Five identical noninteracting particles are place in an...

Five identical noninteracting particles are place in an infinite square well with $L=1.0 \mathrm{nm} .$ Compare the lowest total energy for the system if the particles are $(a)$ electrons and $(b)$ pions. Pions have symmetric wave functions and their mass is $264 m_{e}$.

Key Concepts

Infinite Square Well

This is a basic model in quantum mechanics where a particle is confined within perfectly rigid boundaries with zero potential energy inside and infinite potential energy outside. The model leads to a set of quantized energy levels due to the boundary conditions imposed on the wavefunction, making it a useful system for understanding quantum confinement and energy quantization.

Energy Quantization

In confined systems like the infinite square well, the energy levels are quantized and determined by the well’s dimensions, the mass of the particle, and the quantum number. The energy for each level is given by an expression that includes the square of the quantum number, indicating that as the quantum number increases, the energy increases quadratically.

Mass Dependence in Quantum Systems

The mass of the particle appears in the denominator of the energy level expression for the infinite square well, meaning that heavier particles have lower energy eigenvalues compared to lighter ones when confined in the same potential. This mass dependence is crucial when comparing systems with different particle masses in otherwise identical quantum wells.

Quantum Statistics

Quantum statistics governs how identical particles populate the available quantum states. Fermions and bosons obey different statistical rules; fermions follow the Fermi–Dirac statistics, which include the Pauli exclusion principle that prevents more than one identical fermion from occupying the same quantum state, while bosons obey Bose–Einstein statistics and can all condense into the same state if energetically favorable.

Pauli Exclusion Principle in Fermion Systems

For fermions, such as electrons, the Pauli exclusion principle restricts the occupancy of quantum states. This means that even in a noninteracting system, fermions must fill up the available energy levels starting from the lowest, with a limited occupancy per level (accounting for additional degeneracies like spin), which affects how the system's total ground state energy is determined.

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