Let's break down the solution step-by-step for the simulation described in the image: --- ### **Step 1: Understanding the Problem** - **Goal:** Solve the **Time-Independent Schrödinger Equation (T.I.S.E.)** for an electron in a **1D periodic potential** made by 12 atoms (nuclei). - **Equation:** \[ E\Psi(r) = \left[ -\frac{\hbar^2}{2\mu} \nabla^2 + V(r) \right] \Psi(r) \] - \( \Psi(r) \): Electron wavefunction - \( V(r) \): Periodic potential from nuclei - \( E \): Energy of the electron state --- ### **Step 2: Setting Up the Potential \(V(r)\)** - The **bottom plot** shows the **periodic potential** \(V(r)\) (black curve) representing the positive nuclei. - This potential is **repeated for 12 atoms**. --- ### **Step 3: Solving the Schrödinger Equation** - The Schrödinger equation is **solved numerically** for this potential. - **Boundary conditions:** Typically, periodic or fixed boundary conditions are used. --- ### **Step 4: Interpreting the Solutions** - **Wavefunctions:** The blue curves in the middle are the **real part or square modulus** (\(|\psi|^2\)) of the electron wavefunctions for each state. - **Energies:** Listed on the right in electron volts (eV), these are the **quantized energy levels** for the electron in the periodic potential. --- ### **Step 5: Key Observations** - **Lowest Energy States:** Are **most localized** around the nuclei (where the potential is lowest). - **Higher Energy States:** Wavefunctions spread out more, show more oscillations, and have higher energies. - **Energy Gaps:** There are **clusters of energy levels** (bands) and **gaps** between them, demonstrating the **band structure** of solids. --- ### **Final Answer / Summary Table:** | State (from bottom) | Energy (eV) | Key Feature of Wavefunction | |---------------------|-------------|----------------------------------| | 1 | -23.7 | Most localized, least oscillation| | 2 | -23.6 | Slightly less localized | | ... | ... | ... | | 12 | 0.0 | Most delocalized, most oscillation| - **Bottom wavefunctions:** Strongly localized at potential wells (atomic sites). - **Higher wavefunctions:** More nodes, more spread out, higher energy. --- ## **Explanation** This simulation represents how electrons behave in a **crystal lattice**. The **Schrödinger equation** is solved for a **1D string of 12 atoms**. The solutions are: - **Discrete energy states** (bands and gaps). - **Wavefunctions** that change shape with energy. **This is the foundation of band theory in solid-state physics!** --- ### **In summary:** - The plot shows **quantized electron energies** and their **wavefunctions** in a 1D crystal. - The method: Solve the Schrödinger equation for the given periodic potential. - **Final answer:** The energy levels (as listed) and the corresponding wavefunctions, as shown, are the solution set for the electron in the given periodic crystal potential.