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8/3/2019 Chem 373- Lecture 11: Harmonic oscillator-I
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Lecture 11: Harmonic oscillator-I.
Vibrational motion
(This lecture introduces the classical harmonic oscillator
as an introduction to section 12.4 .
Lecture on-line
The Harmonic Oscillator-I (PDF)The Harmonic Oscillator-I (PowerPoint
Handout for this lecture (PDF)
Supporting material for classical harmonic oscillator (PDF)Supporting material for classical harmonic oscillator
(PDF handout format with 6 slides per page)
Supporting material for quantum mechanical harmonic oscillator (PDF)Supporting material for quantum mechanical harmonic oscillator (PDF
handout format with 6 slides per page)
8/3/2019 Chem 373- Lecture 11: Harmonic oscillator-I
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Tutorials on-line
Basic concepts
Observables are Operators -
Postulates of Quantum Mechanics
Expectation Values -More Postulates
Forming Operators
Hermitian Operators
Dirac Notation
Use of Matricies
Basic math background
Differential EquationsOperator Algebra
Eigenvalue Equations
Extensive account of Operators
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Audio-visuals on-line
Overview of the harmonic oscillator (PDF)(Good overview from the Wilson
group,****)
Overview of the harmonic oscillator (Powerpoint)
(Good overview from the Wilsongroup,****)
Vibrating molecule-I (Quick Time movie 1.4 MB)
(From the CD included in Atkins
,***)
Vibrating molecule-II (Quick Time movie 1.4 MB)
(From the CD included in Atkins ,***)
Slides from the text book (From the CD included in Atkins ,**)
The material in this lecture covers the following in Atkins.
8/3/2019 Chem 373- Lecture 11: Harmonic oscillator-I
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Review : Classical harmonic oscillatorLet us consider a particle of mass m attached to a spring
Equilibrium
x=0,t=0
o
x
Stretchx=x
o
compressx=-xoxo
xo
At the beginning at t = o the particle is at equilibrium,that is no force is working at it , F = 0,
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In general F = -k x . The force propotional to
displacement and pointing in opposite directiono
Equilibriumx=0,F=0
o
x
xo
xo
xo F=-kxo
xo
F= kxo
Review : Classical harmonic oscillator
k is the force constant
of the spring
V x k x( ) =1
2
2
F dVd x
dd x
k x k x= = = ( )22
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We might consider as an other example two particles
attached to each side of a spring
re
A B
F= 0 Equilibrium
r = re
Case I: Equilibrium
Review : Classical harmonic oscillator
8/3/2019 Chem 373- Lecture 11: Harmonic oscillator-I
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re-x
A B
F= -k(-x) Equilibrium
r = re
Case III: Compress
x
A B
F= -kx Stretchr = re+x
Case II: Stretch
r = re+x
Again we have thatthe force F is proportional
to the displacementx and pointing in the
opposite directionF = - k x
Review : Classical harmonic oscillator
8/3/2019 Chem 373- Lecture 11: Harmonic oscillator-I
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Review : Classical harmonic oscillator
x = A sin (k
mt )
V x k x( ) =1
2
2
FdV
d x
d
d x
kx k x= = = ( )2 2
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V(x) = 1/2k2x2V
x-A2 A2k1 > k2
E
V(x) = 1/2k1x2
-A1 A1
k k1 2>
Appendix : Classical harmonic oscillator
The parabolic potential energy V =1/2
kx2 a harmonic oscillator, where x isthe displacement from equilibrium. Thenarrowness of the curve depends onthe force constant k: the larger the
value of k, the narrower the well.
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Harmonic oscillator...Quantum mechanically
We shall now turn to a quantummechnical treatment of the one
dimensional harmonic oscillator
We have
H = E + Ekin pot
Hm
d
dxkx
E kxpot
= +
=
h2 2
22
2
2
1
2
1
2
V x kx( ) =
1
2
2Mass
Displacement
Force constant
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The parabolic potential energy V =1/2
kx2 a harmonic oscillator, where x isthe displacement from equilibrium. Thenarrowness of the curve depends onthe force constant k: the larger the
value of k, the narrower the well.
Harmonic oscillator...Quantum mechanically
V(x) = 1/2k2x2V
x-A2 A2k1 > k2
E
V(x) = 1/2k1x2
-A1 A1
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Harmonic oscillator...Quantum mechanically
=cycles per time unit
1
2
k
m
or
k m= 4 2 2 Thus
d
dx
d
dxmx
ddx
m x
d
dxx
m
H = -2m
kx = -2m
= -2m
(
= - 2m
where =
2 2 2
2
2
2
h h
h
h
h
h
2
2
2
22 2 2
2
2
2 2 22
2
2 2 2
1
22
4
2
+ +
)
( )
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Harmonic oscillator...Quantum mechanically
We must solve
H (x) =E (x)
or
-2m
(x) = E (x)
Thus
(x)(x) = - (x)
(x)(x)=0
2
2
2
( )
( )
h
h
h
d
dxx
d
dxx
mE
d
dx
mEx
2
22 2
2
22 2
2
22 2
2
2
+
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Harmonic oscillator...Quantum mechanically
ddx
mE x
x f x
22
2 2
2
2
2
(x) (x) = 0
Let us look at a solution of
the form(x) = exp(-
2+ ( )
) ( )
h
let us further try to obtain
a power expansion of f(x)
of the form
f(x) = cnm=0
n= xn
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Harmonic oscillator...Quantum mechanically
The x f x
with f x xn
solution
=
(x) = exp(-
c
nm=0
n=2
2 ) ( )
( )
Must satisfy (- ) = ( ) = 0.
We
d
dx
mEx
want to solve :
2
22 22
(x)(x)=0
2+ ( )
h
It thi
E v
is shown in supplementarymaterial that s is only possible if
v = 1,2,3,4...= +h( )1
2 =k
m
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E
1
2h
3
2h
5
2h
72h
x
9
2h
112
h
The energylevels of a
harmonic
oscillator
are evenlyspaced with
separation
, with =
(k/m)1/2.
Even in its
lowest state,
an oscillatorhas an
energy
greater
than zero.
Harmonic oscillator...Quantum mechanically .Energy levels
v = 0
v = 1
v = 2
v = 3
v = 4
v = 5
v = 6E v= +h( )1
2
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We obtain as the solutions to (x)
Gaussian function)
( ) ( ) (
( ) exp ( )
x N polynomial in x bell shaped
x Ny
H yv v v
=
=
2
2
Harmonic oscillator...Quantum mechanically.... Wavefunction
With
mk =h
2 1 4
/y x= /
Where
Nv = 1
2
1
2 v v!
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_________________________
v H
1
1 2y
2 4y - 23 8y - 12y
4 16y - 48y +12
5 32y -160y +120y6 64y - 48y + 72y - 120
_____________________________
v
2
3
4 2
5 3
6 4 2
0
Hermit polynominals
Harmonic oscillator...Quantum mechanically
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What you should learn from this lecture
1
12
12
2
.
) ) .
)
The potential energy in a harmonic oscillator is given
V( x) = V(x x k(x
k is the force constant and x is the position where
the force F = -dV
dx
(x is zero
e 2 e
e
e
= =
=
x k x
Here
k x
2
12
.
( )
You are not required to solve the Schrdinger eq.For the quantum mechanical harmonicoscillator. However you should recall that the energy is given by :
v = 1,2,3,4...E v= +h = km
3. Also recall for the solution (x)
=
v v vx N
yH y y x
mk( ) exp ( ); / ;
/
=
=
2 2 1 4
2
hWhere
Nv =1
2
1
2 vv!