Derive the Schrödinger wave equation for a particle of mass  confined to a one-dimensional box of length .

To derive the Schrödinger wave equation for a particle of mass \(m\) confined in a one-dimensional box of length \(L\), we begin with the time-independent Schrödinger equation:

\[
\hat{H} \psi(x) = E \psi(x)
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