What does the wavefunction actually mean? What are its units? What is the significance of a function that can be imaginary? These are a question that has vexed even the most famous scientists pitting Einstein and Schrödinger on one hand against the Copenhagen group of Bohr, Born, and Heisenberg. The controversy arises because the size of the wavefunction is not determined by Schrödinger's equation. Any multiple N of the wavefunction is also a wavefunction with the same eigenvalue.

Or in other words, the size of a wavefunction is not so important as its shape and that shape is not affected by multiplying by a constant N. The Copenhagen group has chosen to fix the size of the wave function by normalization. A normalized wavefunction has the property that its integral over all space is one:

In the above equation the * means take the complex conjugate. This is included because the wavefunction may have imaginary components. The square of an imaginary number z² =z* z is always real. Born interpreted the square of a normalized wavefunction to represents the differential probability that an electron can be found in the region dx about the point x :

Normalization is just another way of saying that the sum over all space of these individual probabilities dP must equal 1. That is to say that the chance of finding the electron anywhere is 100%.

The physical meaning of the wavefunction is then contained in its square. The square of the wavefunction is interpreted as an electron density or electron distribution function. The square of a one-dimensional wavefunction will have units of probability per unit length. The square of a three-dimensional wavefunction will have units of probability per unit volume. So on a microscopic scale chance and probability are a natural consequence of the quantum interactions between particles. This view of nature was strongly opposed by both Schrödinger and Einstein. To paraphrase Einstein, God would not play dice with the Universe. Einstein even designed many igneous experiments to disprove quantum mechanics. All of these experiments failed. Quantum mechanics and the Copenhagen interpretation is our current best model for the microscopic world of the electron. It is a world of probability waves.

The procedure for normalizing a wavefunction is quite simple. Before I outline the procedure, I am going to introduce a compact notation for integrals introduced to quantum mechanics by Paul Dirac.

The term <F(x)| carries the complex conjugate and is called a bra vector. The other half |G(x)> is called a ket vector. The overlap integral <F(x)|G(x)> has the same function as the dot product between normal Cartesian vectors. A Cartesian dot product tells you how much of the first vector lies in the direction of the second vector. Simarly the overlap integral represents the amount of one function contained in the other function. When the overlap integral is one, as it is between the same normalized function, we can say the two functions occupy exactly the same space.

The graphs at the top of this page give the overlap between a hydrogen 2p and 3p orbital. The product of these orbitals 2p*3p is represented by the dark line. The overlap is the integral or area under this line. In this case the overlap <2p|3p> = 0. The 2p and 3p orbitals span completely different spaces and are said to be orthogonal to each other. They are like perpendicular Cartesian vectors that have a dot product of 0.

The procedure for normalization goes as follows. Let f(x) be an unnormalized eigen function of the Hamiltonian. We would like to multiply f(x) by a normalization constant N such that Nf(x) is normalized. If we integrate a normalized wavefunction the result must be one. Solving for N gives: