Mauro Murzi's pages on Philosophy of Science - Quantum mechanics
State description in quantum mechanics
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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 uop be the corresponding quantum operator. The first Schrödinger equation is first Schroedinger equation. The eigenvalues ui are the admissible values of u.

The complex function Ψ (q) can be represented as a sum of the eigenfunctions:

expansion of Psi function

Complex numbers σi are called the expansion coefficients; the square of their module determines the probability PSui ) that u has the value ui in the state S :   PSui ) = | σi |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 :   PSq ) = | 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:

  1. The eigenvalues of the first Schrödinger equation give the admissible values for that physical quantity.
  2. The representation of Ψ (q) as a sum of the eigenfunctions gives, by means of the expansion coefficients, the probability of every value.
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