State description in classical mechanics


3. Schrödinger equations.A brief presentation of the state description in classical physics precedes the analysis of the rules that associate a quantum operator to a classical physical quantity; then the first Schrödinger equation is introduced and, as an example of application of the equation, the formula for Bohr energy levels is derived. The section ends with an introduction to the state description in quantum mechanics and the role of the second Schrödinger equation. State description in classical mechanics.Consider a classical particle (that is a particle moving according to Newton's laws of motion), with a given mass m, that is subjected to a field of force. Suppose that:
Then, according to classical physics, it is possible to determine:
If the initial position and momentum of a particle are known, with the exact expression
of the forces acting on that particle, then the position and the momentum of the
particle can be calculated at every time in the future and in the past.
Since every classical mechanical quantity is uniquely defined by means of position and
momentum, which are known at every time, then also the value of every classical
mechanical quantity at every time is known.
Relations (3.1)(3.3), derived in a simple classical system, are nevertheless useful in quantum mechanics.
