quantum harmonic oscillator
DESCRIPTION
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 , k is the force constant & parabolic potential - PowerPoint PPT PresentationTRANSCRIPT
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Quantum Harmonic Oscillator
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Harmonic Oscillator
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1D S.H.O. : linear restoring force , k is the force constant
& parabolic potential
.
harmonic potential’s minimum at = a point of stability in a system
2006 Quantum Mechanics Prof. Y. F. Chen
Quantum Harmonic Oscillator
Quantum Harmonic Oscillator
xkxF )(
2/)( 2xkxV
A particle oscillating in a harmonic potential
0x
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Ex : the positions of atoms that form a crystal are stabilized by the pre
sence of a potential that has a local min at the location of each atom
→
∵ the atom position is stabilized by the potential, a local min results in th
e first derivative of the series expansion = 0
∴
→ 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
Quantum Harmonic Oscillator
0
)()(
!
1)(
n
no
xxn
n
xxdx
xVd
nxV
o
2
2
2
)()(
2
1)(
)()()( o
xx
oxx
o xxdx
xVdxx
dx
xdVxVxV
oo
2
2
2
)()(
2
1)()( o
xx
o xxdx
xVdxVxV
o
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for the H.O. potential , the time-indep Schrödinger wa
ve eq. :
use(1) & (2)
→
making the substitution
→ called Hermite functions.
2006 Quantum Mechanics Prof. Y. F. Chen
Schrödinger Wave Eq. for 1D Harmonic Oscillator
Quantum Harmonic Oscillator
2/)( 22 xmxV
)()(2
1
222
2
22
xExxmxd
d
mnnn
xm
n
n
E2
0)(~)(~2
2
2
nn
n
d
d
)()(~ 2/2
nn He
0)(1)(
2)(
2
2
nnnn Hd
dH
d
Hd
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One important class of orthogonal polynomials encountered in QM & las
er physics is the Hermite polynomials, which can be defined by the form
ula
the first few Hermite polynomials are :
in general :
.
2006 Quantum Mechanics Prof. Y. F. Chen
Hermite Functions
Quantum Harmonic Oscillator
,2,1,0,)1()(2
2
nd
edeH
n
nn
n
128)(,24)(,2)(,1)( 33
2210 HHHH
knn
n
k
n knk
nH 2
]2/[
0
)2()!2(!
!)1()(
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the Hermite polynomials come from the generating function :
.
→ Taylor series :
.
→
substituting into :
→ recurrence relation :
2006 Quantum Mechanics Prof. Y. F. Chen
Hermite Functions
Quantum Harmonic Oscillator
tn
tHetg
n
nn
tt ,!
)(),(0
22
tt
g
n
tetg
n tn
nntt ,
!),(
0 0
22
)()1(2
222
0
)(
0
n
u
n
unn
t
tn
n
tn
n
Hud
edee
te
t
g
2 2
0
( , ) ( )!
nt t
nn
tg t e H
n
gtt
g)22(
,2,1,)(2)(2)( 11 nHnHH nnn
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substituting into :
→ recurrence relation :
with &
→ 2nd-order ordinary differential equation for
eigenvalues of the 1D quantum H.O. :
2006 Quantum Mechanics Prof. Y. F. Chen
Hermite Functions
Quantum Harmonic Oscillator
2 2
0
( , ) ( )!
nt t
nn
tg t e H
n
gtx
g2
1
00 !
)(2
!
)(
n
n
nn
n
n tn
Ht
n
H
,2,1,)(2)(
1 nHnd
dHn
n
1 1( ) 2 ( ) 2 ( )n n nH H n H 1
( )2 ( )n
n
dHn H
d
)(nH
0)(2)(
2)(
2
2
n
nn Hnd
dH
d
Hd
2
112 nEn nn
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the eigenfunctions of 1D H.O. :
with the help of , find normalization consta
nt , →
(i) in CM, the oscillator is forbidden to go beyond the potential, beyond t
he turning points where its kinetic energy turns negative.
