Problem 61 - University Physics with Modern Physics

Question

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\(N\) elecurons is \(\hbar N^{2} \sqrt{k^{\prime} / m}\) (Hint: See Section 40.4 and the hint given in Problem \(41.59 .\) .)

Step-by-Step Solution

Answer

The minimum energy is \( \hbar N^2 \sqrt{\frac{k^{\prime}}{m}} \).

Step 1: Understanding the Potential Energy of One Electron

Each electron is subject to a simple harmonic oscillator potential, described by the potential energy function \( U(x) = \frac{1}{2} k^{\prime} x^2 \). This indicates that each electron experiences a restoring force proportional to its displacement from the equilibrium position, characteristic of harmonic motion with a constant \( k^{\prime} \).

Step 2: Relating the Potential to Quantum Mechanics

For quantum mechanical systems such as these electrons in a harmonic oscillator potential, the energy levels are quantized. The energy of each electron in the harmonic oscillator potential is given by the formula \( E_n = \hbar \omega (n + \frac{1}{2}) \), where \( \omega = \sqrt{\frac{k^{\prime}}{m}} \) is the angular frequency of oscillation, and \( n \) is a non-negative integer (quantum number).

Step 3: Applying the Pauli Exclusion Principle

According to the Pauli Exclusion Principle, no two electrons can have the same set of quantum numbers in this system. The ground state (minimum energy configuration) for 2\(N\) electrons will involve filling the quantum states starting from the lowest, ensuring each subsequent state is occupied singly until all 2\(N\) electrons are placed.

Step 4: Calculating the Minimum Energy Configuration

For the 2\(N\) electrons, the lowest possible energy state configuration is arranging the first \(N\) states with the first \(N\) values of \( n \) being 0 through \( N-1 \), each filled with two electrons (one with spin-up and one with spin-down). Thus, the system's total energy is the sum of the energies of these \(N\) states, each filled with 2 electrons.

Step 5: Summing the Energies for All Electrons

The total energy of the system can be calculated as \( E_{total} = 2 \sum_{n=0}^{N-1} \hbar \omega (n + \frac{1}{2}) \). Simplifying this sum, we notice: \( \sum_{n=0}^{N-1} n = \frac{N(N-1)}{2} \) and \( \sum_{n=0}^{N-1} \frac{1}{2} = \frac{N}{2} \). Therefore, the total minimum energy is \( E_{total} = 2\hbar \omega \left( \frac{N(N-1)}{2} + \frac{N}{2} \right) = \hbar \omega N^2 \).

Step 6: Substituting Back the Angular Frequency

Substitute back the expression for \( \omega = \sqrt{\frac{k^{\prime}}{m}} \) into the equation for \( E_{total} \). The minimum total energy of the electron system is thus \( E_{min} = \hbar N^2 \sqrt{\frac{k^{\prime}}{m}} \).

Key Concepts

Harmonic Oscillator

A harmonic oscillator in quantum mechanics refers to a system where particles experience a force that pulls them towards a fixed point. This pulling force is proportional to their displacement from that point, which predicts a type of motion called harmonic motion. Each electron in this exercise faces this kind of potential, described by the potential energy function: \[ U(x) = \frac{1}{2} k^{\prime} x^2 \] Here, \( x \) is the displacement, and \( k^{\prime} \) is a positive constant that embodies the strength of the "spring," or restoring force. In simpler terms, this potential tries to "restore" any displacement back towards the equilibrium point. In the world of quantum mechanics, the quantum harmonic oscillator represents a fundamental problem. It serves not only to understand simple oscillating systems but also approximates more complex potentials, closer to realistic scenarios in particle physics and chemistry.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a quantum mechanical principle that restricts identical fermions from sharing the same quantum state. To break it down, if you have a group of electrons (which are known as fermions because of their half-integer spin), no two electrons in an atom, or any system, can have the same set of quantum numbers. This principle has significant implications for arranging electrons in atomic orbitals and plays a pivotal role here. When configuring the lowest energy state, or ground state, of multiple electrons, we must fill available "quantum states" such that no state holds more than one electron of the same spin. In this exercise, the arrangement ensures each state holds two electrons due to their opposite spins (one spin-up, one spin-down), effectively doubling the occupancy while adhering to the exclusion principle rules.

Minimum Energy Configuration

Minimum energy configuration in a quantum system refers to the arrangement of particles such that their total energy is as low as it could possibly be. For the electrons in our problem, achieving the ground state involves placing them in the lowest possible energy levels or quantum states while respecting constraints like the Pauli Exclusion Principle. Since each state can only hold two electrons based on their spin, the first \( N \) quantum numbers from \( 0 \) to \( N-1 \) are filled with two electrons each. This minimizes the overall energy by arranging electrons in a densely packed order, each new electron taking the next available lowest energy state. This creates a configuration that is not only energy efficient but also crucial for understanding stability in atomic and molecular systems.

Quantized Energy Levels

In quantum mechanics, energy levels are said to be quantized, meaning electrons possess specific discrete energy values rather than a continuous range. Devices or systems ruled by quantum mechanics often exhibit these well-defined, distinct energy levels. In the example of a harmonic oscillator, each electron’s possible energy is given by the equation: \[ E_n = \hbar \omega \left( n + \frac{1}{2} \right) \] Here, \( n \) stands for the quantum number, a non-negative integer, \( \hbar \) is the reduced Planck's constant, and \( \omega \) represents the angular frequency. This quantization denotes that each state, or quantized level, corresponds to a specific \( n \) value.When calculating the total energy in the system of electrons, the sum of these individual energy levels gives us quantized insights, which are crucial to understanding phenomena like electronic transitions, stability, and reactions in atoms.