Quantum Mechanics for Scientists and Engineers

So far, we have introduced quantum mechanics through the example of the Schrödinger equation and the spatial and temporal wavefunctions that are solutions to it. This has allowed us to solve some simple but important problems and to introduce many quantum mechanical concepts by example. Quantum mechanics does, however, go considerably beyond the Schrödinger equation. For example, photons are not described by the kind of Schrödinger equation we have considered so far, though they are undoubtedly very much quantum mechanical.

To prepare for other aspects of quantum mechanics and to make the subject easier to deal with in more complex problems, we need to introduce a more general and extended mathematical formalism. This formalism is actually mostly linear algebra. Readers probably have encountered many of the basic concepts already in subjects such as matrix algebra, Fourier transforms, solutions of differential equations, possibly (though less likely) integral equations, or analysis of linear systems, in general. For this book, we assume that the reader is familiar with, at least, the matrix version of linear algebra – the other examples are not necessary prerequisites. The fact that the formalism is based on linear algebra is because of the basic observation that quantum mechanics is apparently absolutely linear in certain specific ways as we discussed previously.

Thus far, we have dealt with the state of the quantum mechanical system as the wavefunction Ψ(r, t) of a single particle.