5.61 Fall 2004

Lectures #12-15

page 1

THE HARMONIC OSCILLATOR •

Nearly any system near equilibrium can be approximated as a H.O.

• One of a handful of problems that can be solved exactly in quantum mechanics examples m1

m2

A diatomic molecule

E (electric field)

µ (spin magnetic moment)

B (magnetic field)

Classical H.O.

k

m

X0

Hooke’s Law:

X

(

)

f =
k X
X 0 
kx

(restoring force)

d2x f = ma = m 2 =
kx dt



d2x  k  x=0 + dt 2  m 

5.61 Fall 2004

Lectures #12-15

Solve diff. eq.:

()

page 2

General solutions are sin and cos functions

( )

( )

x t = Asin
t + B cos
t

k m


=

or can also write as

()

(

x t = C sin  t +


)

where A and B or C and  are determined by the initial conditions.

()

x 0 = x0

e.g.

()

v 0 =0

spring is stretched to position x0 and released at time t = 0. Then

()

()

()

x 0 = A sin 0 + B cos 0 = x0

()

v 0 =

So

dx dt

()



()

=  cos 0
 sin 0 = 0 x =0

()

B = x0 

A=0

( )

x t = x0 cos
t

Spring oscillates with frequency
=

x

0

and maximum displacement

k m

from equilibrium

Energy of H.O. Kinetic energy  K 2

2 1  dx  1 1 1 K = mv 2 = m  = m 
 x0 sin  t = kx02 sin 2  t 2 2 2 2  dt 

( )

( )

Potential energy  U

()

f x =


dU dx



()

U =
 f x dx =

 ( kx )dx = 2 kx 1

2

=

( )

1 2 kx0 cos2  t 2

5.61 Fall 2004

Lectures #12-15

page 3

Total energy = K + U = E

E=

( )

( )

1 2 kx0 sin2
t + cos2
t  2

E=

1 2 kx 2 0

x (t ) x 0(t ) 0

t

-x0(t) U

1 2 kx 2 0

K E

0

t

Most real systems near equilibrium can be approximated as H.O. e.g.

Diatomic molecular bond

A

B X

U

X0

X A + B separated atoms

equilibrium bond length

5.61 Fall 2004

( )

( )

U X = U X0

Lectures #12-15

dU + dX

(X
X ) 0

X = X0

Redefine x = X
X 0

(

()

(X
X )

2

0

X = X0

)

(

1 d 3U + 3! dX 3

1 d 2U x+ 2 dx 2 x=0

3

X = X0

)

1 d 3U x + 3! dx 3 x =0

x3 +


2

x =0

U

real potential

H.O. approximation

x

At eq.

dU dx

=0 x =0

For small deviations from eq.

()


U x 

x 3