3.4 Two-particle spin states

If we have two indistinguishable^[1] electrons, we can define a two-particle spin state. Due to symmetry and the rules of quantum mechanical addition of angular momentum, there are four possible spin states in total. These spin states are represented using the quantum numbers and , as introduced in Section 3.1, but now the quantum numbers are the sum of the values for the individual electrons. Therefore the spin quantum number is or and, for the case when , the spin magnetic quantum is , while for the case when , we only have .

Such a two-particle spin state therefore can only have an overall spin function which is either symmetric or antisymmetric with respect to exchange of the electrons. The symmetric spin state is referred to as a triplet because there are three possible symmetric combinations:

where the first arrow in each ket refers to particle 1 and the second to particle 2. The antisymmetric spin state is referred to as a singlet because there is only one possible combination:

Equation label:(10)

You can see that the states and can be factorised into and respectively, where the subscripts label each particle. In contrast, the states and cannot be factorised into the product of a particle 1 state multiplied by a particle 2 state. Two-particle states which cannot be factorised are known as entangled states and said to exhibit entanglement.

Exercise 10

Verify that the three spin kets

are symmetric with respect to swapping the labels of the particles.

Answer

Starting with

exchanging the particle labels and then rearranging gives

Since this final expression is identical to the initial expression, this shows is symmetric to swapping particle labels.

For and the particle labels are interchanged and re-ordered (perfectly acceptable!) to get the same expressions as required.

• 1 Particles are indistinguishable when they are identical (i.e. they have the same intrinsic properties like mass, charge, and spin) and they are so close together that their wavefunctions overlap so that we cannot tell them apart, even in principle.