## Density Matrix vs State Vector

The difference between a density matrix and a state vector is that the latter corresponds to having more complete information about the system. Let’s see how this plays out in a simple example.

A density matrix that is diagonal with entries of {1}/{2} means that we are talking of just a two dimensional state space (such as that of a single electron whose position we are ignoring), and that we have chosen a particular observable, such as spin component in the z direction, to determine the basis with respect to which you are representing the state.

In this context there are actually many different pure states for which the probability of having spin up or down in the z direction are equal. It could for example be certain to have a particular value in (say) the x direction, or the y, or any other direction perpendicular to the z axis. Each of these corresponds to a “ket” whose components (relative to the z eigenstates) are of the form {1}/{\sqrt{2}} multiplied by complex phase factors (and the different relative phases correspond to different directions of spin in the xy plane). On the other hand, representing the state by that density matrix means that we have prepared the electron in such a way that its spin has equal probability of being measured up and down in the z direction, but also that all ways of getting that result are equally likely so there is also equal probability of getting up or down in any other direction.

## Half Integer Spins

People often ask what it means for an electron to have spin 1/2.

Here is my attempt at an informal explanation.

It means that electrons (and most other elementary particles) are represented by wave functions or fields whose values are not given just by complex numbers (the “scalar” or “spin zero” case), but instead by complex vectors (of “internal” coordinates) which admit a finite dimensional representation of the rotation group. The action of a rotation on a state then corresponds to the usual change of position in space combined(*) with a reorientation of the “internal” coordinates.

It turns out (due to mathematics that I cannot usefully(*) insert here) that the possible results of measuring angular momentum corresponding to “internal” properties of a particle occur with increments of just half of those corresponding to measurements of the classical orbital angular momentum.

And the electron happens to be an example of the simplest kind of non-scalar field.

(*) – The crux of this can perhaps be inadequately explained by saying that the way the internal and external actions of the rotation group have to be related is such that one full rotation in the position space produces a sign reversal in the internal space and so to bring everything back to where it started actually requires two full rotations in the position space.