(ii) the quantum wave functions extend beyond the potential, and thus th
ere is a finite probability for the oscillator to be found in a classically forb
idden region
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
)()(~ 2/2
nnn HeC
!2)( 22
ndHe nn
nC
!2)( 22
ndHe nn
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n=0 n=1
n=2 n=3
n=4 n=5
n
n=0 n=1
n=2 n=3
n=4 n=5
n=0 n=1
n=2 n=3
n=4 n=5
n=0 n=1
n=2 n=3
n=4 n=5
n
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
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the classical probability of finding the particle inside a region :
.
the velocity can be expressed as a function of :
→
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
2 / ( )( )
2 /cl
t vP
T
( ) sin ( )v A t
22)( Av
2 2
1 1( )clP
A
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(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
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
n=0 n=30n=30
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(1) classically, the motion of the H.O. is in such a manner that the positi
on of the particle changes from one moment to another.
(2) however, although there is a probability distribution for any eigenstat
e in QM, this distribution is indep of time → stationary states
(3) even so, the Ehrenfest theorem reveals that a coherent superpositio
n of a number of eigenstates, i.e., so-called “wave packet state”, will lea
d to the classical behavior
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
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show :
using the generation function , we can have
∵ the orthogonality property, the integration leads to
→
as a consequence, we can obtain
2006 Quantum Mechanics Prof. Y. F. Chen
Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
!2)( 22
ndHe nn
0 0
22
!!)()(
2222
m
mn
nmn
sstt
mn
stHHeeee
0
222)( )(!!
22
nn
nnststts dHe
nn
stedee
0
2
0
)(!!!
2 2
nn
nn
n
n
dHenn
st
n
ts
!2)( 22
ndHe nn
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given a mean rate of occurrence r of the events in the relevant interval,
the Poisson distribution gives the probability that exactly n
events will occur
for a small time interval the probability of receiving a call is .
the probability of receiving no call during the same tiny interval is
given by . the probability of receiving exactly n calls in the total
interval is given by
2006 Quantum Mechanics Prof. Y. F. Chen
The Poisson Distribution
Quantum Harmonic Oscillator
)( nXP
t tr
t
tr1
tt
trtPtrtPttP nnn )(1)()( 1
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rearranging , dividing through by ,
and letting , the differential recurrence eq. can be found and writt
en as
for :
which can be integrated to lead to
with the fact that the probability of receiving no calls in a zero time
interval must be equal to unity :
2006 Quantum Mechanics Prof. Y. F. Chen
The Poisson Distribution
Quantum Harmonic Oscillator
trtPtrtPttP nnn )(1)()( 1 t
0 t
)()()(
1 tPrtPrtd
tdPnn
n
0n )()(
00 tPrtd
tdP
trePtP )0()( 00
)0(0P
tretP )(0
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substituting into for :
, repeating this process, can be found to be
the sum of the probabilities is unity :
the mean of the Poisson distribution :
2006 Quantum Mechanics Prof. Y. F. Chen
The Poisson Distribution
Quantum Harmonic Oscillator
tretP )(0 )()()(
1 tPrtPrtd
tdPnn
n 1n
tretrtP )()(1 )(tPn
( )( )
!
nr t
n
r tP t e
n
1!
)(
!
)()(
000
trtr
n
ntr
n
trn
nn ee
n
tree
n
trtP
rtn
rttree
n
trntnPn
n
ntr
n
trn
nn
1
1
00 !)1(
)()(
!
)()(
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in other words, the Poisson distribution with a mean of is given by :
2006 Quantum Mechanics Prof. Y. F. Chen
The Poisson Distribution
Quantum Harmonic Oscillator
en
Pn
n!
)(
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The Schrödinger coherent wave packet state can be generalized as
with
it can be found that the norm square of the coefficient is exactly
the same as the Poisson distribution with the mean of
2006 Quantum Mechanics Prof. Y. F. Chen
Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator
0
)(~),(n
tEi
nn
n
ect
2/2
!
)( en
ec
ni
n
2|| nc
2
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substituting & into
using
2006 Quantum Mechanics Prof. Y. F. Chen
Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator
1
2nE n
)(!2)(~ 2/2/1 2
n
nn Hen
0
( , ) ( ) :nEi t
n nn
t c e
2 2
2 2
/ 2 / 2 ( 1/ 2)
0
( )
( ) / 2 / 21/ 4
0
( ) 1( , ) ( )
! 2 !
