23: Postulates of Quantum Mechanics

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In order to understand deeper quantum mechanics, scientists have derived a series of axioms that result in what are called postulates of quantum mechanics. These are, in fact, assumptions that we need to make to understand how the measured reality relates with the mathematics of quantum mechanics. It is important to notice that the postulates are necessary for the interpretation of the theory, but not for the mathematics behind it. Regarding of whether we interpret it or not, the mathematics is complete and consistent. In fact, as we will see in the next chapter, several controversies regarding the interpretation of the mathematics are still open, and different philosophies have been developed to rationalize the results. Recall also that there are different ways of writing the equation of quantum mechanics, all equivalent to each other (i.e., Schrödinger’s differential formulation and Heisenberg’s algebraic formulation that we saw in chapter 3). For these reasons, there is not an agreement on the number of postulates that are necessary to interpret the theory, and some philosophy and/or formulation might require more postulates than others. In this chapter, we will discuss the six postulates, as they are usually presented in chemistry and introductory physics textbooks and as they relate with a basic statistical interpretation of quantum mechanics. Regardless of the philosophical consideration on the meanings and numbers of the postulate, as well as their physical origin, these statements will make the interpretation of the theory a little easier, as we will see in the next chapter.

23.1: Postulate 1- The Wave Function Postulate
The state of a quantum mechanical system is completely specified by a function Ψ(r,t) that depends on the coordinates of the particle(s) and on time.
23.2: Postulate 2- Experimental Observables
To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.
23.3: Postulate 3- Individual Measurements
In any measurement of the observable associated with operator A^, the only values that will ever be observed are the eigenvalues a that satisfy the eigenvalue equation.
23.4: Postulate 4- Expectation Values and Collapse of the Wavefunction
If a system is in a state described by a normalized wave function Ψ, then the average value of the observable corresponding to A^ is given by.
23.5: Postulate 5- Time Evolution
The wave function of a system evolves in time according to the time-dependent Schrödinger equation.
23.6: Postulate 6- Pauli Exclusion Principle
The total wave function of a system with N spin-12 particles (also called fermions) must be antisymmetric with respect to the interchange of all coordinates of one particle with those of another.