The Schrodinger equation is a second order differential equation whose solution called the wave function, describes the wavelike properties of a particle. The wave function, also called the quantum state, is the most complete description that can be given to a quantum particle. The solutions to the Schrodinger equation encompass not only subatomic, atomic and molecular states but it can also theoretically describe the complete universe.

When applied to subatomic particles, the Schrodinger equation describes them not as materialistic point particles, but as three-dimensional matter waves, and in the case of the electron, a three-dimensional matter wave which surrounds the atomic nucleus. The classical Bohr orbitals are replaced by electronic “clouds”, smearing out the electron, delocalizing it throughout a small region of space. The electronic cloud is called an orbital and represents the probability of finding an electron in the region of space occupied by the cloud.

"The Schodinger equation is the reason why stars shine at night. The equation is responsible for quantum tunnelling, the process which allows protons to overcome the Coulomb barrier, the electrostatic interaction that protons and atomic nuclei need to overcome so that they can reach close proximity to undero a nuclear fusion." Stardust

Consider a subatomic particle, such as a neutron, in motion with no forces exerting on it. So if there are no forces then the energy potential U is independent of the position x and therefore constant. So, U(x) = 0.

If the neutron is travelling in the +x direction with a momentum of magnitude p, then its kinetic energy is

Such a particle in a definate state of energy is referred to as being in a
stationary state.

From the de Broglie equations the particle has a definate wavelength

The wave function can be written as analogical to travelling mechanical waves
However, this equation does not describe a stationary state but can be transformed to stationary state by replacing B with iA.
Euler’s formula states that for any angle
Comparing this equation
with the equation desciribing the stationary state form
with energy
and spatial wave function
Substituting
Since
the free particle wave function therefore satisfies the Schrodinger equation.

If the potential in the x direction is not constant, as in the two potential wells in the illustration below, then the solutions to the Schrodinger equation are possible only for certain values of Energy E. These values represent the allowed energy levels of the system described by U(x).

Sketch of Erwin Schrödinger by Louisa Gilder