# The Harmonic Oscillator

## 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

Classical H.O.

Hooke’s Law:

(restoring force)

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

or can also write as

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

e.g.

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

Then

So

Spring oscillates with frequency

and maximum displacement   from equilibrium

Energy of H.O.

Kinetic energy º K

Potential energy º U

Total energy = K + U = E

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

e.g.          Diatomic molecular bond

Redefine      and

At eq.

For small deviations from eq.

Total energy of molecule in 1D

COM coordinate describes translational motion of the molecule

QM description would be free particle or PIB with mass M

We’ll concentrate on relative motion (describes vibration)

and solve this problem quantum mechanically.

## THE QUANTUM MECHANICAL HARMONIC OSCILLATOR

K            U

Note: replace m with m (reduced mass) if

Goal: Find eigenvalues En and eigenfunctions yn(x)

Rewrite as:

This is not a constant, as it was for P-I-B,

so sin and cos functions won’t work.

TRY:                     (gaussian function)

or rewriting,

which matches our original diff. eq. if

\

We have found one eigenvalue and eigenfunction

Recall

\

This turns out to be the lowest energy: the “ground” state

For the wavefunction, we need to normalize:

where N is the normalization constant

\

Note  is symmetric. It is an even function:

There are no nodes, & the most likely value for the oscillator displacement is 0.

So far we have just one eigenvalue and eigenstate. What about the others?

with

These have the general form

Normalization                   Gaussian

Hermite polynomial

Energies  are

Note E increases linearly with n.

Þ       Energy levels are evenly spaced

regardless of n

There is a “zero-point” energy

E = 0 is not allowed by the Heisenberg Uncertainty Principle.

Symmetry properties of y’s

are even functions

are even functions

Useful properties:           (even) ×(even) = even

(odd) ×(odd) = even

(odd) ×(even) = odd

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