3: The Schrödinger Equation
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The discussion in this chapter constructs the ideas that lead to the postulates of quantum mechanics, which are given at the end of the chapter. The overall picture is that quantum mechanical systems such as atoms and molecules are described by mathematical functions that are solutions of a differential equation called the Schrödinger equation. In this chapter we want to make the Schrödinger equation and other postulates of Quantum Mechanics seem plausible. We follow a train-of-thought that could resemble Schrödinger's original thinking. The discussion is not a derivation; it is a plausibility argument. In the end we accept and use the Schrödinger equation and associated concepts because they explain the properties of microscopic objects like electrons and atoms and molecules.
- 3.1: Introduction to the Schrödinger Equation
- The Schrödinger equation is the fundamental postulate of Quantum Mechanics.If electrons, atoms, and molecules have wave-like properties, then there must be a mathematical function that is the solution to a differential equation that describes electrons, atoms, and molecules. This differential equation is called the wave equation, and the solution is called the wavefunction. Such thoughts may have motivated Erwin Schrödinger to argue that the wave equation is a key component of Quantum Mechanics.
- 3.2: A Classical Wave Equation
- The easiest way to find a differential equation that will provide wavefunctions as solutions is to start with a wavefunction and work backwards. We will consider a sine wave, take its first and second derivatives, and then examine the results.
- 3.3: Invention of the Schrödinger Equation
- Our goal as chemists is to seek a method for finding the wavefunctions that are appropriate for describing electrons, atoms, and molecules. In order to reach this objective, we need the appropriate wave equation.
- 3.4: Operators, Eigenfunctions, Eigenvalues, and Eigenstates
- he Laplacian operator is called an operator because it does something to the function that follows: namely, it produces or generates the sum of the three second-derivatives of the function. Of course, this is not done automatically; you must do the work, or remember to use this operator properly in algebraic manipulations. Symbols for operators are often (although not always) denoted by a hat ^ over the symbol, unless the symbol is used exclusively for an operator.
- 3.6: The Time-Dependent Schrodinger Equation
- The time-dependent Schrödinger equation, is used to find the time dependence of the wavefunction. This equation relates the energy to the first time derivative analogous to the classical wave equation that involved the second time derivative.
- 3.8: Expectation Values
- These expectation value integrals are very important in Quantum Mechanics. They provide us with the average values of physical properties because in many cases precise values cannot, even in principle, be determined. If we know the average of some quantity, it also is important to know whether the distribution is narrow, i.e. all values are close to the average, or broad, i.e. many values differ considerably from the average. The width of a distribution is characterized by its variance.
- 3.9: Postulates of Quantum Mechanics
- We now summarize the postulates of Quantum Mechanics that have been introduced. The application of these postulates will be illustrated in subsequent chapters.
- 3.E: The Schrödinger Equation (Exercises)
- Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.
David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")