2: Waves and the Schrödinger Equation
- Page ID
- 424922
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The aim of this section is to give a fairly brief review of waves in various shaped elastic media—beginning with a taut string, then going on to an elastic sheet, a drumhead, first of rectangular shape then circular, and finally considering elastic waves on a spherical surface, like a balloon. The reason we look at this material here is that these are “real waves”, hopefully not too difficult to think about, and yet mathematically they are the solutions of the same wave equation the Schrödinger wavefunction obeys in various contexts, so should be helpful in visualizing solutions to that equation, in particular for the hydrogen atom.
- 2.1: The One-Dimensional Wave Equation
- The mathematical description of the one-dimensional waves can be expressed as solutions to the "wave equation." It may not be surprising that not all possible waves will satisfy the wave equation for a specific system since waves solutions must satisfy both the initial conditions and the boundary conditions. This results in a subset of possible solutions. In the quantum world, this means that the boundary conditions are responsible somehow for the quantization phenomena in Chapter 1.
- 2.2: The Schrödinger Equation
- Erwin Schrödinger posited an equation that predicts both the allowed energies of a system as well as address the wave-particle duality of matter. Schrödinger equation for de Broglie's matter waves cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof).
- 2.3: Linear Operators in Quantum Mechanics
- An operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another function.
- 2.4: The Schrödinger Equation is an Eigenvalue Problem
- To every dynamical variable in quantum mechanics, there corresponds an eigenvalue equation . The eigenvalues represents the possible measured values of the operator.