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Quantum Harmonic Oscillator
Quantum Harmonic Oscillator
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Quantum Harmonic Oscillator
1D S.H.O.：linear restoring force
F ( x)  k x , k is the force constant
& parabolic potential V ( x)  k x 2 / 2
.
A particle oscillating in a harmonic potential
harmonic potential’s minimum at x  0 = a point of stability in a system
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Quantum Harmonic Oscillator
Ex：the positions of atoms that form a crystal are stabilized by the
presence of a potential that has a local min at the location of each atom

V ( x)  
1 d nV ( x)
( x  xo ) n
n! dx n x xo
dV ( x)
1 d 2V ( x)
→ V ( x)  V ( xo ) 
( x  xo ) 
dx x  xo
2 dx 2
n 0
( x  xo ) 2  
x  xo
∵ the atom position is stabilized by the potential, a local min results in
the first derivative of the series expansion = 0
∴ V ( x)  V ( xo ) 
1 d 2V ( x)
2 dx 2
( x  xo ) 2  
x  xo
→ a local min in V(x) is only approximated by the quadratic function of a
H.O.
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Schrödinger Wave Eq. for 1D Harmonic Oscillator
for the H.O. potential V ( x)  m 2 x 2 / 2 , the time-indep Schrödinger
 2 d 2
1
2 2
wave eq.： 

m

x   n ( x)  E n  n ( x)
2
2
 2m d x

use(1)  
m

x & (2)  n 
2En

d 2~n ( )
2 ~






 n ( )  0
→
n
2
d
2
making the substitution ~n ( )  e  / 2 H n ( )
d 2 H n ( )
dH n ( )
→

2

  n  1 H n ( )  0 called Hermite functions.
d2
d
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Hermite Functions
One important class of orthogonal polynomials encountered in QM &
laser physics is the Hermite polynomials, which can be defined by the
formula
H n ( )  ( 1) n e
2
n  2
d e
, n  0, 1, 2,
n
d
the first few Hermite polynomials are：
H 0 ( )  1,
H1 ( )  2 ,
H 2 ( )  4 2  2,
H 3 ( )  8 3  12
in general：
H n ( ) 
2006 Quantum Mechanics
[n / 2]

n 0
( 1) k n !
n 2 k
(
2

)
.k ! (n  2k )!
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Hermite Functions
the Hermite polynomials come from the generating function：
g ( , t )  e
t 2  2 t 

tn
  H n ( ) . ,
n!
n 0
→ Taylor series：

tn n g
t  2 t 
g ( , t )  e

n
t 
2
n 0
n g
→ n
t
n
 
(  t ) 2
e
e
 tn
2
t 0
substituting g ( , t )  e
t 
,
n!  t
.
t 0
n u
n  d e
 ( 1) e
d un
2
 H n ( )
2
t 0
t 2  2 t 
u 
g
tn
 ( 2  2t ) g ：
into
  H n ( )
t
n !
n 0

→ recurrence relation：
H n1 ( )  2  H n ( )  2 n H n1 ( ) ,
2006 Quantum Mechanics
n  1, 2,
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Hermite Functions
substituting g ( , t )  e
t 2  2 t 

tn
  H n ( )
n !
n 0
into
g
 2t g ：
x

H n ( ) n
H n ( ) n1
t

2
t


n
!
n
!
n 0
n 0

→ recurrence relation：
dH n ( )
 2n H n 1 ( ) ,
d
with H n1 ( )  2  H n ( )  2 n H n1 ( )
&
n  1, 2, 
dH n ( )
 2n H n 1 ( )
d 
→ 2nd-order ordinary differential equation for H n ( )
d 2 H n ( )
d2
 2
dH n ( )
d
 2n H n ( )  0
1
eigenvalues of the 1D quantum H.O.： n  2n  1  En   n   

