POSTULATES OF QUANTUM MECHANICS

Quantum mechanics is based on a set of postulates or assumptions from which all properties of a system can be derived as theorems. In this respect, quantum mechanics is modeled after Euclid's geometry as are other physical theories such as thermodynamics and classical mechanics.

Postulate I. wave functions represent states

For every state of a system there exists a complete mathematical representation, called the wave function of the state, denoted as Y(q,t). Here q={q₁, q₂,…,q_n} stands for the collection of all dynamical coordinates of all particles in the system and t is the time. Wave functions are required to satisfy several conditions:
(a) Y is continuous and single valued.
(b) Y is differentiable at least twice with respect to coordinates.
(c) The integral of |Y|² over all space exists.
[Note that the wave function varies with time and q is independent of time. Thus, classical orbits and trajectories in the form q(t) are not assumed.]

Postulate II. Linear operators represent observables

For every observable property of a system, say "A", there exists a corresponding operator (or transformation), , on wave functions. These operators are subject to the conditions:
(a) is a linear operator: (aY + bY') = aY + bY'.
(b) Operators that correspond to real observables are "Hermitian". That is, they obey the rule

(c) The following prescription tells how quantum mechanical operators are constructed (this is the "Schroedinger representation")
Write the classical expression for the observable: A(q,p) where q and p are the position and momentum variables of the particles comprising the system. Next replace each component of q by the operator
q_i q_i (multiplication by q_i)
and replace the components of the momentum by
p_i
For instance, the kinetic energy of a single particle moving in three dimensions is given by
,
its potential energy is just and the total energy is

Integrals in quantum mechanics are usually abbreviated in a "bra-ket" or bracket notation. Thus, we write the shorthand:

Postulate III. Eigenvalues are possible measured values

The only possible results of measuring the observable "A" are the eigenvalues of the corresponding operator: . We call a_i an eigen- or characteristic value of "A"; and the set of all eigenvalues, {a_i, i ranges} is the spectrum of "A". Experiments to measure "A" always result in a value from this spectrum.
Furthermore, the collection of eigenfunctions of an observable (any observable) is a complete set or basis for expanding any state function of the system: .
In the laboratory we may prepare the system in a specified state (Y) and then subject it to the measurement of "A". Such a measurement results in one of the values a_i and induces a change of the system state to Y_i. The initial state is a mixture of A-eigenstates having precisely known values of the observable "A". All of this is expressed in the two equations above.

This may be called the "quantization" postulate.

Postulate IV. Average Values from Wave Functions

Repeated measurements of "A" on a system in the state Y, or equivalently measurement of "A" on many duplicate systems in the same state, yield an average value given by the expression
.
Such an average value is defined from the outcomes of individual measurement (a_i, by postulate III) and the frequency, f_i, to obtain such outcomes:
. (The numerator is the sum of measured values of "A" and the denominator is the number of measurements.)
This postulate gives a probabilistic interpretation to the wave function. To see this, use the expansion of Y in the basis of A-eigenstates: . Substitute this expansion into the average value expression and compare the result with the definition of average value. It follows that the quantity

is the probability that a measurement of "A" will yield the result a_i. It is customary to use "normalized" wave functions, those having , so that simply |c_i|² becomes the probability to measure a_i.

In the special case of position measurements, "A" = q, the eigenvalues are continuous and the eigenfunctions are Dirac delta functions. Then it can be shown that for normalized wave functions should be interpreted as the probability to find the system in the element of volume q to q+dq.

Postulate V. Equation of Motion, Schrödinger's Equation

Wave functions satisfy the following "equation of motion": where H is the hamiltonian of the system. This is the time dependent Schrödinger equation. This postulate describes the evolution of wave functions in time.
If H is independent of time then there exist special states called stationary states having the form . Such a product of a spatial factor and a time factor will separate the variables in the time dependent Schrödinger equation. The solution of the time dependence is elementary: , and the spatial equation becomes: . The latter is called the time independent Schrödinger equation.
Energy eigenfunctions and eigenvalues occupy a very important place in quantum mechanics. The energy eigenfunctions correspond to stationary states of a system. Transitions between stationary states are accompanied by radiation that constitutes the characteristic spectrum of the system. Thermal energy is distributed among energy eigenstates as dictated by the Boltzmann formula and all thermodynamic state functions are determined by such distributions.

Postulate VI. Spin and Antisymmetry

Electrons have intrinsic mass , charge, and spin angular momentum s=h/4p.

Wave functions of systems with two or more indistinguishable particles are either symmetric or antisymmetric under interchanges of these particles. Particles with spin angular momentum s=h/4p , s=3h/4p , s=5h/4p , etc. have antisymmetric wave functions, and particles with spin angular momentum s=0, s=2h/4p , s=4h/4p , etc. have symmetric ones.