Quantum mechanics tells us that we must use wavefunctions in order to make accurate predictions about electrons and nuclei. Unfortunately, we are not told how to calculate these functions. The most quantum mechanics can tell is what conditions a wavefunction must satisfy in order to be considered acceptable.
Wavefunctions must satisfy a number of conditions, but the most important one is the Schrödinger wave equation (historical note):
where
The form of the Schrödinger equation shown above is appropriate for one electron moving non-relativistically (this means "much slower than the speed of light") in one dimension (x) under the influence of some potential energy, V(x), that may vary with the x-coordinate of the electron.
The equation says that, to be valid, a wavefunction, ø(x,t), must give the same result when it is operated on by the partial time derivative on the left, and when it is operated on by the partial second derivative and potential energy on the right.
The form of the Schrödinger equation given above is known as the "time-dependent" Schödinger equation because it specifies how a wavefunction must evolve in time. The equation is intimidating. However, if the wavefunction can be factored as follows:
i.e., if the wavefunction can be factored into the product of an x-dependent function and an exponential time function, then a simpler constraint can be placed on the spatial function. We can obtain this simpler constraint by substituting the above form of ø(x,t) into the Schrödinger equation and seeing how it behaves.
We see that factoring the wavefunction in this way allows us to separate the Schödinger equation into two simpler equations. For example, by equating the right hand side of the first line with the fourth line we obtain an equation that depends only on time:
Equating the last two lines, on the other hand, produces an equation that depends only on position x:
The latter equation is known as the "time-independent" Schrödinger equation. It defines the condition that the spatial part of the wavefunction must satisfy to be acceptable, and is the equation most often used by chemists.
and also
The next page explores some of the implications of this equation, and some of the terminology associated with it.
(last updated 6/8/97)