## Postulates

Postulate 1. The state of a quantum mechanical system iscompletely specified by a function that depends on the coordinates ofthe particle(s) and on time. This function, called the wave functionor state function, has the important property that isthe probability that the particle lies in the volume element locatedat attime t.

The wavefunction must satisfy certain mathematical conditionsbecause of this probabilistic interpretation. For the case of asingle particle, the probability of finding it somewhere is 1, sothat we have the normalization condition It is customary to normalize particle wavefunctions to 1. Thewavefunction must be single-valued, continuous, and finite.

Postulate 2. To every observable in classical mechanics therecorresponds a linear, Hermitian operator in quantum mechanics.If we require that the expectation value of an operator isreal, then mustbe a Hermitian operator. Postulate 3. In any measurement of the observable associated withoperator theonly values that will ever be observed are the eigenvalues associatedwith that operator, which satisfy the eigenvalue equation This postulate captures the central point of quantummechanics--the values of dynamical variables can be quantized(although it is still possible to have a continuum of eigenvalues inthe case of unbound states). If the system is in an eigenstateof witheigenvalue thenany measurement of the quantity willyield Although measurements must always yield an eigenvalue, the statedoes not have to be an eigenstate of initially.An arbitrary state can be expanded in the complete set ofeigenvectors of as where the summation may be infinite. In this case we only knowthat the measurement of willyield one of the eigenvalues of butwe don't know which one. However, we do know the probability thateigenvalue willoccur--it is the absolute value squared of the coefficient,

A consequence is that, after measurement of yieldssome eigenvalue thewavefunction immediately collapses'' into the correspondingeigenstate orin the case that isdegenerate, so has more than one corresponding eigenvector,then becomesthe projection of ontothe degenerate subspace). Thus, measurement affects the state of thesystem. This fact is used in many elaborate experimental tests ofquantum mechanics.

Postulate 4. If a system is in a state described by a normalized wavefunction thenthe average value of the observable corresponding to isgiven by Postulate 5. The wavefunction or state function of a system evolves in timeaccording to the time-dependent Schrödinger equation Postulate 6. The total wavefunction must be antisymmetric with respect tothe interchange of all coordinates of one fermion with those ofanother. Electronic spin must be included in this set of coordinates.

The Pauli exclusion principle is a direct result ofthis&nbsp;antisymmetry principle. 