If we consider the equation (6.5)

which represents a state of a particle bouncing between two walls and differentiate twice Ψ_{n}
with respect to *x*, we obtain

From (6.7) we have

so (7.1) becomes

Introducing the symbol

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

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

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

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

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
,
called **eigenvalues** of the operator *H*, and the functions Ψ_{n}, called **eigenfunctions**
or **eigenstates** of *H*.