Lecture 12: Harmonic Oscillator-II.
The material in this lecture covers the following in Atkins.
Section 12.4 The energy levels
Section 12.5 The wavefunction
Lecture on-line
Quantum mechanical harmonic oscillator (properties)
(PowerPoint)
Quantum mechanical harmonic oscillator (properties) (PDF format)
Handout for this lecture
Writeup on Harmonic Oscillator
Harmonic oscillator...
Quantum mechanically .Hamiltonian
Review
We have for the harmonic
oscillator
Force constant
1 2
Epot = kx
Displacement
2
where x is the displacement
from equilibrium.
Thus the hamiltonian is given by :
H = E kin + Epot
2 d2
1 2
h
ˆ
H=−
+ kx
2
2m dx
2
Mass
1 2
V( x ) = kx
2
Harmonic oscillator...
Quantum mechanically .Energy levels
The energy
evels of a
harmonic
oscillator
are evenly
paced with
eparation
ω, with ω =
k/m)1/2.
Even in its
owest state,
an oscillator
has an
nergy
greater
han zero.
v=6
Review
E
1
E = hϖ( + v )
2
v=5
11
hϖ
2
v=4
9
hϖ
2
v=3
v=2
v=1
v=0
7
hϖ
2
5
hϖ
2
3
hϖ
2
1
hϖ
2
x
Harmonic oscillator...Quantum mechanically.... Wavefunction
We have the general solution
 y2 
ψ v ( x ) = N v exp −  H v (y) ; y = x/α
 2 
It is readilly shown that
Nv =
1
1
απ 2 2 v v!
so
ψ v (x) =
1
1
απ 2 2 v v!
 y2 
exp −  Hv ( y )
 2
Harmonic oscillator...Quantum mechanically.... Wavefunction
bell − shaped Gaussian function
The graph of the Gaussian function,
f(x) = e -x2.
Harmonic oscillator...Quantum mechanically.... Wavefunction
v
Hv Hermit polynominals
_________________________
0
1
1
2y
2
4y 2 - 2
3
8y 3 - 12y
4
16y 4 - 48y 2 + 12
5
32y 5 - 160y 3 + 120y
6
64y 6 - 48y 4 + 72y 2 - 120
_____________________________
Note that H v for v odd (1, 3, 5, 7,..)
is odd : H v (y) = - H v ( − y)
Note that H v for v even (0, 2, 4, 6, 8...)
is even : H v (y) = H v ( − y)
Harmonic oscillator...Quantum mechanically.... Wavefunction
Pr operties of Hermit
polynominals :
Here
H"v
− 2yH'v
+ 2vHv = 0
H"v =
Hv +1 = 2yHv − 2vHv −1
(recursionformula)
d2Hv
dy 2
d Hv
'
Hv =
dy
∞
−y
H
'
H
e
dy =
∫ v v
−∞
2
1
π 2 2 v v! δ
vv '
Harmonic oscillator...Quantum mechanically.... Wavefunction
ψ v (x) =
1
1
απ 2 2 v v!
 y2 
exp −  Hv ( y )
 2
For the groundstate v = 0
of the harmonic oscillator
we have the wavefunction
 y2 
1
ψ o (x) =
exp −  H0 ( y )
1
2

απ 2
 y2 
1
exp − 
=
1
2

απ 2
The normalized wavefunction and
probability distribution (shown also by
shading) for the lowest energy state of
a harmonic oscillator.
Harmonic oscillator...Quantum mechanically.... Wavefunction
ψ v (x) =
1
1
απ 2 2 v v!
 y2 
exp −  Hv ( y )
 2
For v = 1
ψ 1( x ) = 2
1
1
απ 2 2
 y2 
exp −  y
 2
The normalized wavefunction and
probability distribution (shown also by
shading) for the first excited state of a
harmonic oscillator.
Harmonic oscillator...Quantum mechanically.... Wavefunction
ψ v (x) =
1
1
απ 2 2 v v!
 y2 
exp −  Hv ( y )
 2
The normalized wavefunctions for the first five
states of a harmonic oscillator. Note that the number
of nodes is equal to v and that alternate
wavefunctions are symmetrical or antisymmetrical
about y = 0 (zero displacement).
Harmonic oscillator...Quantum mechanically.... Wavefunction
Particle can
be found
outside
clasical region
 y2 
ψ v ( x ) = Nv exp  −
 Hv ( y )
2




