Understanding the System
The system consists of 12 electrons constrained to move on a circular ring of radius R. Electrons exhibit quantum behavior, resulting in quantized energy levels.
Quantum States on a Ring
- The electrons can be described using quantum mechanics, where their motion is represented by wave functions.
- For a circular ring, the allowed wave functions are given by the solutions to the Schrödinger equation, leading to quantized angular momentum states.
Energy Levels
- The energy levels for a single electron on a ring are given by the formula: E_n = (ħ^2 * n^2) / (2mR^2), where n is the quantum number (n = 0, ±1, ±2, ...).
- Since we are considering 12 electrons, we must account for the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state.
Filling the Energy Levels
- The lowest energy states will be filled first. The first 12 electrons will occupy the quantum states corresponding to n = 0, ±1, ±2, ±3, ±4, and ±5.
- Specifically, the state n = 0 can hold one electron (spin up/down), and states n = ±1 to ±5 can hold two electrons each (due to spin).
Calculating the Ground State Energy
- The ground state energy can be computed by summing the energies of the occupied states:
- E_total = E_0 + E_1 + E_{-1} + E_2 + E_{-2} + E_3 + E_{-3} + E_4 + E_{-4} + E_5 + E_{-5}
- Substituting in the energy formula for each occupied state gives the total ground state energy.
Conclusion
The ground state energy of this system reflects the quantized nature of electrons confined to a circular ring, highlighting the interplay between quantum mechanics and particle statistics.