/ 21 ( )
!
i ni n t
nnn
ni t
i tn
n
et e H e e
n n
ee e H
n
2 2
0
( , ) ( ) :!
nt t
nn
tg t e H
n
2 2
2 2
2( ) / 2 / 2 ( ) ( )
1/ 4
( ) / 2 / 2 2 2( ) ( )1/ 4
1( , ) exp / 2 2
1 exp / 2 2
i t i t i t
i t i t i t
t e e e e
e e e e
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as a result, the probability distribution of the coherent state is given by :
it can be clearly seen that the center of the wave packet moves in the p
ath of the classical motion
2006 Quantum Mechanics Prof. Y. F. Chen
Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator
2 2( ) 2
2 2 2
2
1( , ) ( , ) ( , ) exp cos[2( )] 2 2 cos( )
1 exp 2 cos ( ) 2 2 cos( )
1 exp [ 2 cos( )]
P t t t e t t
t t
t
)cos(2 t
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with , &
the operator acting on the eigenstate
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
xm
1 1( ) 2 ( ) 2 ( )n n nH H nH 21/ 2
/ 2( ) 2 ! ( )nn nn e H
x )(~ n
)(~)(~12
1
)()(2
1!2
)(!2
)(!2)(~ˆ
11
112/2/1
2/2/1
2/2/1
2
2
2
nn
nnn
nn
nn
n
nnm
HnHenm
Henm
Henm
x
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in a similar way, the operator acting on the eigenstate
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
)(~ nxp
)(~)(~12
1
)()(2
1!2
)()()(!2
)(!2
)(!2)(~ˆ
11
112/2/1
2/2/2/1
2/2/1
2/2/1
2
22
2
2
nn
nnn
nnn
nn
nn
nx
nnmi
HnHenmi
HeHenmi
Henmi
Henx
ip
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→
&
consequently, it is convenient to define 2 new operators :
&
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
)(~1)(~ˆ1
ˆ2
11
nnx np
mix
m
)(~)(~ˆ1
ˆ2
11
nnx np
mix
m
xp
mix
ma ˆ
1ˆ
2
1ˆ†
xp
mix
ma ˆ
1ˆ
2
1ˆ
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the operator is the increasing (creation) operator :
this means that operating with on the n-th stationary states yields a s
tate, which is proportional to the higher (n +1)-th state
the operator is the lowering (annihilation) operator :
this means that operating with on the n-th stationary states yields a s
tate, which is proportional to the higher (n -1)-th state
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
†a
)(~1)(~ˆ 1† nn na
†a
a
)(~)(~ˆ 1 nn na
a
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in terms of & , the operators & can be expressed as :
&
we can find the commutator of these 2 ladder operators :
which is the so-called canonical commutation relation
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
a †a x xp
†ˆˆ2
ˆ aam
x
†ˆˆ2
ˆ aam
ipx
1ˆ,ˆˆ,ˆ2
1
ˆ1
ˆ,ˆ1
ˆ2
1]ˆ,ˆ[ †
xpi
pxi
pm
ixm
pm
ixm
aa
xx
xx
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is the hermitian conjugate :
proof :
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
†a a
1
†221 |ˆ||ˆ| aa
1 2 1 2
1 2 1 2
2 1 2 1
2 1
1 1ˆ ˆ ˆ| |
2
1 1ˆ ˆ
2
1 1ˆ ˆ
2
1 1ˆ ˆ
2
x
x
x
x
ma x i p
m
mx i p
m
mx i p
m
mx i p
m
†2 1ˆ | |a
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with , &
the operator acting on the eigenstate
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
xm
1 1( ) 2 ( ) 2 ( )n n nH H nH 21/ 2
/ 2( ) 2 ! ( )nn nn e H
x )(~ n
)(~)(~12
1
)()(2
1!2
)(!2
)(!2)(~ˆ
11
112/2/1
2/2/1
2/2/1
2
2
2
nn
nnn
nn
nn
n
nnm
HnHenm
Henm
Henm
x
Quantum Harmonic Oscillator
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in a similar way, the operator acting on the eigenstate
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
)(~ nxp
)(~)(~12
1
)()(2
1!2
)()()(!2
)(!2
)(!2)(~ˆ
11
112/2/1
2/2/2/1
2/2/1
2/2/1
2
22
