In the last lecture, we saw that the Bohr model is able to predict the allowed energies of any single-electron atom or cation. However, the Bohr model is, by no means, a general approach and, in any case, it relies heavily on classical ideas, clumsily grafting quantization onto an essentially classical picture, and therefore, provides no real insights into the true quantum nature of the atom.
Any rule that might be capable of predicting the allowed energies of a quantum system must also account for the particle-wave duality and include a wave-like description for particles. In 1926, the Austrian physicist Erwin Schrödinger posited an equation that predicts both the allowed energies of a system as well as the probability of finding a particle in a given region of space. The equation, known as the Schrödinger wave equation, does not yield the probability directly, in fact, but rather the probability amplitude alluded to in the previous lecture. This amplitude function is, in general, a complex function denoted (for a single particle in one spatial dimension) and is referred to as the wave function. It is related to the probability as follows:
The Schrödinger equation cannot be derived from any more fundamental principle. However, in order to motivate it, let us use the assumption that should have a wave-like form. Thus, consider a free particle of mass and momentum . Recall the de Broglie hypothesis stating that the particle has a wavelength given by
We can always set up the Schrödinger equation via the following simple prescription. Start with an expression for the classical energy. In this case, for a free particle
Since is a probability, we require that the probability of finding the particle somewhere in space be exactly 1. That is, we require that the probability that be 1, which means
Finally, we need to specify how behaves at the physical boundaries of the space we are working in. These conditions are known as boundary conditions.
Once we specify the Schrödinger equation, the boundary conditions on and the normalization condition, we have all the information we need to calculate both the allowed energies and the wave function . We will see shortly how this prescription is applied to a few simple examples.
Before we embark on this, however, let us pause to comment on the validity of quantum mechanics. Despite its weirdness, its abstractness, and its strange view of the universe as a place of randomness and unpredictability, quantum theory has been subject to intense experimental scrutiny. It has been found to agree with experiments to better than 10% for all cases studied so far. When the Schrödinger equation is combined with a quantum description of the electromagnetic field, a theory known as quantum electrodynamics, the result is one of the most accurate theories of matter that has ever been put forth. Keeping this in mind, let us forge ahead in our discussion of the quantum universe and how to apply quantum theory to both model and real situations.