# Doubt/Confusion in De Broglie Wavelength of a particle in a box setup

So I am pretty new to quantum mechanics (mostly related to atomic structures), and have some difficulties in the following setup:

Consider a particle of mass $$m$$ confined to a one-dimensional box of length $$L$$. The particle moves in the box with momentum $$p$$ colliding elastically with the walls. We consider the quantum mechanics of this system.

At each energy state, the particle may be represented by a standing wave given by the de Broglie hypothesis. Express its wavelengths $$λ_{dB}$$ in terms of $$L$$ in the $$n^{\text{th}}$$ energy state.

First of all, (it may seem silly), what is energy state here? In atomic structure, I know it corresponds to electrons permissible energy in shells, as per Bohr's quantization. But here in this setup, I couldn't grasp the meaning of 'energy state'.

Secondly, question asked to express particle as a standing wave. I know the standing wave concept but from a classical mechanics point of view, not quantum mechanical. I couldn't figure which concept should I apply here.

Interestingly, the answer is $$\lambda_{dB}=\frac{2L}{n}$$, which is the same as the general formula of standing wave (classical) wavelength as shown in the figure. I don't know how the $$n$$ in the figure of standing waves and the $$n$$ corresponding to energy state in our state are so related, so this is my major doubt. I may be confused between these formulae, but I can't help myself here.

Well the energy for a particle in a box is as follows: $$E_n = \frac{n^2h^2}{8mL^2}$$ Where each n represent the state, so $$E_n$$ represents the energy per state.
To come back to your question of where this formula came from: The momentum is known to be $$p=m v \longrightarrow v = \frac{p}{m}$$ Inside a one-dimensional box there's kinetic energy: $$E= \frac{1}{2}mv^2$$. We can rewrite this as $$E = \frac{p^2}{2m}$$. since $$p = \frac{h}{\lambda} = \frac{hn}{2L}$$ So we get $$E_n = \frac{n^2h^2}{8mL^2}$$ when we plug all of this in.
• Can you please, tell that how you wrote $E_n$? I am pretty novice in quantum mechanics. I know nothing beyond basic quantum mechanical model of atom. Nov 2, 2021 at 18:13
• By the way, $p = \frac{h}{\lambda}$ because $p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$ Nov 2, 2021 at 18:30
• One more doubt is that wavelength gets shorter for increasing value of $n$, so is it sort of frequency like factor? Also how are we exciting particle, through external means? Nov 2, 2021 at 18:30