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# Chapter 7: The Schrödinger Equation

In the previous chapter we began our discussion of quantum mechanics with wave functions, energy eigenstates, and Born's probabilistic interpretation. We now consider the process of how to find wave functions that are energy eigenfunctions, the energy of these states, and also how to determine their time evolution. To do so, we need look no further than the Schrödinger equation.

# Table of Contents

## Sections

- Section 7.1: Classical Energy Diagrams.
- Section 7.2: Eigenfunction Shape for Piecewise-constant Potentials.
- Section 7.3: Eigenfunction Shape for Spatially-varying Potentials.
- Section 7.4: Exploring Energy Eigenfunctions Using the Shooting Method.
- Section 7.5: Exploring Many Steps in Infinite and Finite Wells.
- Section 7.6: Exploring Energy Eigenfunctions and Potential Energy.
- Section 7.7: Time Evolution.
- Section 7.8: Exploring Complex Functions.
- Section 7.9: Exploring Eigenvalue Equations.

## Problems

- Problem 7.1: Match the energy eigenfunction with the correct potential energy function.
- Problem 7.2: Determine the energy eigenfunction for a ramped potential energy well.
- Problem 7.3: A time-dependent energy eigenfunction for a plane wave is shown.
- Problem 7.4: Determine the complex wave function given in the top graph.

## Alternate Visualizations

- Section 7.7: Time Evolution.
- Section 7.8: Exploring Complex Functions.
- Problem 7.3: A time-dependent energy eigenfunction for a plane wave is shown.
- Problem 7.4: Determine the complex wave function given in the top graph.