Non-interacting Many-Particle Systems
Quantum mechanical properties of a single particle are an important starting point for studying quantum mechanics, but in real experimental and practical situations, you will rarely deal with just a single particle. Most frequently you encounter systems consisting of many (from two to infinity) interacting particles. The main difficulty in dealing with many-particle systems comes from a significantly increased dimensionality of space, where all possible states of such systems reside. In Sect. 9.4 you saw that the states of the system of two spins belong to a four-dimensional spinor space. It is not too difficult to see that the states of a system consisting of N spins would need a 2 N -dimensional space to fit them all. Indeed, adding each new spin 1/2 particle with two new spin states, you double the number of basis vectors in the respective tensor product, and even the system of as few as ten particles inhabits a space requiring 1024 basis vectors. More generally, imagine that you have a particle which can be in one of M mutually exclusive states, represented obviously by M mutually orthogonal vectors (I will call them single-particle states), which can be used as a basis in this single-particle M-dimensional space. You can generate a tensor product of single-particle spaces by stacking together M basis vectors from each single-particle space. Naively you might think that the dimension of the resulting space will be M N , but it is not always so. The reality is more interesting, and to get the dimensionality of many-particle states correctly, you need to dig deeper into the concept of identity of quantum particles.