From (2.13), the graph of the potential energy of
a one-dimensional harmonic oscillator with respect to the distance *x*
from its equilibrium position, is a parabola with vertex at the origin O and up concave.

A particle with finite total energy that moves with harmonic motion is contained into a infinitely deep potential well so the probability to find it at infinite distance is zero.

In this case the Schrödinger equation is

and, taking *U(x)* from (2.13),

This last equation can be simplified by the variable subtitution

from which

so, from (10.2) we have

We can have a more simple notation of (10.2) if we let

so

The solution of this equation must approach 0 as ξ approaches positive or negative infinity: we may express it as

where *φ(ξ)* is a function to be determined later.

By substituting this expression and its second derivative with respect to *ξ* in (10.5) we have

Now we assume that the solution φ of (10.7) may be expressed by a series expansion

and therefore

By substituting these values in (10.7) we have

Since φ does not have singularities, we cannot consider negative exponents, so

The sum (10.10) is identically zero if and only if all its coefficients are zero

The equation (10.11) establishes a recursive relationship between the coefficients of the series expansion(10.8)
of *φ(ξ)*

Moreover, since *φ(ξ)* must be finite for any *ξ*, there must be an index *n*
of the expansion such that the not-zero n-th term would produce the (n+2)-th term equal to 0 an so all the
following terms with indexes (n+4), (n+6) and so on. Therefore, from (10.11) we obtain

that is, from (10.4)

If we use the Planck constant instead of the Dirac constant we have

The equation (10.14) represents a fundamental result in the history of Physics of the twentieth century,
because it provides a theoretical justification to the Planck conjecture and improves it:
harmonic oscillators, contrary to that provided by Classical Mechanics, may not have any energy,
but their energy is quantized.
The minimum value of the energy of a quantum oscillator id given by (10.14) with *n*=0.
All other possible energy values differ from that for multiples of *hν*.
So a harmonic oscillator can emit or absorb energy only if it exchanges with the outside
blocks of energy multiples of *hν*.