GRE Physics GR 1777 Problem Solution
013. Quantum Mechanics (Infinity potential wall)
Solution
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 $latex m$ inside a one-dimensional box are
$latex E_n = \\frac{\\hbar^2 k^2}{2m}$.
And the boundary condition ($latex u(a) = 0$) implies that
$latex kL = n\\pi$ where $latex n = 1,2,3, ..$
Thus,
$latex E_n = \\frac{\\hbar^2 k^2}{2m} = \\frac{\\hbar^2 \\pi^2 n^2}{2m a^2} $.
In the ground state ($latex n=1$), it becomes
$latex E_1 = \\frac{\\hbar^2 \\pi^2}{2m a^2}$.
Answer
(D)
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