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 that is the probability that the particle lies in the volume element located at at time .
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
Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.This postulate comes about because of the considerations raised in section 3.1.5: if we require that the expectation value of an operator is real, then must be a Hermitian operator. Some common operators occuring in quantum mechanics are collected in Table 1.
Postulate 3. In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues , which satisfy the eigenvalue equation
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 of (
An important second half of the third postulate is that, after measurement of yields some eigenvalue , the wavefunction immediately ``collapses'' into the corresponding eigenstate (in the case that is degenerate, then becomes the projection of onto 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 function , then the average value of the observable corresponding to is 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. We will later see that Slater determinants provide a convenient means of enforcing this property on electronic wavefunctions.