Question

Consider an electron that is confined to the \(x y\) -plane by a two-dimensional rectangular infinite potential well. The width of the well is \(w\) in the \(x\) -direction and \(2 w\) in the \(y\) direction. What is the lowest energy that is shared by more than one distinct state, that is, where two different states have the same energy?

Short answer

The lowest energy shared by more than one distinct state for an electron in a two-dimensional rectangular infinite well is given by the formula \(E_\text{lowest} = \frac{5 \hbar^2 \pi^2}{4 m w^2}\), where \(\hbar\) is the reduced Planck constant, \(m\) is the electron mass, and \(w\) is the well width in both \(x\) and \(y\) directions.

Step-by-step solution

Determine the energy equation for an electron in a rectangular well.

The energy of an electron in a two-dimensional rectangular infinite potential well can be written as follows: \[E_{n_x, n_y} = \frac{\hbar^2}{2m} \left[\left(\frac{n_x \pi}{w}\right)^2 + \left(\frac{n_y \pi}{2w}\right)^2\right]\] where \(E_{n_x, n_y}\) is the energy of the electron in the well, \(\hbar\) is the reduced Planck constant, \(m\) is the electron mass, and \(n_x\) and \(n_y\) are the energy state quantum numbers for the \(x\) and \(y\) directions, respectively.

Find the degenerate energy states.

Degenerate states are those states with the same energy. We can find them by making \(E_{n_x, n_y}\) equal to \(E_{n_x', n_y'}\) for different quantum numbers \((n_x, n_y)\) and \((n_x', n_y')\).

Calculate the energy of the lowest degenerate states.

To find the lowest energy shared by more than one distinct state, let's first try the simplest combination of quantum numbers that are not equal, e.g., \((n_x, n_y) = (1, 2)\) and \((n_x', n_y') = (2, 1)\). We then determine if these two states have the same energy: \[E_{1, 2} = \frac{\hbar^2}{2m} \left[\left(\frac{1 \cdot \pi}{w}\right)^2 + \left(\frac{2 \cdot \pi}{2w}\right)^2\right] = \frac{\hbar^2 \pi^2}{2m w^2} (1 + 1)\] \[E_{2, 1} = \frac{\hbar^2}{2m} \left[\left(\frac{2 \cdot \pi}{w}\right)^2 + \left(\frac{1 \cdot \pi}{2w}\right)^2\right] = \frac{\hbar^2 \pi^2}{2m w^2} (4 + \frac{1}{4})\] It is clear that \(E_{1, 2} \neq E_{2, 1}\). Let's try a different combination of quantum numbers: \((n_x, n_y) = (1, 3)\) and \((n_x', n_y') = (3, 1)\). \[E_{1, 3} = \frac{\hbar^2}{2m} \left[\left(\frac{1 \cdot \pi}{w}\right)^2 + \left(\frac{3 \cdot \pi}{2w}\right)^2\right] = \frac{\hbar^2 \pi^2}{2m w^2} (1 + \frac{9}{4})\] \[E_{3, 1} = \frac{\hbar^2}{2m} \left[\left(\frac{3 \cdot \pi}{w}\right)^2 + \left(\frac{1 \cdot \pi}{2w}\right)^2\right] = \frac{\hbar^2 \pi^2}{2m w^2} (9 + \frac{1}{4})\] Now we have found degenerate states: \(E_{1, 3} = E_{3, 1}\). The lowest energy shared by more than one distinct state can be calculated using one of these degenerate states: \[E_\text{lowest} = E_{1, 3} = \frac{\hbar^2 \pi^2}{2m w^2} \left(1 + \frac{9}{4}\right) = \frac{5 \hbar^2 \pi^2}{4 m w^2}\] Thus, the lowest energy shared by more than one distinct state in this two-dimensional rectangular infinite potential well is \(\frac{5 \hbar^2 \pi^2}{4 m w^2}\).

Key concepts

Infinite Potential Well

Imagine a region where a particle is completely free to move without any external forces, but upon reaching certain boundaries, it cannot escape. Such an idealized concept is known as an infinite potential well. In quantum mechanics, it means the potential energy outside the well is infinite, ensuring that a particle within it is 'trapped' and can only occupy specific energy levels.

The infinite potential well is a cornerstone in understanding quantum states. When applied to the exercise provided, a two-dimensional well confines an electron to the plane, creating quantized states that the electron can occupy. A profound implication of this is the quantization of energy, meaning that the electron can only have specific energies, not just any arbitrary value — a principle distinctly different from classical mechanics.
Through mathematical analysis, we can determine exact solutions for the energy levels of a particle in such a well, which helps us comprehend how the confinement affects the behavior of microscopic particles, like the electron in our example.

Quantum States

In the microcosm of quantum mechanics, the notion of quantum states represents the discreet and distinct ways in which a system, like a particle, can exist. Each state corresponds to a particular set of attributes like energy, position, momentum, and spin, and is described using wavefunctions.

Illustrating this through our exercise, the electron in the two-dimensional box has quantized states designated by quantum numbers, namely n_x and n_y. These numbers are essentially the 'addresses' of the electron's quantum states, describing where the electron 'lives' energetically within the confines of the well. As the electron transitions between different quantum states, it can only do so in 'jumps', absorbing or emitting energy equal to the difference between states. This stuttered, quantized behavior is the hallmark of quantum systems and makes quantum mechanics such a unique and intriguing field of study.

Particle in a Box

Delving into the concept of a particle in a box, in the context of our exercise, refers to a theoretical model where we consider a particle that's confined in all directions by impenetrable barriers. For a two-dimensional box, like our infinite potential well with width w and height 2w, the model simplifies the complex reality into something mathematically solvable, revealing the quantized nature of the particle's behavior.

This model is fundamental in quantum mechanics because it is one of the simplest systems that clearly demonstrates principles like wave-particle duality and quantization. When teaching these concepts, it is essential to emphasize that while the box is a simplification, the insights gained from it apply to real-world quantum systems, providing a bridging point between abstract quantum theory and tangible atomic and molecular phenomena.

Degenerate Energy Levels

The term degenerate energy levels refers to the fascinating situation in quantum mechanics where two or more different quantum states have the exact same energy. As seen in the exercise, not all combinations of quantum numbers n_x and n_y led to degeneracy. But for the particular quantum numbers n_x = 1, n_y = 3 and n_x' = 3, n_y' = 1, their respective states were indeed degenerate, having the same energy.

The existence of degenerate energy levels can lead to interesting phenomena, such as greater stability in molecules and the potential for unique reactions in chemical processes. In technological applications, degeneracy is explored in the design of quantum computers and semiconductors. It's a reminder that in quantum mechanics, energy is not just a value but a ladder of discrete steps, with some steps wide enough for multiple states to comfortably 'stand' together.