Problem 26 - Fundamentals of Physics

Question

A rectangular corral of widths \(L_{x}=L\) and \(L_{y}=2 L\) holds an electron. What multiple of \(h^{2} / 8 m L^{2}\) where \(m\) is the electron mass, gives (a) the energy of the electron's ground state, (b) the energy of its first excited state, (c) the energy of its lowest degenerate states, and (d) the difference between the energies of its second and third excited states?

Step-by-Step Solution

Answer

(a) \(\frac{5}{4}\), (b) \(\frac{17}{4}\), (c) \(2\), (d) \(\frac{17}{32}\).

Step 1: Understand the Problem

We need to find the energies of an electron in different states within a rectangular corral with dimensions \(L_x = L\) and \(L_y = 2L\). The goal is to express these in terms of multiples of \(h^2/8mL^2\), where \(m\) is the electron's mass.

Step 2: Set Up the Energy Formula

The energy levels of an electron in a rectangular corral are given by: \( E_{n_x,n_y} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} \right) \). Here, \(n_x\) and \(n_y\) are quantum numbers for each dimension.

Step 3: Calculate Ground State Energy

For the ground state, \(n_x = 1\) and \(n_y = 1\). Substitute these into the energy formula: \[ E_{1,1} = \frac{h^2}{8m} \left( \frac{1^2}{L^2} + \frac{1^2}{(2L)^2} \right) = \frac{h^2}{8mL^2} \left( 1 + \frac{1}{4} \right) = \frac{5h^2}{32mL^2} \]. This is \( \frac{5}{4} \times \frac{h^2}{8mL^2} \).

Step 4: Calculate First Excited State Energy

The first excited state occurs when either \(n_x\) or \(n_y\) is increased. Choose \(n_x = 2, n_y = 1\) (as it gives the lowest increase): \[ E_{2,1} = \frac{h^2}{8m} \left( \frac{2^2}{L^2} + \frac{1^2}{(2L)^2} \right) = \frac{h^2}{8mL^2} \left( 4 + \frac{1}{4} \right) = \frac{17h^2}{32mL^2} \]. This corresponds to \( \frac{17}{4} \times \frac{h^2}{8mL^2} \).

Step 5: Calculate Lowest Degenerate Energy States

Degenerate states have the same energy. Check combinations \((n_x=2, n_y=1)\) and \((n_x=1, n_y=2)\):\( E_{1,2} = \frac{h^2}{8m} \left( \frac{1^2}{L^2} + \frac{2^2}{(2L)^2} \right) = \frac{h^2}{8mL^2} \left( 1 + 1 \right) = \frac{h^2}{4mL^2} \). This corresponds to \(2\times \frac{h^2}{8mL^2} \). This energy is equal to that of \((n_x=2,n_y=1)\).

Step 6: Calculate Energy Difference between Second and Third Excited States

Second excited state could be \((n_x=3, n_y=1)\) or \((n_x=1,n_y=3)\) and third might be \((n_x=2, n_y=2)\). Calculate \(E_{3,1}\) and \(E_{1,3}\): \( E_{3,1} = \frac{h^2}{8mL^2} \left(9 + \frac{1}{4}\right) = \frac{37h^2}{32mL^2}\), and \(E_{2,2} = \frac{h^2}{8mL^2} \left(4 + 1\right) = \frac{5h^2}{8mL^2}\). The difference is \(E_{3,1} - E_{2,2} = \frac{37h^2}{32mL^2} - \frac{20h^2}{32mL^2} = \frac{17h^2}{32mL^2}\).

Key Concepts

Energy Levels

In quantum mechanics, energy levels are a fundamental concept that describes the specific energies that a confined electron can occupy. These energy levels are quantized, meaning an electron can only possess certain discrete energy values.
A simple analogy is to think of these levels as steps on a staircase. An electron moving in a fixed space, such as within a rectangular corral, can only "stand" on particular steps. Each step represents a distinct energy state, and moving an electron from one step to another requires a specific amount of energy input or release.
Mathematically, energy levels are determined by quantum numbers, which denote the allowed states that the system can be in. The equation used in our exercise, \(E_{n_x,n_y} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} \right)\), helps us calculate these energy values for different combinations of quantum numbers. Here, each combination corresponds to a different energy level, allowing us to explore how the electron behaves under quantum constraints.

Electron Confinement

Electron confinement describes how electrons are trapped within a defined space, such as a potential well or a corral. When electrons are confined, they are restricted in movement, leading to quantized energy states.
In our example, electrons are constrained in a rectangular space with dimensions \(L_x = L\) and \(L_y = 2L\), mimicking a tiny box or corral. Inside this box, electrons can only reside at specific positions, which in turn affects their energy and other properties.
Confinement plays a crucial role in determining an electron's energy levels. Within this small, restricted space, rather than moving freely, electrons "bounce" between the corral's walls. This motion creates standing wave patterns, which directly influence the quantization of energy levels. The smaller the space, the greater the energy required for electrons to fit within these quantized states.

Quantum States

Quantum states refer to the set of all possible states that an electron can inhabit in a quantum system. Each state is characterized by specific quantum numbers, such as \(n_x\) and \(n_y\) as seen in our problem, representing different angular momenta or spatial quantizations.
These quantum states determine the unique properties of an electron, including its energy level, position probability, and momentum.

• Ground State: This is the lowest energy state an electron can occupy, with all quantum numbers set to their minimum allowed values.
• Excited State: Any state with higher energy than the ground state, where one or more of the quantum numbers has increased.

Understanding quantum states is key because they dictate how electrons transfer energy, participate in chemical reactions, and influence electrical properties in materials. Quantum state knowledge enables us to predict specific behaviors crucial for fields like material science and electronics.

Degeneracy in Quantum Systems

Degeneracy occurs in quantum systems when two or more independent quantum states share the same energy level. This phenomenon is quite common in systems that have symmetrical structures or specific dimensional constraints.
In our exercise example, the states \((n_x=2, n_y=1)\) and \((n_x=1, n_y=2)\) are degenerate because they both yield the same energy value according to the provided formula.
Degeneracy is significant because it can affect the statistical behavior of systems, particularly at the macroscopic level, leading to important thermal, electronic, and magnetic properties. It also provides insight into the system's symmetry—it hints at deeper underlying principles dictating the system's structure and behavior. Understanding degeneracy helps in predicting the occupancy of energy states and is vital when dealing with complex quantum systems like those found in semiconductor technologies.