# Particle in One-Dimensional Box Problem [Quantum Mechanics]

• michaelmolli

#### michaelmolli

Homework Statement
a) Determine the ratio (Em/En) between two energy states of a particle in a one dimensional box of length l.
b) Show that this is consistent with the non-relativistic low-energy limit.

The attempt at a solution

I have figured out a) using the de broglie wave-particle duality and the energy momentum relationship to get:
Em/En = m2/n2 but I am unsure what this non-relativistic low-energy limit is.
I though about using classical kinetic energy to (ie. non-relativistic) of the particle and using similar equations (de broglie wavelength, standing wave) but I realized using the assumption of a De Broglie wavelength implies its accordance with relativity and I find myself somewhat lost.

Determine the ratio (Em/En) between two energy states of a particle in a one dimensional box of length l.
b) Show that this is consistent with the non-relativistic low-energy limit.

one should take a one dimensional box say of length L and write out the schrodinger equation -then calculate the wave function -apply the boundary condition at L=0 and L=L, the wave function must vanish at the boundary as the particle is confined.
that should yield the energy eigen values - take the ratio of two states say mth and nth state its a non relativistic situation.
check the energy ratio .

The attempt at a solution
I have figured out a) using the de broglie wave-particle duality and the energy momentum relationship to get:
Em/En = m2/n2 but I am unsure what this non-relativistic low-energy limit is.
What's this energy momentum relationship you refer to?

I though about using classical kinetic energy to (ie. non-relativistic) of the particle and using similar equations (de broglie wavelength, standing wave) but I realized using the assumption of a De Broglie wavelength implies its accordance with relativity and I find myself somewhat lost.
I don't think the assumption of a de Broglie wavelength implies SR. Why do you think it does?

You should solve it as drvrm described. Using the assumption of a De Broglie wavelength implies that the particle is free (no potential), your particle is free only inside the box.