Question

A system of 8 non-interacting electrons is confined by a 3-dimensional potential $V(r)=\frac{1}{2}m\omega^2 r^2$. The ground state energy of the system in units of $\hbar\omega$ is ............ (Specify your answer as an integer.)

Hint:

Remember that 3D harmonic oscillator shells follow degeneracy $(n+1)(n+2)/2$, each doubled by spin.

Answer 1

Correct Answer: 18

Step 1: Energy levels of 3D harmonic oscillator.
For a 3D harmonic oscillator, energy levels are $E_n = \left(n+\frac{3}{2}\right)\hbar\omega$ with degeneracy $(n+1)(n+2)/2$. Each level can hold $2$ electrons (spin).

Step 2: Fill electrons into shells.
$n=0$: degeneracy 1 → can hold 2 electrons. Energy: $(3/2)$ for each.
$n=1$: degeneracy 3 → can hold 6 electrons. Energy: $(5/2)$ for each.

Step 3: Total energy.
Electrons in $n=0$: $2 \times \frac{3}{2} = 3$.
Electrons in $n=1$: $6 \times \frac{5}{2} = 15$.
Total = $3 + 15 = 18$.