The Framework of Quantum Mechanics

Rays in a vector space are simply one-dimensional subspaces. The state space axiom therefore says that states of a quantum systems are identified with an entire one-dimensional subspace tα|ψy : α P Cu, where |ψy is some unit vector. When performing calculations the convention is to represent physical states by unit vectors, and so henceforth we will assume that all kets are normalized unless otherwise stated. It is understood that multiplying this ket by an overall scalar does not change the physical state it represents. In particular, multiplying |ψy by an overall phase eiφ does not change any of the experimental outcomes predicted in quantum mechanics. This will be explained in greater detail when we discuss the Measurement Axiom. There is another type of phase that does distinguish one state from another and which can lead to different experimental predictions. For a state |ψy decomposed in a linear combination |ψy = α|0y+ β|1y, a relative phase is a factor eiφ that is multiplied to just one of the kets but not both. For example, we would say that the vector |ψ1y = α|0y+ βeiφ|1y differs from the vector |ψy by an relative phase, and the two represent different physical states of the system. The State Space Axiom does not tell you the dimension of the Hilbert space associated with a given physical system. Nor does the State Space Axiom tell you which physical states of the system correspond to which rays in the Hilbert space. How both of these assignments are made depends on experimental observation. To understand this better, we need a way to describe observation and measurement in quantum mechanics. The Measurement Axiom will stipulate how this is done, but first we address of how quantum systems evolve in time.