2
2
nn
nnn
nnn
nn
nn
nx
nnmi
HnHenmi
HeHenmi
Henmi
Henx
ip
Quantum Harmonic Oscillator
![Page 29: Quantum Harmonic Oscillator](https://reader036.vdocuments.site/reader036/viewer/2022081603/56815a95550346895dc812e8/html5/thumbnails/29.jpg)
→
&
consequently, it is convenient to define 2 new operators :
&
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
)(~1)(~ˆ1
ˆ2
11
nnx np
mix
m
)(~)(~ˆ1
ˆ2
11
nnx np
mix
m
xp
mix
ma ˆ
1ˆ
2
1ˆ†
xp
mix
ma ˆ
1ˆ
2
1ˆ
Quantum Harmonic Oscillator
![Page 30: Quantum Harmonic Oscillator](https://reader036.vdocuments.site/reader036/viewer/2022081603/56815a95550346895dc812e8/html5/thumbnails/30.jpg)
the operator is the increasing (creation) operator :
this means that operating with on the n-th stationary states yields a s
tate, which is proportional to the higher (n +1)-th state
the operator is the lowering (annihilation) operator :
this means that operating with on the n-th stationary states yields a s
tate, which is proportional to the higher (n -1)-th state
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
†a
)(~1)(~ˆ 1† nn na
†a
a
)(~)(~ˆ 1 nn na
a
Quantum Harmonic Oscillator
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in terms of & , the operators & can be expressed as :
&
we can find the commutator of these 2 ladder operators :
which is the so-called canonical commutation relation
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
a †a x xp
†ˆˆ2
ˆ aam
x
†ˆˆ2
ˆ aam
ipx
1ˆ,ˆˆ,ˆ2
1
ˆ1
ˆ,ˆ1
ˆ2
1]ˆ,ˆ[ †
xpi
pxi
pm
ixm
pm
ixm
aa
xx
xx
Quantum Harmonic Oscillator
![Page 32: Quantum Harmonic Oscillator](https://reader036.vdocuments.site/reader036/viewer/2022081603/56815a95550346895dc812e8/html5/thumbnails/32.jpg)
is the hermitian conjugate :
proof :
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
†a a
1
†221 |ˆ||ˆ| aa
1 2 1 2
1 2 1 2
2 1 2 1
2 1
1 1ˆ ˆ ˆ| |
2
1 1ˆ ˆ
2
1 1ˆ ˆ
2
1 1ˆ ˆ
2
x
x
x
x
ma x i p
m
mx i p
m
mx i p
m
mx i p
m
†2 1ˆ | |a
Quantum Harmonic Oscillator
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with
&
→
using the commutation relation
→
define the so-called number operator :
→ the H.O. Hamiltonian takes the form :
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
††††††2
ˆˆˆˆˆˆˆˆ4
ˆˆˆˆ42
ˆaaaaaaaaaaaa
m
px
††††††22 ˆˆˆˆˆˆˆˆ4
ˆˆˆˆ4
ˆ2
1aaaaaaaaaaaaxm
aaaaxmm
pH x ˆˆˆˆ
2ˆ
2
1
2
ˆˆ ††222
1ˆˆˆˆ]ˆ,ˆ[ ††† aaaaaa
2
1ˆˆˆ †aaH
aaN ˆˆˆ †
2
1ˆˆ NH
Quantum Harmonic Oscillator
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the eigenstates of can be found to be coherent states :
coherent states have the minimum uncertainty
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
a );0,(
0
2/||0
ˆ2/|| )(~!
)(~);0,(2†2
nn
na
neee
†ˆ ˆ ˆ( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2
2 ( ) cos
2
i x a am
m m
22 †
2 2
ˆ ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2
( 1)2
x a am
m
22 2ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2
x x xm
![Page 35: Quantum Harmonic Oscillator](https://reader036.vdocuments.site/reader036/viewer/2022081603/56815a95550346895dc812e8/html5/thumbnails/35.jpg)
as a consequence, we obtain the minimum uncertainty state :
2006 Quantum Mechanics Prof. Y. F. Chen
Creation & Annihilation Operators
Quantum Harmonic Oscillator
†ˆ ˆ ˆ( ) ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2
( ) 2 sin2
x
mii p i a a
mi m
22 †
2 2
ˆ ˆ ˆ( ,0; ) ( ,0; ) ( ,0; ) ( ,0; )2
( 1)2
x
mp a a
m
2);0,(ˆ);0,();0,(ˆ);0,( 222 m
ppp xxx
2
xpx