# Difference between Particle in a Box and Harmonic Oscillator approximations

Why in the particle in a box model do the values of n begin at 1 but in the harmonic oscillator they begin at 0? I understand what the wave-functions and their corresponding probabilities look like and that the PIB has 0 nodes for n=1 which means the number of nodes is n-1, so for n = 0 it would have -1 nodes which is physically unreasonable. But I do not understand why the PIB cannot have n=0 whereas the Harmonic oscillator can.

• PIB cannot have n=0 precisely for the reason that you said yourself just a sentence ago: that the corresponding $\psi$ would not be physically meaningful. If that is not enough of a justification for you, then what is? Dec 4, 2019 at 14:48
• This still doesn't answer the question of why PIB cannot have an n=0 whereas the HO can. All you've stated is that we can change the rules by redefining things. But redefining things doesn't change the facts. PIB can never have n = 0 because this would mean the energy is zero, which means the particle is not moving so momentum is 0 and we can know the position, but this violates the uncertainty principle. But why can HO have an n=0?? Dec 4, 2019 at 22:46
• Ivan Neretin, an explanation to WHY something is not physically meaningful. Thanks Dec 4, 2019 at 22:47
• $n=0$ is not a fact; it is a convention, and as such, can be changed, like Buck Thorn said. $n$ is not a physical value at all. Also, look at the HO state with $n=0$: its energy is not $0$. Dec 5, 2019 at 5:28
• Do you realize that there is a zero point energy associated with the harmonic oscillator? That is, when n=0 the energy is not zero? Dec 5, 2019 at 17:56

You could just as well define the quantum number and associated energy levels for a 1D pib as \begin{align} E_{n'}&=k(n'+1)^2 \\&=k n'^2 +2kn' + k\end{align} where $$k=\frac{\pi^2 \hbar^2}{2mL^2}$$ Here $$n'$$ defines a "new" quantum number related to the original one as $$n'=n-1$$ and therefore ranges from 0 to $$\infty$$.
$$E_n = \hbar \omega n'$$
where $$n'$$ ranges in integer increments from $$\frac12$$ to $$\infty$$.