The determinant wavefunction and identical particles

Photo Credit: Samuel Monnier

The determinant wavefunction and identical particles

This article explains how the Pauli principle evolves naturally to the idea of a single determinant wavefunction and how that it turn comes from the idea of identical particles.

Pauli principle

The Pauli principle states that when the coordinates of any pair of electrons is swapped the sign of the wavefunction is reversed,

Pauli proved from relativistic quantum mechanics that this property holds for any set of identical fermions i.e. any particle with half-integer spin. He also showed that for identical bosons – particles with integer spin – swapping the coordinates does not change the sign pf the wavefunction.

Why is Pauli’s principle true?

The real reason is not well understood: the famous physicist Richard Feynman certainly thought so.

However, it is easy to rationalize the Pauli principle. Keep reading.

As you know, the probability of finding two particles at positions \( \B{x}\sub{1} \) and \( \B{x}\sub{2} \) is given by

Now suppose the particles are identical. Then, if we swap their positions, we will not be able to tell that anything has happened. Therefore we expect that reversing the positions for identical particles leads to the same probability,

Naively, the above equation has two solutions:

The solution with the negative sign corresponds to fermions, while the positive sign corresponds to bosons. It is not too difficult to polish up this argument properly by introducing a Hermitian “coordinate swap” operator. Proving that the signs are related to spin is much harder.

Hartree’s orbital-product wavefunction

The simplest wavefunction we can think of is Hartree’s orbital-product wavefunction

Think of this as wavefunction representing electron 1 in orbital \( \phi_1 \), electron 2 in orbital \( \phi_2 \), and so on for all \( N \) particles. Note: if we swap coordinates \( \textbf{x}_i \) and \( \textbf{x}_j \) in the above product, it is the same as leaving the coordinates unchanged but swapping the functions \( \phi_i \) and \( \phi_j \). So we see that after the swap we we will not have the same wavefunction — the electrons will be in different orbitals. So the wavefunction is not the same — let alone obey the Pauli principle where the wavefunction is supposed to be the same except for sign reversal.

How can we make a wavefunction which satisfies the Pauli principle?

Let’s make the problem a bit easier. Suppose we wanted a wavefunction where swapping electrons led to no change. How to achieve that?

Symmetric orbital-product wavefunction

One way is to take any wavefunction, permute the particle coordinates in all ways, then add all these permuted-coordinate wavefunctions together. We call this a symmetrized wavefunction because if we swap any pair of coordinates in this sum it will remain unchanged. It remains the same because in the list of all permuted wavefunctions a particular term always occurs with a unique partner where the two coordinates are swapped. We write the symmetrized wavefunction like this:

The factor \( N! \) is required to ensure the wavefunction is normalized (assuming the orbitals \( \phi_i \) are normalized also). Incidentally, such a wavefunction is appropriate to describe a boson wavefunction which obeys the Pauli principle for bosons. But we are interested in electrons, which are fermions, so the wavefunction should change sign. Since the permuted list contains all terms in pairs, it should be clear to you that for the wavefunction to change sign every term must be subtracted from its permuted partner if we want the wavefunction to change sign. The result is

Antisymmetric orbital-product wavefunction

The \( \sigma_u \) is either +1 or -1 depending on whether the permutation \( P_u \) if even or odd. The whole wavefunction is antisymmetrized.

Note that any wavefunction can be symmetrized or antisymmetrized, not just the Hartree orbital-product wavefunction, as we have done.

Determinant wavefunction

Finally, the antisymmetrized orbital-product wavefunction can be written in terms of a determinant like this

This is called a single determinant wavefunction. It forms the basis of many methods for approximately solving the Schrodinger equation.