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 operatorisreal, thenmustbe a Hermitian operator. Postulate 3. In any measurement of the observable associated withoperatortheonly values that will ever be observed are the eigenvaluesassociatedwith 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 eigenstateofwitheigenvaluethenany measurement of the quantitywillyield
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 ofas
where the summation may be infinite. In this case we only knowthat the measurement ofwillyield one of the eigenvalues ofbutwe don't know which one. However, we do know the probability thateigenvaluewilloccur--it is the absolute value squared of the coefficient,
A consequence is that, after measurement ofyieldssome eigenvaluethewavefunction immediately ``collapses'' into the correspondingeigenstateorin the case thatisdegenerate, so has more than one corresponding eigenvector,thenbecomesthe projection ofontothe 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 wavefunctionthenthe average value of the observable corresponding toisgiven 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 antisymmetry principle.