# GRE GR 1777, Problem 013 Solution

Physics GRE GR 1777

13. Quantum Mechanics (Infinity potential wall)

Infinity potential wall problem is one of the basic problems in quantum mechanics. So I recommend you to remember some equation about it.

The energy eigenvalues of a particle of mass $m$ inside a one-dimensional box are

$E_n = \frac{\hbar^2 k^2}{2m}$.

And the boundary condition ($u(a) = 0$) implies that

$kL = n\pi$ where $n = 1,2,3, ..$

Thus,

$E_n = \frac{\hbar^2 k^2}{2m} = \frac{\hbar^2 \pi^2 n^2}{2m a^2}$.

In the ground state ($n=1$), it becomes

$E_1 = \frac{\hbar^2 \pi^2}{2m a^2}$.

Detailed Solution will be posted in a week.

Reference)
https://en.wikipedia.org/wiki/Particle_in_a_box
3.3 The Eigenvalue Problem for a Particle in a Box, Quantum Physics (third edition), Stephen Gasiorowicz