are made until it is
possible to perform a separation of variables. The final form of the
wave function is given by
1
4
 
 reln (x,t)   
  
2
1
n
2 n!
H n [ (x   )]e

  x  2
2
e

i
 L C dt E t
n
e

ix Ý
which is very similar to the wave function of the unperturbed
harmonic oscillator.
,
The Complete Wave Function
From the product of the center of mass wave function and relative
wave function, the total wave function for our system can be written
as
1
2 4
 
n (x,R,t)  A 
  
1
n
2 n!
H n [ (x   )]e

 x  2
2
e

M
ix Ý iR uÝ
e
e
ic(Ru)
.e

i 
1 2
 Ldt 
MuÝ dt (E E n )t 


2

To summarize, the wave function of two particles connected by a
harmonic potential and subject to a time-dependent force is the
product of a center of mass wave function akin to that of a free
particle and a relative motion wave function akin to that of the
quantum harmonic oscillator. Furthermore, we have been able to
produce this equation using purely analytic methods.
.
The Gaussian Pulse
We will now specify the force as a Gaussian pulse, which will allow us
to calculate such quantities as the probability distribution and the
average energy. The form of our force is
F(t)  F0e (tt0 ) .
2
This particular form for the force
 it models a
is of interest because
real world force that has no
discontinuities and could be
applied in the laboratory.
Furthermore, it is easily
differentiated and integrated
and it effectively only acts for a
limited period of time.

Calculating u(t) and (t)
Now we may calculate the functions from our Galilean
transformations and determine how the new frames move relative to
the old frame.
The following differential
equation was solved to find u(t)
 (tt 0 )2
F(t)  F0e
The following differential
equation was solved to find (t)

Ý(t).
 MuÝ
m1

2
Ý
Ý(t)   2 (t).
F(t)  F0e (tt0 )  
The Probability Distribution
A quick look at the wave function yields the probability distribution:
1
2
2
A   2
x 
   n   H n (x )e
.
2 n!   
2
2
*
For a particle that is in the ground state prior to the kick, the
probability distribution looks like:

The wave packet retains the
Gaussian distribution
characteristic of the
unperturbed quantum harmonic
oscillator, but also oscillates
translationally with time. This is in
accordance with the
displacement-operator
definition of coherent states.
Another Check of Coherent States
Now we will confirm that we do, indeed, have coherent states, using
the following expression from the annihilation-operator definition
2m
C m  e z
2
2
z
.
m!
We calculate the probability that our system is in a stationary state
after the pulse using
 C 2   (x,t)  (x,t) 2 .
m
m
rel
This expression yields

 2 2 
1 
  2 2  2 2 
 
2 2 
m
2 2 



1 1 2 2 
2
Cm   e
    2 2  .
m! 2 
 
Thus, our relative wave function meets the annihilation-operator
definition of coherent states.
The Average Energy
According to Ehrenfest’s theorem, the expectation value of the
energy should show the same behavior as its classical equivalent. We
calculate the average energy as
E  H  
   HdxdR




*

E tot 

2
E

1
1
1 Ý 2
  2 (t)2  2ME uÝ(t)  MuÝ(t)2  
(t) .
m1 2
2
2
The General Potential and the Delta Kick
We have shown that any time-limited force acting on our model
produces coherent states. Now, we would like to generalize the
potential in which our masses interact. In order to make our
calculations more simple, we will assume a Dirac delta function form
for the force, such that
F(t)  F0(t).
Although this form is not exceedingly realistic, it allows us to expand
our analysis of the system considerably. We will now develop two
new methods for finding the resultant wave function of our two
particle model will a general potential and a delta-function force.
The first step in developing these techniques is to return to the relative
motion Schrödinger equation
p 2


 rel
.
  V (x)  F0 (t)x  C rel  i
m1
t
2

The General Potential, cont.
Let us write the Hamiltonian altogether such that the Schrödinger
equation becomes
i
rel
 H(t)rel (x,t).
t
Now we will consider a very small interval of time   t   such that
we may write the wave function and its first derivative at t=0 as

1
rel (x,0)  rel (x,)  rel (x,)
2
rel (x,0) 1
 rel (x,)  rel (x,).

t
2
These substitutions allow us to find the wave function right after the
pulse in terms of the wave function right before the pulse.

rel (x,0 )  e
i
F0 x
m1
(x,0 ).
The General Potential, cont.
To obtain the resultant wave function for any time, we let the
following operator act on our expression
i
 H0t
rel (x,t)  e
e
i
F0 x
m1
(x,0 ),
where H0 is the time-independent part of our Hamiltonian. Next, we
multiply the right side of our expression by a quantity equal to one,
such that 
i
i
F0 x  i H 0 t  i H 0 t 
 H 0t

 rel (x,t)  e
e m1 e
e
 (x,0 ).


