Word Count : 342

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To calculate the ground state energy for a system of five identical spin-\(\frac{1}{2}\) particles in a one-dimensional simple harmonic oscillator potential, we need to consider both the quantum mechanics of the system and the Pauli exclusion principle for fermions.

The total energy of a particle in a one-dimensional simple harmonic oscillator potential is given by:

\[
E_n = \left( n + \frac{1}{2} \right) \hbar \omega
\]

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