Read Engel and Reid
Section 14.1
Probability interpretation of the wavefunction
Assume that we can solve for ψ for a particular problem. What does it mean? In what ways can we learn about the properties of the particular system, be it the H atom, water molecule or a complex organic dye?
The answer to these questions, suggested by Max Born in 1926, showed that a wavefunction gives a description of a particle in terms of probabilities. Specifically, for a one-dimensional system, the probability of finding the particle in the region of space from x to x + dx is given by ψ2(x)dx (often this is written allowing for the complex nature of ψ - for example ψ(x) = Ae2x - as a product of ψ and its complex conjugate ψ*).
Thus ψ2 is a probability density (density, since it must be multiplied by infinitesimal length dx to get a probability), and ψ itself is called a probability amplitude.
We can then write down the probability of finding a particle (described by the real wavefunction ψ(x)) in a particular region of space, say between x = a and x = b, a < b,
P(a ≤ x ≤ b) = ∫ab&psi2(x)dx
i.e. the sum over all infinitesimal probabilities P(x)dx over the region.
Born's interpretation imposes very important constraints on the nature of the wavefunction, and it should now be clear why wavefunctions must be bounded (i.e. not reach infinity), single-valued, and continuous. A very important condition on a physically meaningful wavefunction is that of normalisation. Clearly, if we extend the limits of integration to (in words, over all space), the integral must equal unity: the particle must exist somewhere!
∫-∞∞ψ2(x)dx = 1(14.2)
Wavefunctions obtained from the Schrödinger equation will always have a multiplicative constant, N, the normalisation constant.