If we make the following substitutions, an identity of exponentials
allows us to rewrite our equation as
it
A  H 0  H 0  a
Bx

iF  i H 0 t i H 0 t   i H 0 t
rel (x,t)  exp 0 e
xe
 (x,0 )
e
 m1

iF0 a  a   i H 0 t
 exp
e xe e
 (x,0 ).
 m1

The Heisenberg Picture Method
As opposed to the Schrödinger picture, in the Heisenberg picture, the
operators carry the time dependencies. For our system, we have the
following Heisenberg operators
xH  e
i
 H 0t
i
xe
H 0t
i
i
i
i
H 0 t  1   H 0 t
H 0 t 
dx H i   H 0 t
pH
 e
[H 0, x]e
pe
 e
 
dt


  

 i
i
i
i
H 0 t    H 0 t V
H 0 t  V
dp H i   H 0 t
 e
[H 0, p]e
e
 e
 H .
dt
x

 
 x H
Therefore, the wave function may be written
as
iF0
x H  i H0t
rel (x,t)  e m1 e
(x,0 ),
where xH may be solved for explicitly in terms
of t, x, and p, given the following equation

d2xH
V H
F 

.
2
dt
x H
The Infinite Series Method
Another method for evaluating our wave function expression is to
take advantage of the identity
xH  e
 a

xe  
a
j
j 0
j!
C j,
where the Cj’s are commutators given by

C0  x
C j 1  [a,C j ].
Calculating a few of these value suggests that the general form for
Cj is given by
  j1 (x)p   j 2 (x) p2  ...  j( j 2) (x) p j 2 ,
C j   j 0 (x)
which allows us to rewrite our equation for the resultant wave function
as
i



 
h
rel (x,t)  exp Ch p e
h 0

H0t
 (x,0 ).
The Delta-Kicked Harmonic Oscillator
We will now test our two new methods on our first model of the two
masses interacting in a harmonic potential. By our first method


Ý
Ý( t)   2 (t)
F(t)

F0 (t)  
Ý(t)
F(t)  F0 (t)  muÝ
u(t) 
m1
F0 t
 (t).
M
m1
 (t) 

F0
sin( t) (t).
m1

Therefore, the relative wave function is given by
1
4
( x )
i
 2  1

  C dt E n t 
2
 reln (x,t)   
H
[

(x


)]e
e
n
n
   2 n!
 iF

it
0
exp
(F0 sin( t)  2m1x).
2  (t)e
 2m1 

2
The Heisenberg Picture Method
This method does not yield any information about a general potential,
but is useful when V(x) is specified in a form that may be integrated.
We will test this method, therefore, on the harmonic potential case,
where our wave function becomes
rel (x,t)  e
iF0
xH
m1
 n (x,t).
The Heisenberg x-operator can be found from Newton’s Second Law,
such that
p
x H  x cos(t) 
sin( t).


The expression for the resultant wave function is then
 rel (x,t)  e d x d p n (x,t)  e De d xe d p n (x,t)  e D e d x n (x  d1,t),

0
1
0
1
0
which meets the displacement-operator definition of coherent states.
The wave packet will show harmonic oscillator while keeping its
 shape.
The Infinite Series Method
The infinite series method applied to the harmonic potential yields the
Cj values such that
C0  x
i
C1 
p
C2k  (  ) 2k C0
C2k 1  (  ) 2k 1C1 .

These substitutions yield


2k 1
(t) 2k iC1
k (t)
x H  e xe  C0  (1)

(1)


(2k)!
 k 0
(2k  1)!
k 0

iC
 C0 cos(t)  1 sin( t)
 a
a
k
 x cos(t) 
p


sin( t),
which, as was shown previously, yields a coherent states form for the
relative wave function, given by

 rel (x,t)  e D e d x n (x  d1,t).
0
In Conclusion …
• The model of two masses interacting in a harmonic potential and
subject to any time-limited force produces coherent states. Except
in the case where the force goes to zero when the oscillator is in its
equilibrium position with zero velocity.
• The wave function of this model is the product of a center of mass
wave function and a relative motion wave function.
• We developed two methods of
analyzing the similar model of a
delta-kicked oscillator with a
general potential. We found that
this model will not produce the
displacement-operator definition of
coherent states.
• However, our new methods agree
that the model with the harmonic
potential does indeed yield
coherent states.