FOUNDATIONS OF QUANTUM MECHANICS

III
THE POSTULATES
The sciences do not try to explain, they hardly even try to interpret, they mainly make
models. By a model is meant a mathematical construct which, with the addition of certain
verbal interpretations, describes observed phenomena. The justification of such a
mathematical construct is solely and precisely that it is expected to work [. . . ]
— John von Neumann
It would seem that the theory is exclusively concerned about ‘results of measurement’,
and has nothing to say about anything else. [. . . ] To restrict quantum mechanics to be
exclusively about piddling laboratory operations is to betray the great enterprise.
— John Bell
In this chapter we will formulate and discuss Von Neumann’s postulates. Next, we will extend the
quantum mechanical concept of ‘pure’ states by adding ‘mixed’ states, and show how quantum
mechanics treats states of subsystems of composite physical systems. Finally, we apply these
concepts to spin 1/2 particles and we derive some formulas needed in subsequent chapters.
III. 1
VON NEUMANN’S POSTULATES
We are now ready to give, in some cases in simplified fashion, Von Neumann’s postulates of
quantum mechanics, which link the physical concepts of the theory to the mathematical concepts of
its formalism.
1. State postulate, pure states. Every physical system has a corresponding Hilbert space H, the
states of the system are completely described by unit vectors in H. A composite physical
system corresponds to the direct product of the Hilbert spaces of the subsystems.
2. Observables postulate. Every physical quantity A of the system corresponds to a self - adjoint
operator A in H. Dirac called the quantities ‘observables’.
3. Spectrum postulate. The only possible outcomes which can be found upon measurement of a
physical quantity A, corresponding to an operator A, are values from the spectrum of A.
4. Born postulate, discrete case. If the system is in a state |ψ⟩ ∈ H, and a measurement is made
of a physical quantity A, corresponding to an operator A with a discrete spectrum Spec A,
probability to find the outcome a i ∈ Spec A, is equal to
Prob |ψ⟩ (a i ) = ⟨ψ | P ai | ψ⟩, (III. 1)