1.7: The Wavefunction

The wave-like properties of electrons are an example of the "wave-particle duality". Indeed, in the early 20th century, quantum mechanics revealed that a combination of wave and particle properties is a general property of everything at the size scale of an electron.

Without addressing the broader implications of this unusual observation, we will simply note that our purposes require a suitable mathematical description for the electron that can describe both its particle and wave-like properties. Following the conventions of quantum mechanics, we will define a function known as the wavefunction, \(\psi(x,t)\), to describe the electron. It is typically a complex function and it has the important property that its magnitude squared is the probability density of the electron at a given position and time.

\[ P(x,t) = |\psi(x,t)|^{2} = \psi^{*}(x,t)\psi(x,t) \nonumber \]

If the wavefunction is to describe a single electron, then the sum of its probability density over all space must be 1.

\[ \int^{+\infty}_{-\infty}P(x,t)dx=1 \nonumber \]

In this case we say that the wavefunction is normalized such that the probability density sums to unity.