2006 Quantum Mechanics
2
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
the eigenfunctions of 1D H.O.： ~n ( )  Cn e  / 2 H n ( )
2

with the help of


e  H n ( ) d  2 n n !  , find normalization
2
2
constant Cn , →



e  H n ( ) d  2 n n ! 
2
2
(i) in CM, the oscillator is forbidden to go beyond the potential, beyond
the turning points where its kinetic energy turns negative.
(ii) the quantum wave functions extend beyond the potential, and thus
there is a finite probability for the oscillator to be found in a classically
forbidden region
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
n=0
n=1
n=2
n=3
n=4
n=5
 n  

2006 Quantum Mechanics

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
the classical probability of finding the particle inside a region  ：
Pcl ( ) 
t 2 ./ v( )

T
2 / 
the velocity v( )  A sin (t ) can be expressed as a function of  ：
v( )  
1
P
(

)



→ cl

2006 Quantum Mechanics
1
A
2

2

 A2   2 

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
(i) the difference between the two probabilities for n=0 is extremely
striking ∵there is no zero-point energy in CM
(ii) the quantum and classical probability distributions coincide when the
quantum number n becomes large
(iii) this is an evidence of Bohr’s correspondence principle
n=0
2006 Quantum Mechanics
n=30
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
(1) classically, the motion of the H.O. is in such a manner that the
position of the particle changes from one moment to another.
(2) however, although there is a probability distribution for any
eigenstate in QM, this distribution is indep of time → stationary states
(3) even so, the Ehrenfest theorem reveals that a coherent
superposition of a number of eigenstates, i.e., so-called “wave packet
state”, will lead to the classical behavior
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
show



2
e  H n ( ) d  2 n n !  ：
2
using the generation function , we can have
e
 2
e
t 2  2 t 
e
s2 2 s 


  e
 2
m 0 n 0
H n ( ) H m ( )
t nsm
n! m!
∵ the orthogonality property, the integration leads to

→


e
 (  s  t ) 2
e

2tsn
n 0
n!
 
2t s
d    e

t nsn
n 0
n! n!




2t s
t nsn
n 0
n! n!




e  H n ( ) d
2
2
e  H n ( ) d
2
as a consequence, we can obtain
2006 Quantum Mechanics

2



e  H n ( ) d  2 n n ! 
2
2
Prof. Y. F. Chen
Quantum Harmonic Oscillator
The Poisson Distribution
given a mean rate of occurrence r of the events in the relevant interval,
the Poisson distribution gives the probability P ( X  n) that exactly n
events will occur
for a small time interval  t the probability of receiving a call is r  t .
the probability of receiving no call during the same tiny interval  t is
given by 1  r  t . the probability of receiving exactly n calls in the total
interval t   t
is given by
Pn (t   t )  Pn (t )1  r  t   Pn1 (t ) r  t
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
The Poisson Distribution
rearranging Pn (t   t )  Pn (t )1  r  t   Pn1 (t ) r  t , dividing through by  t ,
and letting  t  0 , the differential recurrence eq. can be found and
written as
dPn (t )
dt
 r Pn1 (t )  r Pn (t )
for n  0 ： dP0 (t )  r P (t )
dt
0
which can be integrated to lead to P0 (t )  P0 (0)e  r t
with the fact that the probability P0 (0) of receiving no calls in a zero time
interval must be equal to unity：
P0 (t )  e  r t
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
The Poisson Distribution
dPn (t )
substituting P0 (t )  e  r t into
 r Pn1 (t )  r Pn (t )
dt
for n  1 ：
P1 (t )  (r t )e  r t , repeating this process, Pn (t ) can be found to be
(r t ) n  r t
Pn (t ) 
e
n !
the sum of the probabilities is unity：


 P (t )  
n 0
n
n 0
(r t ) n
n!
e
r t
e
r t

(r t ) n
n 0
n!

