Coordinates in Wave Mechanics

The second problem with visualizing wave functions is that they typically exist in a space of very many dimensions; three dimensions for each particle in the system. A hydrogen atom, with a proton and an electron, is already a six-dimensional system, and it only gets worse as more particles are added. This is a somewhat intractable problem, but inroads can be made. To begin, it is always possible to define three ``center of mass'' coordinates for the system as a whole. These coordinates describe the system, considered as a single entity, from the perspective of an external observer. The remaining coordinates describe the ``internal'' properties of the system. For example, in the standard treatment of the hydrogen atom, the ``orbitals'' are functions of the internal coordinates, while the atom as a whole has a center-of-mass wave function. We may suppress two of the center-of-mass coordinates, say y and z, and the external wave function can then be depicted as a complex-valued function of a single variable, as on the previous page.

Such a wave function is an eigenstate of momentum. That is, the momentum operator, $-\imath \hbar \frac{\partial}{\partial x}$, operating on the wave function $\Phi = exp^{-\imath kx}$, produces a constant times that wave function, $\Phi = \hbar k exp^{-\imath kx}$. The constant $\hbar k$ is the momentum eigenvalue.