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[3. Schrödinger equations.]
First Schrödinger equation.
Let u be a given classical mechanical quantity; let u_{op}
the quantum operator associated to u; the first Schrödinger equation
for u is
.
For example, if u = E then the first Schrödinger equation for
the energy E is
.
In the first Schrödinger equation the complex function φ (q) is the unknown
and u is a parameter. For every value of u there is a complex function
φ (q) that is a solution of the equation. However, not every φ (q) is
acceptable from a physical point of view, so some restrictions are imposed on φ (q).
One of such restrictions is the requirement that the quadratic integral of φ (q)
exists and is finite, that is φ (q) is acceptable as a solution only if the integral
∫ φ (q)² dx, extended over the domain to which x belongs,
is finite. Another requirement is that φ (q) must be finite for all values
of its argument q.
With such requirements, not for every value of u has the equation an acceptable
solution φ (q). The physical meaning of first Schrödinger equation is precisely
that only for certain values of u the equation has a solution φ (q): those
values are the admissible values for the physical quantity u.
Consider the equation (3.6) (first Schrödinger equation for energy E):
it admits solutions, compatible with the requirements on φ (q), only for a discrete
set of values of E, not for every value of E. Equation (3.6) thus determines the
admissible value for mechanical energy: they are the values giving a solution to the equation.
While in classical physics every value for the energy E is admissible, in quantum
mechanics only those values of the energy that satisfies the first Schrödinger equation
are admissible: this is the root of the quantization of the energy levels.
A brief recapitulation. Let u be a physical quantity; every value of u
is acceptable in classical physics. In quantum mechanics an operator u_{op}
is defined according to certain rules, so to construct the equation
u_{op}φ (q) = u·φ (q).
Some conditions of regularity are imposed on φ (q). The equation has solutions
only for certain values of u: those are the only acceptable values of u.
The physical quantity u has been quantized.
The values u_{i} for which the first Schrödinger equation
u_{op}φ (q) = u·φ (q)
has admissible solutions are called eigenvalues and the corresponding solutions
φ_{i} (q) are called eigenfunctions.
This words derives from the German terms eigenvert and eigenfunktion,
utilized by Schrödinger in his four articles entitled
Quantisierung als Eigenwertproblem (1926).