 e r t  e r t  1
the mean of the Poisson distribution：


(r t ) n
n 0
n!
 n   nPn (t )   n
n 0
2006 Quantum Mechanics
e
r t
e
r t

(r t ) n1 (rt )
n 1
(n  1) !

 rt
Prof. Y. F. Chen
Quantum Harmonic Oscillator
The Poisson Distribution
in other words, the Poisson distribution with a mean of is given by：
Pn ( ) 
2006 Quantum Mechanics
n
n!
e 
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Schrödinger Coherent States of the 1D H.O.
The Schrödinger coherent wave packet state can be generalized as

En
i t
( , t )   cn~n ( )e 
n 0
with c n 
( e i ) n
n!
e 
2
/2
it can be found that the norm square of the coefficient | cn | 2 is exactly
the same as the Poisson distribution with the mean of  2
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Schrödinger Coherent States of the 1D H.O.

1

substituting En   n    & ~n ( )  2n n ! 
2


( , t )   cn n ( )e

i
En
t

1/ 2
e  / 2 H n ( ) into
2
:
n 0

 ( , t )  
n 0
2
( ei ) n  2 / 2
1
e
H n ( )e / 2 e i ( n1/ 2) t
n !
2n n ! 
n

1
 1/ 4
e
 ( 2  2 ) / 2  i t / 2

1
 1/ 4
1
 1/ 4

n 0
using g ( , t )  e
 ( , t ) 
e

t 2  2 t 

tn
  H n ( )
:
n !
n 0
e  (
2
 2 ) / 2  i t / 2
e  (
2
 2 ) / 2  i t / 2
2006 Quantum Mechanics
e
e
 ei ( t  ) / 2 

 H ( )
n
n !

2
exp   e  i ( t  ) / 2   2 e i ( t  )
exp
 
2
e i 2( t  ) / 2  2 e i ( t  )


Prof. Y. F. Chen
Quantum Harmonic Oscillator
Schrödinger Coherent States of the 1D H.O.
as a result, the probability distribution of the coherent state is given by：
P( , t )    ( , t ) ( , t ) 

1

1


1

e  (
2
 2 )


exp  2 cos[2(t   )]  2 2 cos(t   )

exp  2  2 2 cos 2 ( t   )  2 2  cos( t   )

exp [  2 cos( t   )]2


it can be clearly seen that the center of the wave packet moves in the
path of the classical motion
  2 cos( t   )
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
with  
m


x , H n1 ( )  2 H n ( )  2nH n1 ( ) &  n ( )  2 n ! 
n

1/ 2
e 
2
/2
H n ( )
the operator x̂ acting on the eigenstate ~n ( )
1 / 2
2
   n
ˆx~n ( )  

e  / 2 H n ( )
  2 n !  
 m 
 
 
 m
1 / 2
2
 n
 2 n !  
e  / 2  H n ( )

 
 
 m
1 / 2
2
 n
1
 2 n !  
e  / 2  H n 1 ( )  n H n 1 ( )

2

 
 
 m
 1

 2
2006 Quantum Mechanics

n  1 ~n 1 ( )  n ~n 1 ( )

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
in a similar way, the operator p̂ x acting on the eigenstate ~n ( )
1 / 2
2

  n
 2 n !  
pˆ x ~n ( )    i
e  / 2 H n ( )
x

1 / 2

  n
 2 / 2




   i m 
2
n
!


e
H n ( )





  i m 
2
  i m 
2
  i m 

2006 Quantum Mechanics
n
1
2
n
n !  
n !  

1 / 2
1 / 2
e
e 
 2 / 2
2
/2
( ) H n ( )  e 
2
/2
H n ( )

 1 H ( )  n H ( )
n 1
 2 n 1

n  1 ~n 1 ( )  n ~n 1 ( )

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
→

1  m
1
xˆ  i
pˆ x  ~n ( ) 

2  
m 

n  1 ~n 1 ( )
&

1  m
1
xˆ  i
pˆ x  ~n ( ) 

