Problem 23 - Fundamentals of Physics

Question

A cubical box of widths \(L_{x}=L_{y}=L_{z}=L\) contains eight electrons. What multiple of \(h^{2} / 8 m L^{2}\) gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Step-by-Step Solution

Answer

The energy of the ground state system is 36 times \(\frac{h^{2}}{8mL^{2}}\).

Step 1: Understanding the Problem

We have eight electrons in a cubical box with equal side lengths. The problem is asking for the ground state energy of the system as a multiple of \(\frac{h^{2}}{8mL^{2}}\). Electrons are fermions and follow the Pauli exclusion principle, meaning each electron state can be occupied by at most two electrons (due to spin).

Step 2: Energy in a 3D Box

In a three-dimensional box, the energy levels for a particle can be calculated using \[E_{n_x,n_y,n_z} = \frac{h^{2}}{8mL^{2}} (n_x^2 + n_y^2 + n_z^2)\] where \(n_x, n_y, n_z\) are quantum numbers that can take positive integer values. For non-interacting electrons, we populate these states starting from the lowest energy upwards.

Step 3: Applying Pauli Exclusion Principle

Each energy state \((n_x, n_y, n_z)\) can contain 2 electrons due to their spin states. The lowest energy state, \(E_{1,1,1}\), can hold 2 electrons, followed by \(E_{1,1,2}, E_{1,2,1}, E_{2,1,1}\) etc.

Step 4: Determining Quantum Numbers for the Ground State

For eight electrons, distribute them in the following order of increasing energy: 1. \((n_x, n_y, n_z) = (1,1,1)\) with 2 electrons2. \((n_x, n_y, n_z) = (1,1,2)\) with 2 electrons3. \((n_x, n_y, n_z) = (1,2,1)\) with 2 electrons4. \((n_x, n_y, n_z) = (2,1,1)\) with 2 electrons.

Step 5: Calculate Energy Levels

Calculate energy for each level using the formula from Step 2:- For \((1,1,1)\), energy \(E_{1,1,1} = 3\frac{h^{2}}{8mL^{2}}\)- For \((1,1,2)\), \((1,2,1)\), and \((2,1,1)\), energy \(E = 6\frac{h^{2}}{8mL^{2}}\).- Each of these states hold 2 electrons.

Step 6: Calculate Total Energy

The total energy of the system is the sum of individual energies:\[ E_{total} = 2 \times 3\frac{h^{2}}{8mL^{2}} + 2\times(6\frac{h^{2}}{8mL^{2}}) + 2\times(6\frac{h^{2}}{8mL^{2}}) + 2\times(6\frac{h^{2}}{8mL^{2}}) = 36\frac{h^{2}}{8mL^{2}}.\]

Step 7: Simplify Total Energy

Since every term is divisible by \(\frac{h^{2}}{8mL^{2}}\), simplify the expression:\[ E_{total} = 36 \times \frac{h^{2}}{8mL^{2}}.\]Therefore, the total energy in terms of \(\frac{h^{2}}{8mL^{2}}\) is 36.

Key Concepts

Pauli Exclusion Principle

The Pauli exclusion principle is a fundamental principle in quantum mechanics that explains the behavior of fermions, a class of particles that includes electrons. According to this principle, no two fermions can occupy the same quantum state simultaneously. For electrons, which also possess a property called "spin," this means that each energy state in an atom or system can be occupied by at most two electrons—one with "spin-up" and the other with "spin-down."

• Quantum States: These are defined by a set of quantum numbers that describe the energy, angular momentum, and other properties of electrons.


• Spin: A quantum property of particles that comes in two possible states: +1/2 (up) or -1/2 (down).


• Application: In a confined system like a box, the principle determines the distribution of electrons across different energy levels.

The exclusion principle ensures that electrons fill up available energy states in a way that no two electrons share the same set of quantum numbers, keeping the system stable and adhering to this unique quantum behavior.

Electron Spin

Electron spin is an intrinsic form of angular momentum carried by electrons, which plays a crucial role in quantum mechanics and the structuring of atoms. Unlike classical angular momentum, spin does not correspond to any physical spinning motion. It's a purely quantum mechanical property.

• Two Spin States: Electrons can exist in one of two spin states: "spin-up" or "spin-down," which can also be represented as +1/2 and -1/2.


• Magnetic Moment: The spin of an electron creates a small magnetic moment, meaning it can interact with magnetic fields, contributing to magnetism in some materials.


• Filling Energy Levels: In a multi-electron system, spin allows two electrons to occupy the same energy level as long as they have opposite spins.

In our example of electrons in a box, each spatial energy state can hold two electrons due to spin, adhering to the Pauli exclusion principle. This spin-state doubling is critical for determining how electrons distribute themselves in different quantum systems.

Particle in a Box

The "particle in a box" model is a fundamental quantum mechanics problem that describes a particle free to move in a confined space but not beyond its boundaries. This model helps in understanding quantized energy levels, as opposed to continuous energies in classical physics.

• Quantum Numbers: Each possible state of the particle is associated with a set of quantum numbers corresponding to its energy levels.


• Quantization of Energy: In a three-dimensional box with sides of length \(L\), the possible energies are given by \[E_{n_x,n_y,n_z} = \frac{h^{2}}{8mL^{2}} (n_x^2 + n_y^2 + n_z^2)\], where \(n_x, n_y, n_z\) are integers.


• Electrons in Boxes: When applied to electrons in a cubical box (as in the exercise), the concept allows us to calculate the energy levels these electrons can occupy, respecting quantum rules.

By calculating these energies and filling them according to the Pauli exclusion principle, we determine the ground state energy as a sum of the energies associated with occupied states. This visualization simplifies learning about quantum confinement and energy quantization.