Each of 2N electrons (mass m ) is free to move along the x...

Each of 2$N$ electrons (mass $m )$ is free to move along the $x$ -axis. The potential-energy function for each electron is $U(x)=\frac{1}{2} k^{\prime} x^{2},$ where $k^{\prime}$ is a positive constant. The electric and magnetic interactions between electrons can be ignored. Use the exclusion principle to show that the minimum energy of the system of 2$N$ electrons is $\hbar N^{2} \sqrt{k^{\prime} / m} .$ Hint: See Section 40.5 and the hint given in Problem $41.47 . )$

Key Concepts

Fermionic Ground State Energy and Fermi Level

For a system of fermions in a quantized potential, the electrons fill the lowest available energy levels due to the exclusion principle. As electrons are added, the highest occupied energy level defines the Fermi level. The total ground state energy of the system is obtained by summing the energies of all filled states. In the specific case of electrons in a one-dimensional harmonic oscillator potential, this summation leads to a collective ground state energy that scales with the square of the number of electrons, reflecting how the exclusion principle forces electrons into increasingly higher energy states as their number increases.

Quantum Harmonic Oscillator

The quantum harmonic oscillator describes a particle confined in a quadratic potential, where the energy levels are quantized. The energy eigenvalues are given by E? = ??(n + 1/2), with n being a non-negative integer and ? the oscillator frequency, which in this case is related to the constants of the system. Each electron in the system experiences such a quantized spectrum, and the properties of these discrete levels are crucial for understanding the total energy when the electrons fill up the available states.

Pauli Exclusion Principle

This principle states that no two fermions (particles with half?integer spin, such as electrons) can occupy the same quantum state simultaneously. In the context of many-electron systems, it forces electrons to fill up the available energy levels one by one, with at most one electron per state (or two with opposite spins in a spin-degenerate system). Consequently, for a system of electrons confined in a potential, the ground state energy is determined not just by the lowest single-particle state but by the cumulative energy of all occupied states up to the Fermi level.

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