2  
m 

n ~n 1 ( )
consequently, it is convenient to define 2 new operators：
aˆ † 
&
aˆ 

1  m
1
xˆ  i
pˆ x 

2  
m 


1  m
1
xˆ  i
pˆ x 

2  
m 

2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
the operator â † is the increasing (creation) operator：
aˆ † ~n ( ) 
n  1 ~n1 ( )
this means that operating with â † on the n-th stationary states yields a
state, which is proportional to the higher (n +1)-th state
the operator â is the lowering (annihilation) operator：
aˆ ~n ( ) 
n ~n1 ( )
this means that operating with â on the n-th stationary states yields a
state, which is proportional to the higher (n -1)-th state
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
in terms of â & â † , the operators x̂ & p̂ x can be expressed as：

xˆ 
2m
aˆ  aˆ † 
& pˆ x  i
 m
2
aˆ  aˆ † 
we can find the commutator of these 2 ladder operators：
[aˆ , aˆ † ] 

m
1  m
1
1
ˆ
ˆ
ˆ
ˆ
x

i
p
,
x

i
p
x
x

2 
m 

m 

1 i
i
  xˆ, pˆ x    pˆ x , xˆ   1

2 

which is the so-called canonical commutation relation
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
â † is the hermitian conjugate â ：
 1 | aˆ |  2   2 | aˆ † |  1

proof：
 1 | aˆ |  2 
1
1
2
m 
xˆ  i
1
pˆ x  2
m 
1
1  m 
 1 xˆ  2  i


m 
2 
1

1
1  m 
 2 xˆ  1  i


m 
2 

1 
2

2

  2 | aˆ † |  1
2006 Quantum Mechanics
m 
2

xˆ  i
1
m 

ˆpx  2 

pˆ x  1

ˆpx  1 






Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
with  
m


x , H n1 ( )  2 H n ( )  2nH n1 ( ) &  n ( )  2 n ! 
n

1/ 2
e 
2
/2
H n ( )
the operator x̂ acting on the eigenstate ~n ( )
1 / 2
2
   n
ˆx~n ( )  

e  / 2 H n ( )
  2 n !  
 m 
 
 
 m
1 / 2
2
 n
 2 n !  
e  / 2  H n ( )

 
 
 m
1 / 2
2
 n
1
 2 n !  
e  / 2  H n 1 ( )  n H n 1 ( )

2

 
 
 m
 1

 2
2006 Quantum Mechanics

n  1 ~n 1 ( )  n ~n 1 ( )

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
in a similar way, the operator p̂ x acting on the eigenstate ~n ( )
1 / 2
2

  n
 2 n !  
pˆ x ~n ( )    i
e  / 2 H n ( )
x

1 / 2

  n
 2 / 2




   i m 
2
n
!


e
H n ( )





  i m 
2
  i m 
2
  i m 

2006 Quantum Mechanics
n
1
2
n
n !  
n !  

1 / 2
1 / 2
e
e 
 2 / 2
2
/2
( ) H n ( )  e 
2
/2
H n ( )

 1 H ( )  n H ( )
n 1
 2 n 1

n  1 ~n 1 ( )  n ~n 1 ( )

Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
→

1  m
1
xˆ  i
pˆ x  ~n ( ) 

2  
m 

n  1 ~n 1 ( )
&

1  m
1
xˆ  i
pˆ x  ~n ( ) 

