FOUNDATIONS OF QUANTUM MECHANICS

42 CHAPTER III. THE POSTULATES
where P ai is the projector from the spectral decomposition (II. 57) of A.
5. Schrödinger postulate. As long as no measurements are made on the system, the time evolution
of the system is described by a unitary transformation,
|ψ(t)⟩ = U (t, t 0 ) |ψ(t 0 )⟩. (III. 2)
6. Projection postulate, discrete case. If the system is in a state |ψ⟩ ∈ H and a measurement is
made on a physical quantity A corresponding to an operator A with discrete spectrum, and the
outcome of the measurement is the eigenvalue a i ∈ Spec A, the system is, immediately after
the measurement, in the eigenstate
|ψ⟩ P a i
|ψ⟩
. (III. 3)
∥P ai |ψ⟩∥
The first four postulates connect the (undefined) concepts ‘physical system’, ‘state’, ‘quantity’
and ‘measurement’ to mathematical concepts. In the literature the postulates 3 and 4 are sometimes
combined into the so - called measurement postulate. The last two postulates determine the evolution
of the states in time.
Ad 1. The state postulate implies that systems with the same |ψ⟩ are in the same physical state.
The way in which this state vector |ψ⟩ is produced, is thus unimportant. Also the fact that two systems
which are described by the same |ψ⟩ can, upon measurement, have different outcomes, which is
allowed according to the measurement postulate, is no reason to regard their states as being different.
On the other hand, not every pair of mutually different unit vectors also represent different states.
Usually it is assumed that vectors whose only difference is their phase factor e iθ , with θ ∈ R, describe
the same physical state, because they predict the same probability distributions for outcomes of all
possible measurements. Such vectors form a so - called unit ray.
The statement that all unit vectors of H describe physical states also need not be true in general.
Notice that the set of unit vectors is extremely large. Even for a particle in one spatial dimension the
Hilbert space is infinite - dimensional. Furthermore, some types of superposition, linear combinations
of two or more eigenstates, do not occur in nature, for instance superpositions of states with different
charges, i.e., electrical, baryonic etc., or superpositions of states with different spin.
It is possible to prohibit these superpositions in the theory by introducing so - called superselection
rules. The requirement that, for identical particles, only states are allowed which are symmetric or
antisymmetric under permutation of the particles is an example of such a superselection rule. In the
presence of a superselection rule the class of allowed states breaks up into in a direct sum of the
eigenspaces of the superselection operator,
H = ⊕ j=1
H j . (III. 4)
Within one such subspace H j , called a coherent sector, superpositions of all states are allowed.