Harmonic oscillator...Quantum mechanically.... Wavefunction
ψ v (x) =
1
1
απ 2 2 v v!
 y2 
exp −  Hv ( y )
 2
The probability distributions for the first five states
of a harmonic oscillator represented by the density
of shading. Note how the regions of highest
probability (the regions of densest shading) move
towards the turning points of the classical motion as
v increases.
Harmonic oscillator...Quantum mechanically .Energy levels
Potential energy
increases more
suddenly for
particle in a box
1 2
V = kx
2
V=0
Particle confined
in box
Particle confined
in harmonic potential
Harmonic oscillator...Quantum mechanically .Energy levels
Comparison of energy levels in harmonic oscillator
and particle in a box
E
2
Energy levels
25 h 2
8mL
n=5
in particle in
box
2 2
E=
n h
n=4
2
8mL
n = 1, 2, 3
h2
8mL2
2
4 h 2
8mL
h2
8 mL 2
9
n=3
∆E =
(2n + 1)
2
h
16
8mL2
h
2
n=2
8mL2 n=1
Particle-in-box
11 hϖ
2
v=5
9 hϖ
2
Energy levels for
harmonic oscillator
1
h
ϖ
+ v)
E
=
(
v=4
2
7 hϖ
2
v=3 v = 0, 1, 2, 3
5 ϖ
2h
v=2
3 ϖ
h
2
v=1
1 ϖ
h
2
v=0
Spacing
∆E = hϖ
Harmonic oscillator
Zero-point Energy
Harmonic oscillator...Quantum mechanically.... properties
Expectation values
∞
Ω = ∫ ψ *v ( x )Ωψ v ( x )dx
∞
 y2 
ψ v ( x ) = N v exp  −
 H v ( y ); y = x/α
 2 
-∞
x = ∫ ψ *v ( x )xψ v ( x )dx
-∞
∞
2
2
y
y
= N v2 ∫ exp[ − ]H v (y)x exp[ − ]H v (y)dx
2
2
-∞
∞
2
2
y
x
y
x
2
2
= N vα ∫ exp[ − ]H v (y)( )exp[ − ]H v (y)d( )
2
α
2
α
-∞
∞
2
2
y
y
= N v2α 2 ∫ exp[ − ]H v (y)y exp[ − ]H v (y)dy
2
2
-∞
∞
2
2
= N vα ∫ exp[ − y 2 ]H v (y)yH v (y)dy
-∞
Harmonic oscillator...Quantum mechanically.... properties
∞
Properties of Hermit
2
2
2
= N vα ∫ exp[ − y ]H v (y)yH v (y)dy
polynominals :
-∞
H v +1 = 2 yH v − 2 vH v −1
H"v − 2 yH'v + 2 vH v = 0
1
yH v = H v +1 + vH v −1
2
H v +1 = 2 yH v − 2 vH v −1
(recursionformula)
∞
1
= N v2α 2 ∫ exp[ − y 2 ]H v (y)H v +1 (y)dy
2 -∞
∞
2
2
+ N vα v ∫ exp[ − y 2 ]H v (y)H v −1 (y)dy
-∞
= ∂ v,v+1 + ∂ v,v-1 = 0 + 0
<x> = 0
∞
2
−
y
dy =
∫ H v 'H v e
−∞
1
π 2 2 v v! δ '
vv
Harmonic oscillator...Quantum mechanically.... properties
You will show in assigned questions :
∞
x 2 = ∫ ψ *v ( x )x 2 ψ v ( x )dx
-∞
hϖ 1
( + v)
=
k 2
We note that < x 2 > increases with v
as the probability to find the particle at
the turning points increases.
Also < x 2 > decreases with k
Harmonic oscillator...Quantum mechanically.... properties
It follows
∞
1
1
V = k x 2 = k ∫ ψ *v ( x )x 2 ψ v ( x )dx
2
2 -∞
1
1 1 1
1
1
= khϖ(v + ) = h(v + )ϖ = E
2
2 k 2
2
2
We can find the average kinetic energy from
1
1
V + T =< E >⇒ E + T = E ⇒ T = E
2
2
2
ˆ
p
1
E
2
x
ˆ
ˆ
< pˆ x >
;
We also have T =
thus < T > =
2
2m
2m
1
or < pˆ 2x > = 2mE; < pˆ 2x > = 2mhω( + v )
2
You are being asked to shown in assigned problems
< px > = 0
Harmonic oscillator...Quantum mechanically.... properties
In general for a potential
V = axb
It can be shown that
b
T =
V "Virial Theorem"
2
2
V =
E
b+2
b
T =
E
b+2
Taylor expansion
Harmonic oscillator...Quantum mechanically..
Vibration Spectroscopy
0
0
dV
V(R) = V(R e ) + ( ) ∆R e
dR
small
2
3
1 d V
1 d V
2
+ ( 2 ) ∆R e + ( 3 ) ∆R e 3 + ...
2 dR
8 dR
2V
1 d2V
1
d
V(R) = ( 2 ) ∆R e 2 = k∆R e 2 ;( 2 ) = k
2 dR
2
dR
Harmonic oscillator...Quantum mechanically
2V
1 d2V
1
d
V(R) = ( 2 ) ∆R e 2 = k∆R e 2 ;( 2 ) = k
2 dR
2
dR
The force constant is a measure of the
curvature of the potential energy close
to the equilibrium extension of the
bond. A strongly confining well (one
with steep sides, a stiff bond)
corresponds to high values of k.
Harmonic oscillator...Quantum mechanically
We note relation between bond energy D ;
bond order and force constant k
Harmonic oscillator...Quantum mechanically
Alternative descriptions of the vibrations of CO2. (a)
The stretching modes are not independent, and if
one CO group is excited the other begins to vibrate.
(b) The symmetric and antisymmetric stretches are
independent, and one can be excited without
affecting the other: they are normal modes. (c) The
two perpendicular bending motions are also normal
modes.
Harmonic oscillator...Quantum mechanically
The three
normal
modes of
H2O. The
mode v2 is
predominant
ly bending,
and occurs
at lower
wavenumber
than the
other two.
Harmonic oscillator...Quantum mechanically
What you should learn from this lecture
1. You are not required to remember the hermit polynomials
and their relations. However you should be able to make use of the
two tables
Pr operties of Hermit
v
Hv
polynominals :
_________________________
Hermit polynominals
0
1
H"v − 2yH'v + 2vHv = 0
1
2y
2
4y 2 - 2
3
8y 3 - 12y
4
16y 4 - 48y 2 + 12
5
32y 5 - 160y 3 + 120y
6
4
Hv +1 = 2yHv − 2vHv −1
(recursionformula)
2
6
64y - 48y + 72y - 120
_____________________________
∞
−y
H
'
H
e
dy =
∫ v v
−∞
2
1
π 2 2 v v! δ
vv '
What you should learn from this lecture
2. You should remember Hv is odd for v odd and even for v even.
You should understand the meaning of odd and even functions
3. You should understand the problem assigned to this
lecture on the vibrating diatomic molecule A - B