1 - Second quantization

Summary

Quantum mechanics of one particle

In quantum mechanics the physical state of a particle is described in terms of a ket |Ψ〉. This ket belongs to a Hilbert space which is nothing but a vector space endowed with an inner product. The dimension of the Hilbert space is essentially fixed by our physical intuition; it is we who decide which kets are relevant for the description of the particle. For instance, if we want to describe how a laser works we can choose those energy eigenkets that get populated and depopulated and discard the rest. This selection of states leads to the well-known description of a laser in terms of a three-level system, four-level system, etc. A fundamental property following from the vector nature of the Hilbert space is that any linear superposition of kets is another ket in the Hilbert space. In other words we can make a linear superposition of physical states and the result is another physical state. In quantum mechanics, however, it is only the “direction” of the ket that matters, so |Ψ〉 and C|Ψ〉 represent the same physical state for all complex numbers C. This redundancy prompts us to work with normalized kets. What do we mean by that? We said before that there is an inner product in the Hilbert space.