THE HARMONIC OSCILLATOR - MIT OpenCourseWare

5.61 Fall 2007

Lectures #12-15

THE HARMONIC OSCILLATOR

page 1

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

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One of a handful of problems that can be solved exactly in quantum

mechanics

examples

m1

m2

A diatomic molecule

E (electric field)

B (magnetic field)

? (spin magnetic moment)

Classical H.O.

m k

X0

X

Hooke's Law:

( ) f = -k X - X0 -kx

(restoring force)

f

=

ma =

d2x m

=

-kx

dt 2

d2x dt 2

+

k m

x

=

0

5.61 Fall 2007

Lectures #12-15

page 2

Solve diff. eq.: General solutions are sin and cos functions

x(t) = Asin(t) + Bcos(t)

or can also write as

x(t) = C sin(t + )

= k m

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

( ) ( ) e.g.

x 0 =x 0

v 0 =0

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

Then

x(0) = A sin(0) + Bcos(0) = x0 B = x0

v(0) = dx = cos(0) - sin(0) = 0 A = 0 dt x=0

So

x(t) = x0 cos(t)

Mass and spring oscillate with frequency: = k m

and maximum displacement x0 from equilibrium when cos(t)= ?1

Energy of H.O. Kinetic energy K

( ) ( ) K

=

1 2

mv2

=

1 2

m

dx 2 dt

=

1 2

m

-

x 0

sin

t

2

=

1 2

kx 2 0

sin2

t

Potential energy U

( ) ( ) ( ) ( ) f x = - dU U = - f x dx = kx dx = 1 kx2 = 1 kx2 cos2 t

dx

2

2 0

5.61 Fall 2007

Lectures #12-15

Total energy = K + U = E

x(t) x0(t)

( ) ( ) E

=

1 2

kx 2 0

sin2

t

+ cos2

t

E = 1 kx2 2 0

0

t

-x0(t)

1 2

kx02

U

K

E

page 3

0

t

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

e.g.

Diatomic molecular bond

A

B

X U

X

X0

A + B separated atoms

equilibrium bond length

5.61 Fall 2007

Lectures #12-15

page 4

( ) ( ) ( ) ( ) ( ) U

X

= U

X0

+ dU dX X = X0

X - X0

+ 1 d 2U 2 dX 2 X = X0

X - X 2 + 1 d 3U

0

3! dX 3 X = X0

X - X0 3 +!

( ) ( ) Redefine x = X - X0 and U X = X0 = U x = 0 = 0

( ) U x = dU x + 1 d 2U x2 + 1 d3U x3 + !

dx x=0

2 dx2 x=0

3! dx3 x=0

U

real potential

H.O. approximation x

At eq.

dU = 0 dx x=0

For small deviations from eq.

x3 ................
................

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