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, ..
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}.
Therefore, the answer is (D).
Detailed Solution will be posted in a week.
3.3 The Eigenvalue Problem for a Particle in a Box, Quantum Physics (third edition), Stephen Gasiorowicz

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