Question

Twenty non-interacting spin \( \frac{1}{2} \) particles are trapped in a three-dimensional simple harmonic oscillator potential of frequency \( \omega \). The ground state energy of the system, in units of \( \hbar \omega \), is ......... .

Hint:

For non-interacting particles in a quantum harmonic oscillator, the ground state energy is proportional to the number of particles.

Answer 1

Correct Answer: 60

Step 1: Understanding the energy levels.
The ground state energy for each spin \( \frac{1}{2} \) particle in a three-dimensional simple harmonic oscillator is given by the formula \[ E_{\text{ground}} = \frac{3}{2} \hbar \omega. \] Step 2: Calculate the total energy for 20 particles.
Since there are 20 particles and they are non-interacting, the total energy is simply the sum of the individual energies of the particles. Therefore, the total ground state energy is \[ E_{\text{total}} = 20 \times \frac{3}{2} \hbar \omega = 30 \hbar \omega. \] Final Answer: The ground state energy of the system, in units of \( \hbar \omega \), is \( \boxed{30} \).