VI–29
Harmonic Oscillator (Notes 9) – use pib - “think through” – 2014
Particle in box; Stubby box;
Properties of going to finite potential  w/f penetrate walls, w/f oscillate,
# nodes inc. with n,
E n
-levels less separate with n, for E>V
0
, spacing collapse, E n
continuous
Consider one ball (mass) on a spring
Hook’s law states that restoring force
F = -k (x
V =
1
/
2
– x e q =
 x = x - x
) = -kq e
 displacement x
F = -

V/
 q [recall: d(kq
2
(linear force ) F
)/dq = 2kq] e d
kq
2
k = force constant F u
T =
H

 
2  m 2  x
2
2
=



2 m
2 d 2 dq 2
=
 
2  m 2  q
2
2
 (1/2) kq 2 here dq = dx



= E

T V
 a) like a box with sloped sides
– soft potential - expect penetrate b) still a well - expect oscillator c) must be integrable – expect damped ,
i.e.

(x)  0 at q =

Trial:

~

(q)e
–  q
Fail: q(+) OK, blows up q(-) side (or opp.

-)
29

VI–30
Solution:
 
~

(q)e
–  q2
works both side - (

must positive )

(q) – polynomial form works f
0
~c onst – orthog: f
1
~q (odd-even)
 orthog.
if

2
(q) ~ q
2
-const
Result
–
(Atkins p.300-06
– confusing) – see House Ch. 6 not to derive, just get idea
Wave function (from Engel  )
– modify variable to simplify:
 

(y) = H

(y) e
-y2/2
  = 0, 1, 2, … (quantize) recursion formula :
H

(y) = (-1)
 e y2
d

/dy

(e
-y2
) y =
  q

=2
 mk h = mk  q non-classical: see wavefunction penetrate potential
[Engel - difference – classical: sin q & turning point at V(q m
)]
Hermite polynomials (H


: ex: H
0
= 1 note: – odd - even progression
H
H
H
H
H
H
4
5
6
1
= 2y
2
= 4y
2 – 2
3
= 8y
3
= 16y
= 32y
= 64y
– 12y
4
- 48y
2
+ 12
– alternate exponents
– 
number solution,
– add exponential term,
2 exp[-y
6
5
-160y
3
+ 120y
- 480y
4
+ 720y
2 – 120

= 0,1,
/2], to get damping
…∞ thus:


= A

H

(y)e
-y2/2
y =
  q A

=(2
 
!)
-1/2
(

)
1/4
30

VI–31
Homework: insert


into Schrödinger equation to get:
E

= (

+ ½)   
= k m
= (

+ ½) h
 
=

/2
 note: – even (constant) energy level spacing: 
E = h

– zero point energy: ½ h

– heavier mass,  ↓ 
E

0 classical
– weaker force constant,  ↓ 
E

0
Shapes: wave fcts probabilities (House) (Atkins)
31

VI–32
Probabilities: low
 – high in middle; high
 – high at edge
This fits classical, turnaround points  slower motion
A.
Solutions for

= 0-4
B.

(x)*

(x), probabilities for

= 0,4,8 compared to classical result (---)
Plots of

( probabilities ) for

= 0 – 4 and n = 12, from Engel
To describe two masses on a spring (relate to molecules) need to change variables q = (x
2
- x
1
) – (x
= r – r eq r = x
2
– x
1
0
2
- x
0
1
)
(3-D representation) relative 1-D position in this case:

= m
1 m
2
/(m
1
+ m
2
)  reduced mass goes into

2-mass harmonic oscillator:
 
= [( 2
/2
 
d
2
/dq
2
+
1
/
2
kq
2
]


= E


 and get E

= (

+
1
/
2
)  
(

+
1
/
2
) h
 
= k


=
1
/
2

Use to model vibration of a diatomic molecule – low  k

32

harmonic (ideal) spacing regular
VI–33
E-levels collapse in
 real molec.
 anharmonic anharmonic probability distribution , even
 multiatom 3n - 6 relative coord – complex but separable
Two-dimensional Harmonic oscillator:
H = T + V
T =  2
/2m (
 2
/
 x
2
+
 2
/
 y
2
)
V = V(x, y) expand about x = 0,y = 0 (Taylor series)
= V(0, 0) +

V/
 x

+ ½
 f(x) =

 2
V/
 y
2 
0
y n

1/n!]d
2
0
+ ½
 n f/dx n
| x0
(x-x
0
)
x +

V/
 y

0
y + ½

2
V/
 x
 y
 n
(here expand for x
0
=0)
2
V/
 x
0
xy +1/6

2 
0
x
2
3
V/
 x
3 
0 x
3 …etc.
– more complex potential – expands to
∞ powers, dec. in size
– not separated -- cross-terms like
 2
V/
 x
 y mix variables
– V(0, 0) = 0 – arbitrary constant – just shift E, can ignore
– 
V/
 x

0
=

V/
 y

0
= 0 – evaluate derivative at min. choose x = 0, y = 0 as the minimum, so 1 st
term in q disappear
– same as choosing q = 0 as origin for q = x-x e
Then V = ½(
 2
V/
 x
2
)
0
x
2
+½ (
 2
V/
 y
2
)
0
y
2
+½ (

+ ½ k xy xy + …
2
V/
 x
 y)
0
xy
= ½ k x
x
2 where k x
= (
 2
+ ½ k y y
2
V/
 x
2
)
0 etc. – 2-D force constants, many
– same form as harmonic oscillator, if neglect terms, like x
3
33

VI–34 solvable if can separate variables
– do change of variable x, y 
q
1
, q
2
where q
1
, q
2
chosen so that potential is not coupled, mixed coord.
– V(q
1
, q
2
) = ½ k
1
q
1
2
+ ½ k
2 q
2
2 q n
– normal coordinates
– call this potential “diagonalized” – use matrix approach can do to arbitrary accuracy
– also works for n-dimensions: (3n - 6) vibration
Basis for vibrational spectroscopy – IR and Raman
We will discuss further in spectroscopy section at end
34