Question

Make rough sketches of the wave functions at the top of the \(n=2\) band and the bottom of the \(n=3\) band for a one-dimensional "crystal" consisting of seven finite wells. Explain why these two states of roughly equal wave number have vastly different energies.

Short answer

The wave function at the top of the \(n=2\) band has 2 nodes per well and that at the bottom of the \(n=3\) band has 3 nodes per well, both representing roughly equal wave numbers. However, the \(n=3\) state has significantly more energy than the \(n=2\) state due to it being in a higher energy band and its ability to penetrate farther into the barriers between the wells, allowing it to sample a larger segment of the energy continuum.

Step-by-step solution

Sketch the Wave functions

Begin by sketching the wave functions at the top of the \(n=2\) band and the bottom of the \(n=3\) band. It's important to remember that in the model of a one-dimensional crystal with seven finite wells, the wave functions represent the likelihood of finding an electron within the crystal. At the top of the \(n=2\) band, the wave function will have 2 nodes (points where the wave function equals to zero) per well, while at the bottom of the \(n=3\) band, the wave function will have 3 nodes per well.

Comparing the Wave functions

Upon comparing the wave functions, it is observed that both states have roughly equal wave numbers, represented by the number of times that the wave function goes through a full periodicity cycle in its spatial distribution per well. However, the energy associated with each state is vastly different. This is due to the fact that higher bands correspond to higher energy levels.

Explain the Energy differences

The reason for the difference in energy levels, despite having roughly equal wave numbers, stems from the nature of quantum confinement in the finite wells. In quantum physics, a particle confined to a finite well has quantized energy levels. The \(n=3\) state resides in a higher energy band and hence has more energy than the \(n=2\) state, despite having the same wave number. This is because the wave function for the \(n=3\) state can penetrate farther into the barriers between the wells, which allows it to sample a larger segment of the energy continuum, resulting in a greater overall energy.

Key concepts

Wave Functions

A wave function is a fundamental concept in quantum mechanics, representing a mathematical description of the quantum state of a particle or system, like an electron in a 'crystal' made up of finite potential wells. In essence, it's a tool to calculate the probability of finding a particle in a particular position upon measurement.

The wave function is complex and can take on different shapes, determined by quantum numbers, such as principal quantum number (\( n \)). In one-dimensional systems, such as the crystal mentioned in our exercise, each wave function corresponds to a certain energy level and the number of nodes - points where the probability is zero - increases with the state's energy. A node is akin to a silent point in the oscillation of a plucked guitar string, indicating where the probability of finding an electron is zero. In the exercise, the top of the band for state \( n=2 \) has two nodes per well, while the bottom of the band for state \( n=3 \) has three nodes per well, indicating distinct quantum states.

Understanding the nuances of wave functions is critical in predicting how electrons will behave in different quantum states, which has massive implications in the field of electronics and material science.

Finite Potential Wells

The concept of a finite potential well is a model for describing particles, like electrons, that are confined to specific regions in space. If we visualize a well as a pit, the walls of this pit represent potential energy barriers that the electron must have enough energy to overcome to escape. In a one-dimensional crystal consisting of multiple finite wells, these 'pits' are periodic potentials that an electron can be trapped in.

In our exercise, seven finite wells represent the periodic potential in a simplified one-dimensional crystal model. The electron can be found within these wells, but it encounters barriers as it tries to move from one well to another. This confinement within the wells quantizes the energy of the system, leading to discrete energy levels. The electron's wave function reflects this confinement, showing areas of high probability within the wells and low probability as it penetrates the barriers.

This becomes especially relevant in semiconductor physics, where the control of electron motion through such potential wells is the cornerstone of designing electronic components.

Quantum Energy Levels

Quantum energy levels are like the rungs of a ladder that an electron can occupy in a quantum system such as an atom or a quantum well. Each level corresponds to a specific amount of energy the electron has when it's in that state. Just like a person can only stand on the rungs of the ladder and not in between, an electron can only exist at specific energy levels, which are determined by the system's boundaries and the forces acting on the electron.

In the context of the finite well 'crystal' in the exercise, the distinct energy bands reflect the allowed energy ranges for the electrons. At the bottom of each band, the energy is at a minimum for that quantum state. As the state number \( n \) increases, the energy also increases. Therefore, the taller the ladder, the higher the energy level an electron can occupy. It's crucial for understanding quantum mechanics to appreciate that, despite having similar wave numbers, the \( n=3 \) state has more energy than the \( n=2 \) state due to quantum confinement affecting the energy levels differently, even within the same system.