Postulate 1. The state of a quantum mechanical system is completely specified by a function that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property thatis the probability that the particle lies in the volume elementlocated atat time t.

The wavefunction must satisfy certain mathematical conditions because of this probabilistic interpretation. For the case of a single particle, the probability of finding it somewhere is 1, so that we have the normalization condition

It is customary to normalize particle wavefunctions to 1. The wavefunction must be single-valued, continuous, and finite.

Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.

If we require that the expectation value of an operatoris real, thenmust be a Hermitian operator.

Postulate 3. In any measurement of the observable associated with operatorthe only values that will ever be observed are the eigenvaluesassociated with that operator, which satisfy the eigenvalue equation

This postulate captures the central point of quantum mechanics--the values of dynamical variables can be quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states). If the system is in an eigenstate ofwith eigenvaluethen any measurement of the quantitywill yield

Although measurements must always yield an eigenvalue, the state does not have to be an eigenstate of initially. An arbitrary state can be expanded in the complete set of eigenvectors ofas

where the summation may be infinite. In this case we only know that the measurement ofwill yield one of the eigenvalues ofbut we don't know which one. However, we do know the probability that eigenvaluewill occur--it is the absolute value squared of the coefficient,

A consequence is that, after measurement ofyields some eigenvaluethe wavefunction immediately ``collapses'' into the corresponding eigenstateor in the case thatis degenerate, so has more than one corresponding eigenvector, thenbecomes the projection ofonto the degenerate subspace). Thus, measurement affects the state of the system. This fact is used in many elaborate experimental tests of quantum mechanics.

Postulate 4. If a system is in a state described by a normalized wave functionthen the average value of the observable corresponding tois given by

Postulate 5. The wavefunction or state function of a system evolves in time according to the time-dependent Schrödinger equation

Postulate 6. The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. Electronic spin must be included in this set of coordinates.

The Pauli exclusion principle is a direct result of this antisymmetry principle.