7. Schrödinger equation.


If we consider the equation (6.5)

Eqn001.gif

which represents a state of a particle bouncing between two walls and differentiate twice Ψn with respect to x, we obtain

Eqn002.gif

Eqn003.gif

From (6.7) we have

Eqn004.gif

so (7.1) becomes

Eqn005.gif

Eqn006.gif

Introducing the symbol

Eqn007.gif

(said Dirac constant or reduced Planck constant) in (7.3), we get

Eqn008.gif

Since the equality (7.5) holds for every n, we can write

Eqn009.gif

The partial differential equation (7.6) is said Schrödinger equation of the analyzed physical system.

The Schrödinger equation plays a fundamental role in the study of the behavior of atomic and subatomic particles called Quantum Mechanics. It is as important as the Newton laws in Classical Mechanics.

The (7.6) is a very simple expression of the Schrödinger equation, because it applies only a static, one-dimensional system without potential energy. More generally, if a static system is three-dimensional and has potential energy, the Schrödinger equation must be written as

Eqn010.gif

Finally, to simplify the notation, we can introduce the symbol H, said Hamiltonian operator,

Eqn011.gif

Eqn012.gif

In conclusion, to understand the behavior of a physical system, one writes and tries to resolve the equation (7.9). The solution of this equation allows to obtain the values Eqn013.gif, called eigenvalues of the operator H, and the functions Ψn, called eigenfunctions or eigenstates of H.