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[3. Schrödinger equations.]
State description in quantum mechanics.
The state S of a quantum system is completely described by a complex function
Ψ (q) depending from position q.
Let u be a classical physical quantity and let u_{op} be the
corresponding quantum operator. The first Schrödinger equation is
.
The eigenvalues u_{i} are the admissible values of u.
The complex function Ψ (q) can be represented as a sum of the
eigenfunctions:
Complex numbers σ_{i} are called the expansion coefficients;
the square of their module determines the probability
P_{S}( u_{i} ) that u has the value
u_{i} in the state S :
P_{S}( u_{i} ) =  σ_{i} ² .
The square of the module of Ψ (q) gives the probability that, in the
state S, the system is to be found in position q :
P_{S}( q ) =  Ψ (q) ² .
The probability of every physical quantity, in the state S, is determined by the complex function
Ψ (q) together with the first Schrödinger equation, by means of the
following two steps:
 The eigenvalues of the first Schrödinger equation give the admissible
values for that physical quantity.
 The representation of Ψ (q) as a sum of the eigenfunctions gives,
by means of the expansion coefficients, the probability of every value.
