The Lowest Energy Level of a Quantum Mechanical Oscillator

The one-dimensional quantum mechanical oscillator is a theoretical system that models the behavior of a particle confined to a potential well. It exhibits quantized energy levels, meaning that the energy of the system can only take on certain discrete values.

Understanding the Energy Levels
In the case of the quantum mechanical oscillator, the energy levels are given by the equation:

E_n = (n + 1/2)ħw

where E_n is the energy of the nth level, n is the quantum number (0, 1, 2, ...), ħ is the reduced Planck's constant (h/2π), and w is the angular frequency of the oscillator.

Population of Energy Levels
According to the Pauli exclusion principle, each energy level of the oscillator can accommodate a maximum of two electrons, with opposite spins. Therefore, if we have three electrons in the system, they will occupy the lowest three energy levels.

Determining the Lowest Energy
To find the lowest energy of the system, we need to determine the energy of the lowest occupied energy level. In this case, since we have three electrons, the lowest energy level will be the third energy level (n = 2). Plugging this value into the energy equation, we get:

E_2 = (2 + 1/2)ħw = (5/2)ħw

Explanation of the Result
The lowest energy of the system with three electrons is given by (5/2)ħw. This means that the energy of the system is quantized and cannot take on any value between energy levels. The energy levels are equally spaced, with a separation of ħw, and the lowest energy level is offset by (1/2)ħw from zero.

Conclusion
In summary, the lowest energy of a one-dimensional quantum mechanical oscillator with three electrons is given by (5/2)ħw. This result is obtained by considering the quantized energy levels of the oscillator and the population of these levels by the electrons. The energy levels are spaced equally and the lowest energy level is offset from zero by (1/2)ħw.