
(1) 
Schroedinger introduced
a wave equation
which the de Broglie wave
accompanying the motion of
a substance particle
should satisfy.
It was called the
Schroedinger equation
and became
the fundamental equation
of quantum mechanics.


(2) 
Heisenberg thought
that, in the microscopic world,
the position and momentum
coordinates, q
and
p ,
are not such ordinary variables
as those in the classical mechanics
but that they are matrices.
Thereby he proposed
the matrix mechanics.
It was clarified that
these two types of mechanics,
Schroedinger's wave mechanics
and
Heisenberg's matrix mechanics,
are equivalent
to each other.


(3) 
It had been clarified
in the microscopic world that
it is in principle
impossible to measure
the position and momentum
of a particle more precisely
than the limitations
prescribed by
Heisenberg's uncertainty
principle.
Accordingly,
it was made clear that,
in the microscopic world,
the concept of "orbit"
of a particle,
which can be described
by a curved "line",
must be abandoned.


(4) 
The probability interpretation
of the wave function
was proposed.
It claims that
the square of the absolute
value of the wave function
denotes the probability density
that the particle
will be found.
It was elucidated
that the tunnel effect,
based on this idea,
can explain the alpha decay
of atomic nuclei.
This was one of the strong
proofs of the validity
of quantum mechanics.
