Derive an expression for calculating energy values corresponding a 3 for a particle in one dimensional box.

In quantum mechanics, the energy values of a particle in a one-dimensional box can be derived using the Schrödinger equation. For a particle constrained within a box with potential \( V(x) = 0 \) inside and infinite outside, the wave function \( \psi(x) \) must satisfy the time-independent Schrödinger equation:

\[
-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x)
\]

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