1. The text to the right has been
copied from a GSU webpage
“Hyperphysics” – the whole
thing can be viewed if you
click on this link:
Quantum Harmonic Oscillator

2. Why is the quantum simple harmonic oscillator important?
For many reasons – for instance, any potential having a
Local minimum can be approximated by a parabolic potential
In the neighborhood of that minimum.

3. Important examples of potentials with minima are the potentials
associated with the interactions between atoms in molecules. The
interaction between two atoms in a molecule is pretty well descri-
bed by the so-called “Morse Potential” which is definitely NOT a
harmonic potential.
However, in the vicinity
of the minimum it can be
approximated by a para-
bolic harmonic potential,
and such an approximation
is known to work quite well
If the oscillation energy is
relatively small.

4. Let’s play for a moment with a classical simple harmonic oscillator…
The picture displays a classical har-
monic oscillator, at the equilibrium
position (A), at the positions of maxi-
mum displacement to the left (B) and
to the right (C).
Suppose that the angular frequency of
this oscillator is ω . What is the equa-
tion describing the displacement as a
function of time? (suppose that at t=0
the displacement x is zero).

5. Now, we want to solve the following problem.
Suppose that the oscillator is placed in a dark room.
You cannot see it. But you have a camera with a flash.
At some moment, you take a picture of the oscillator.
What is the probability that on the picture the mass
is positioned between x and x+dx ?

6. The probability of finding the mass between
x and x+dx is proportional to (answer): …….
Yes! You are right! It is proportional to dt , i.e.,
to the time the mass spends between these
two points.
But how can we find dt ? Please answer:

8. Now, please compare the profile of the function we have
found with the square of the wavefunction of the SHO
quantum state with n=100 !
The striking similarity of the two
functions, the classical one and the
quantum one, is a beautiful illustra-
tion of the Bohr’s
Principle of Correspondence