Question

What is the minimum possible energy for five (noninteracting) spin- \(\frac{1}{2}\) particles of mass \(m\) in a onedimensional box of length \(L ?\) What if the particles were spin-1? What if the particles were spin- \(\frac{3}{2} ?\)

Short answer

The minimum energy for \(5\) spin-\(\frac{1}{2}\) particles in a one-dimensional box of length \(L\) is given by \[E_{min, 1/2} = 2 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}} + 2 \cdot \frac{4\pi^{2}\hbar^{2}}{2mL^{2}} + 1 \cdot \frac{9\pi^{2}\hbar^{2}}{2mL^{2}}\]If the particles were spin-1 (bosons), the minimum energy would be \[E_{min, 1} = 5 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}}\]And if the particles were spin-\(\frac{3}{2}\) (fermions), the minimum energy would be \[E_{min, 3/2} = 4 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}} + 1 \cdot \frac{4\pi^{2}\hbar^{2}}{2mL^{2}}\]

Step-by-step solution

Determine the energy levels for spin-\(\frac{1}{2}\) particles

The energy levels for particles in a one-dimensional box are given by the formula \[E = \frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}\]where \(n\) is the quantum number that can take integer values starting from 1, \(\hbar\) is the reduced Planck's constant, \(m\) is the mass of the particle and \(L\) is the length of the box. Since the particles are spin-\(\frac{1}{2}\) they are fermions and follow the Pauli Exclusion Principle. Therefore, each energy level can fit two particles, one with spin up and one with spin down. The minimum energy is achieved when the first three energy levels are filled, which requires 2, 2, 1 particles respectively.

Calculate the minimum energy for five spin-\(\frac{1}{2}\) particles

If we substitute the values \(n=1, 2, 3\) into the formula we get:\[E_{\text{min, 1/2}} = 2 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}} + 2 \cdot \frac{4\pi^{2}\hbar^{2}}{2mL^{2}} + 1 \cdot \frac{9\pi^{2}\hbar^{2}}{2mL^{2}}\]

Determine the minimum possible energy for spin-1 particles

Now, for the spin-1 particles, these are bosons. Bosons do not have the restriction of one particle per state, so all five particles could be in the state with \(n=1\). Therefore, \[E_{\text{min, 1}} = 5 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}}\]

Determine the minimum possible energy for spin-\(\frac{3}{2}\)

The spin- \(\frac{3}{2}\) particles are again fermions, but they have four possible states for each energy level (two due to the spin - \(\frac{3}{2}\) and \(\frac{1}{2}\), and two more because each of these spins can be either up or down). So the minimum energy is achieved when the first two energy levels are filled, which requires 4, 1 particles respectively.\[E_{\text{min, 3/2}} = 4 \cdot \frac{\pi^{2}\hbar^{2}}{2mL^{2}} + 1 \cdot \frac{4\pi^{2}\hbar^{2}}{2mL^{2}}\]

Key concepts

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental concept in quantum mechanics. It states that no two identical fermions can occupy the same quantum state within a quantum system simultaneously. Fermions are particles like electrons, protons, and neutrons. They have half-integer spins, such as spin-\(\frac{1}{2}\), spin-\(\frac{3}{2}\), etc.

This principle has significant implications, especially in systems where particles are confined, like electrons in an atom or particles in a box. It leads to the structure and complexity of atomic shells, influencing everything from chemistry to the way stars burn their fuel. In the case of spin-\(\frac{1}{2}\) particles, the Pauli Exclusion Principle insists that each quantum state in a given energy level can be occupied by two particles, one with spin up and the other with spin down.

When we consider a one-dimensional box, the minimum energy configuration for fermions is achieved by filling up available states from the lowest energy upwards, respecting the exclusion principle. This leads to a pattern where particles must occupy increasingly higher energy levels as lower ones are filled.

fermions and bosons

In quantum mechanics, particles are classified into two categories: fermions and bosons. This classification is determined by their intrinsic angular momentum or "spin". Fermions have half-integer spins (such as \(\frac{1}{2}\), \(\frac{3}{2}\), etc.), whereas bosons have integer spins (like 0, 1, 2, etc.). The distinction between these two types of particles leads to different statistical behaviors.

Fermions, following the Pauli Exclusion Principle, cannot occupy the same quantum state at the same time within a system. This results in the 'stacking' effect seen in atomic structures and particle boxes. Consequently, in our exercise, spin-\(\frac{1}{2}\) fermions exhibit layered energy levels with their distribution based on filling principles, leading to discrete energy calculations.

In contrast, bosons can share quantum states, meaning multiple particles can occupy the same state without any restriction. Thus, when dealing with bosons like spin-1 particles, all could theoretically reside in the lowest energy state available. This can profoundly lower the energy of the configuration and leads to phenomena like Bose-Einstein condensates in certain conditions.

quantum numbers

Quantum numbers are critical in defining the state of a particle in a quantum system. They provide a set of numerical values that describe various properties, such as energy, angular momentum, and magnetic moment. In a one-dimensional box scenario, as given in the exercise, the principal quantum number \(n\) plays a crucial role.

The principal quantum number \(n\) determines the energy level of the particle residing in a quantum state. The formula \(E = \frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}\) highlights this relationship, showing how energy levels increase with \(n^2\).

Each different type of particle (fermion or boson) will populate these energy levels differently due to spin and statistical behaviors. In fermionic systems, as the quantum number increases, higher energy levels are occupied according to the exclusion principle. Conversely, for bosons, multiple particles can comfortably occupy the same quantum state corresponding to a single quantum number value.