2  
m 

n ~n 1 ( )
consequently, it is convenient to define 2 new operators：
aˆ † 
&
aˆ 

1  m
1
xˆ  i
pˆ x 

2  
m 


1  m
1
xˆ  i
pˆ x 

2  
m 

2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
the operator â † is the increasing (creation) operator：
aˆ † ~n ( ) 
n  1 ~n1 ( )
this means that operating with â † on the n-th stationary states yields a
state, which is proportional to the higher (n +1)-th state
the operator â is the lowering (annihilation) operator：
aˆ ~n ( ) 
n ~n1 ( )
this means that operating with â on the n-th stationary states yields a
state, which is proportional to the higher (n -1)-th state
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
in terms of â & â † , the operators x̂ & p̂ x can be expressed as：

xˆ 
2m
aˆ  aˆ † 
& pˆ x  i
 m
2
aˆ  aˆ † 
we can find the commutator of these 2 ladder operators：
[aˆ , aˆ † ] 

m
1  m
1
1
ˆ
ˆ
ˆ
ˆ
x

i
p
,
x

i
p
x
x

2 
m 

m 

1 i
i
  xˆ, pˆ x    pˆ x , xˆ   1

2 

which is the so-called canonical commutation relation
2006 Quantum Mechanics
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
â † is the hermitian conjugate â ：
 1 | aˆ |  2   2 | aˆ † |  1

proof：
 1 | aˆ |  2 
1
1
2
m 
xˆ  i
1
pˆ x  2
m 
1
1  m 
 1 xˆ  2  i


m 
2 
1

1
1  m 
 2 xˆ  1  i


m 
2 

1 
2

2

  2 | aˆ † |  1
2006 Quantum Mechanics
m 
2

xˆ  i
1
m 

ˆpx  2 

pˆ x  1

ˆpx  1 






Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
with
pˆ x2
2m
1
&
→
2


4
aˆ  aˆ † aˆ  aˆ †     aˆaˆ †  aˆ † aˆ  aˆaˆ  aˆ † aˆ † 
4
m  2 xˆ 2 
Hˆ 
pˆ x2
2m

1
2

4
aˆ  aˆ † aˆ  aˆ †     aˆaˆ †  aˆ † aˆ  aˆaˆ  aˆ † aˆ † 
m 2 xˆ 2 
4

2
aˆaˆ †  aˆ † aˆ 
using the commutation relation [aˆ, aˆ † ]  aˆaˆ †  aˆ † aˆ  1
1
→ Hˆ    aˆ † aˆ  

2
define the so-called number operator： Nˆ  aˆ † aˆ
1
→ the H.O. Hamiltonian takes the form： Hˆ    Nˆ  

2006 Quantum Mechanics
2
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
the eigenstates of â can be found to be coherent states  ( ,0; ) ：
 ( ,0;  )  e

n ~
|  |2 / 2
~
 0 ( )  e
 n ( )

n!
n 0
|  |2 / 2  aˆ †
e
coherent states have the minimum uncertainty
(i )  ( , 0;  ) xˆ  ( , 0;  ) 

2m
2m
 ( , 0;  ) xˆ 2  ( , 0;  ) 

 ( , 0;  ) aˆ †  aˆ  ( , 0;  )
(    ) 
2m
2
cos 
m
 ( , 0;  )
 aˆ
†
 aˆ   ( , 0;  )
2
 2   2  (   1)    
2m
 x 2   ( , 0;  ) xˆ 2  ( , 0;  )   ( , 0;  ) xˆ  ( , 0;  )
2006 Quantum Mechanics
2

2m
Prof. Y. F. Chen
Quantum Harmonic Oscillator
Creation & Annihilation Operators
(ii )  ( , 0;  ) pˆ x  ( , 0;  )  i
m
2
 i
m
2
 ( , 0;  ) aˆ  aˆ †  ( , 0;  )
(    )  2m
sin 
2
m
 ( , 0;  )  aˆ  aˆ †   ( , 0;  )
2
m
 2   2  (   1)    
 
2
 ( , 0;  ) pˆ x2  ( , 0;  )  
 p x2  ( ,0; ) pˆ x2 ( ,0; )  ( ,0; ) pˆ x ( ,0; )
2

m 
2
as a consequence, we obtain the minimum uncertainty state：
 x   px 
2006 Quantum Mechanics

2
Prof. Y. F. Chen