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Physlet® Quantum Physics 2E: Quantum Theory
Much of introductory quantum mechanics involves finding and understanding the solutions to Schrödinger's wave equation and applying Born's probabilistic interpretation to these solutions.1 We begin by first reviewing some of the basic properties of classical probability distributions before discussing quantum-mechanical probability and expectation values.2
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.
Our study of one-dimensional quantum mechanics begins with what you may think is a simple problem: that of a free particle. The Schrödinger equation for this system is as simple as it gets, after all, V(x) = 0. However, it is not so simple to construct a particle-like solution should we wish to compare quantum mechanics to classical mechanics. In order to do this we must add together an infinite number of individual solutions to the free Schrödinger equation (energy eigenfunctions) to get a (Gaussian) wave packet that in many ways behaves like a classical free particle.
We now consider another one-dimensional problem, the scattering problem. In doing so we need to consider scattering-type solutions and what they mean. For standard scattering situations, the wave functions we use are usually those valid for regions of constant potential energy such as complex exponentials (plane waves) when E > V0 and real exponentials when E < V0.1
The infinite square well is the prototype bound-state quantum-mechanical problem. Despite this being a standard problem, there are still many interesting subtleties of this model for the student to discover. Our understanding of this problem, whether it be in regards to energy eigenfunctions in position or momentum space, time evolution, or the dynamics of wave packets, will be useful for the study of more realistic problems.
Having studied the infinite square well, in which V = 0 inside the well and V = ∞ outside the well, we now look at the bound-state solutions to other wells, both infinite and finite. The wells we will consider can be described as piecewise constant: V is a constant over a finite region of space, but can change from one region to another. We begin with the finite square well (we studied scattering-state solutions to the finite well in Section 9.6 where V = |V0| inside the well and V = 0 outside the well. Solutions are calculated by piecing together the parts of the energy eigenfunction in the two regions outside the well and the one region inside the well.
In this chapter we will consider eigenstates of potential energy functions that are spatially varying, V(x) ≠ constant. We begin with the most recognizable of these problems, that of the simple harmonic oscillator, V(x) = mω2x2/2, is perhaps the most ubiquitous potential energy function in physics. Several systems in nature exactly exhibit the harmonic oscillator's potential energy, but many more systems approximately exhibit the form of the harmonic oscillator's potential energy.1
Thus far we have concerned ourselves with one-dimensional (non-relativistic) problems in quantum mechanics. We now consider the extension to systems with more than one degree of freedom in more than one dimension. Doing so extends our discussion of quantum-mechanical systems to include more real-world-like situations. We finish with the Coulomb potential which is the potential energy function responsible for basic atomic structure.