The Postulates of Quantum
Mechanics
1
Postulate I: the Board and the Game Tools
The system state can be described by a wavefunction defined in the
domain of all possible states of the system.
► For a single particle the possible state space is the set of the
spatial coordinates in which the particle can reside*
► For a system composed of numerous particles the state space
includes all the states of all the particles. The aspiration is to
move to a reduced description of the system (for example:
moving to the center of mass coordinates )
* In the future an additional coordinate will be considered. For
example the spin coordinate.
2
Postulate II: the Rules of the Game
The change of the system state in time can be described by the
evolution operator:
ˆ (t0 )(q, t )
(q, t  t0 )  U
► The evolution operator is a linear operator, which fulfills
properties of a group (Unitary Operator)
► The generator of the evolution group is the energy operator (The
Hamiltonian)

ˆ ( q, t ) ; H
ˆ  ( q)  E   ( q)
i ( q, t )  H
n
n
n
t
► The eigenfunctions of the Hamiltonian operator are also the
eigenfunctions of the evolution operator (stationary states)
3
Postulate III: the Trial Interface
A measuring action performed on the system can be described as an
action of filtering the wavefunction into its components.
 Each measurable quantity has a corresponding Hermitian
Operator, whose eigenfunctions constitute a basis set for filtering.
Each eigenfunction has a corresponding real eigenvalue which
constitutes a possible result of the measurement:
Ô | n   n  | n 
 The state of the system can be written as a superposition of the
basis states:
|    gn  | n     n |   | n 
n
n
4
Postulate III: the Trial Interface
 The result of the measurement is the realization of the system in
only one of its basis states. The measuring process causes a
reduction of the superposition (wave-like) to a single component
(particle-like)
 The probability for a system in a state | to be realized in a
basis state|n is:
Pn  g n   n | 
2
2
 The measurement produces for each particle an experimental
single value, equal to the eigenvalue n of the realized
eigenstate |n
5
The Normalization Requirement
The following must be fulfilled for the square of the wavefunction
to have a probability density significance:
  |     ( q)  ( q) dq   ( q) dq  1
2
*
Q
Q
On the other hand:
  |     |  gn | n    gn  n |    gn g
*
n
n
*
n
n
And therefore:
g
2
n
1
n
6
Average and Expected Values in Statistics
When a fair dice whose faces are marked by the digits 1,2,2,4,4,4 is
tossed once, the domain of possible events is:
x1  1, x2  2, x3  4
If the dice is tossed N times, the average value obtained is:
1
x
N
3
n  x
i
i 1
i
After numerous tosses the value ni/N converges to the probability
for event Pi and the expected value is:
3
lim x   x    Pi  xi   1   2   4  2
N 
i 1
1
6
2
6
3
6
5
6
7
Average and Expectation Values in Quantum
Mechanics
For each single measurement an eigenvalue of the measuring
operator is obtained:
Ô | n   n  | n 
The domain of possible events is {n}
An experimental measurement is composed of a huge number of
single measuring actions, and therefore the average equals the
expectation value. The experimental result is called “the
expectation value of the observation”:
2
ˆ
 O   Pn  n   g n  n   g n   | n   n
n
n
n
ˆ |     | O
ˆ | 
   |  g n  n | n     |  g n  O
n
n
n
8