Each of 2 N 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 $h N^{2} \sqrt{k^{\prime} / m} .$ (Hint: See Section 40.5 and the hint given in Problem $41.47 .)$

Key Concepts

Quantum Harmonic Oscillator

The quantum harmonic oscillator is a fundamental model in quantum mechanics used to describe a particle subject to a restoring force proportional to its displacement. Its energy levels are quantized and given by E? = (n + 1/2)??, where ? is the oscillator’s angular frequency. This model is central to understanding systems with quadratic potentials and provides the basic energy spectrum that is used when considering particles in a potential like U(x) = (1/2)k'x².

Pauli Exclusion Principle

The Pauli exclusion principle states that no two identical fermions, such as electrons, can occupy the same quantum state simultaneously. This principle forces electrons in a multi-electron system to occupy different energy levels, particularly when confined in a potential. It is the key reason for the sequential filling of energy levels in many-electron systems, leading to a characteristic ground state energy that includes contributions from higher excited states.

Fermi Energy

Fermi energy refers to the highest occupied energy level of a system of fermions at absolute zero temperature. In a confined system where electrons are filling discrete energy levels, the Fermi energy characterizes the 'top' of the filled states. The knowledge of the Fermi energy is essential to calculate the ground state energy of the system because it defines how many energy levels are occupied, which is especially important when applying the Pauli exclusion principle.

Ground State Energy Calculation in Many-Fermion Systems

Calculating the ground state energy in many-fermion systems involves summing the energies of all filled quantized states given the restrictions imposed by the Pauli exclusion principle. In a system like the one-dimensional harmonic oscillator for electrons, each electron fills a unique energy level. The total ground state energy is obtained by summing the individual energy contributions from these occupied levels, a process integral to understanding the energy scaling with the number of particles in the system.

Transcript

00:01 Okay, so in this problem we have a two -n, sorry, we have two -n oscillators.

00:08 Okay, so i say that this is an oscillator because it's an electron that is free to move along the x -axis.

00:18 Therefore, we can consider this a harmonic oscillator because of the potential energy for each one of the electrons.

00:28 So i say this is an oscillator because the potential energy is one half of kx square.

00:38 Therefore, this is an harmonic oscillator in one -dimensional.

00:42 And we have two n electrons.

00:49 So we have to consider it in this problem the exclusion principle to show that the minimum energy will be the energy described the problem and how we actually is going to do this first of all let's remember what is the energy for one harmonic oscillator and let's see the energy e to the n of one harmonic oscillator is the plank constant multiplied by the frequency that multiply n which is the state of the electron plus one half.

01:34 So this is the energy of one harmonic oscillator and one dimension.

01:42 And what is the exclusion principle? well, we must remember that electrons cannot occupy the same state, the same energy state.

01:55 Therefore, when we're considering the energy state, n equals 1, for example, this state allows, two electrons, only two electrons.

02:09 And that's because we have to also consider the spin.

02:14 And the spin, which is ms, has values of plus and minus one half.

02:23 Therefore, each n quantum number allows two different electrons.

02:32 Okay.

02:32 Okay, so this is really important.

02:35 Each n allows two electrons...