The Equation of Interest
The energy equation derived from the "Particle in a 1-D box" problem is as follows:
$E=\dfrac{n^2h^2}{8mL^2}$
where $n$ is the principle quantum number, $h$ is Planck's constant, $m$ is the mass of the particle (e.g. electron), and $L$ is the length of region of space (i.e. box).
The Goal
I would like to illustrate the harmonious nature of this energy equation derived from the quantum mechanical nature of the system and classical mechanics. In theory we should be able to define a mass and region of space for a clear comparison.
The Example: Electron vs. Car
I want to compute the lowest possible energy of an electron using the equation above. We define $m=9.11\times10^{-31} \textrm{kg}$ and the region of space as a reasonable value for the diameter of an atom, $L=0.2 \textrm{nm}$.
Plugging this into our equation along with Planck's constant ($h=6.626\times10^{-34} \textrm{ m}^2 \textrm{kg s}^{-1}$) we obtain
$E_{\textrm{electron}} = \dfrac{(6.626\times10^{-34} \textrm{ m}^2 \textrm{kg s}^{-1})^2}{8 *9.11\times10^{-31} \textrm{ kg} * (2\times 10^{-10} \textrm{m})^2} = 1.51 \times 10^{-18} \textrm{ J}$.
Now let us imagine a car with $m=2000 \textrm{ kg}$ and a region of space as $L=1 \textrm{ m}$. The resulting energy is
$E_{\textrm{car}}=2.74\times10^{-71} \textrm{ J}$.
Now in both cases, the energy is so small that we might as well say its zero (from a classical standpoint anyway). Of course $E_{\textrm{car}} << E_{\textrm{elecron}}$.
*So we cannot stop here as quantum theory and classical mechanics are virtually indistinguishable (with respect to raw energies) at this point correct?
We can, however, convert these energies (which are purely kinetic energies) into something which is clearly distinguishable such as velocity by using the equation
$E = \dfrac{1}{2}mv^2$
Alternatively,
$v = \sqrt{\dfrac{2*E}{m}}$
where $m$ is the mass of the object and $v$ is the velocity. If we plug in our previously determined energies we get the following results...
$v_{\textrm{electron}} = 1.8\times 10^6 \textrm{ m s}^{-1}$
and
$v_{\textrm{car}} = 1.7\times 10^{-37} \textrm{ m s}^{-1}$
Here we see that the velocity of the car is essentially zero (or technically speaking, so slow that it wouldn't ever be observed as moving in billions of lifetimes). This jives with classical theory. However, we see that the velocity of the electron is staggering, even though the energy of the electron is 'essentially zero'
The Conclusions
So, in order to properly relate the energy equation derived from the quantum mechanical system of a particle in a 1-D box, one cannot simply compare the raw energies of the particle (i.e. electron) to that of an ordinary object such as a car because both energies result in a number that is essentially zero which defeats the point. As such, one must relate the energy to something appropriate such as the velocity of the object to properly illustrate how the energy equation still relates to everyday life.
The Question
Are my conclusions correct or can the raw energetics simply be used to make a proper comparison between quantum/classical theory without resorting to recasing the results into a physical property such as velocity?
#not-homework