TIME-INDEPENDENT SCHRÖDINGER EQUATION

What is it?

Simply stated, Schrödinger's equation can be interpretted as the following statement: when the hamiltonian operator $\hat{H}$ that describes a system is applied to the wavefunction of a stationary state of a matter wave confined in that system $\psi$, the wavefunction is scaled by the energy of that wavefunction, $E$.

The following statements are also true about the time-independent schrödinger equation:

• The wavefunction $\psi$ does not cancel out by being on both sides of the equation, because the hamiltonian operater contains a derivative that must first be applied
• The hamiltonian operator contains the complete description of the system the matter wave is confined within. More specifically, it describes all the forces acting upon the matter wave (in the potential energy term of the hamiltonian) and the mass of the matter wave (in the kinetic energy term of the hamiltonian)
• A 'solution' to schrödinger's equation is a wavefunction that is a stationary state of the system defined by the hamiltonian operator
• For a given system, there are generally an infite number of solutions to schrödinger's equation. That does not, however, mean that any wavefunction is a solution to schrödinger's equation.
• If the hamiltonian operator is applied to a wavefunction that is not a stationary state of the system described by the hamiltonian operator, the wavefunction will change shape as well as scale, and you will know that wavefunction was not a solution to the equation.

The hamiltonian operator

The hamiltonian operator describes the system: the bit of matter you are trying to calculate wavefunctions and energies for, as well as all of the forces acting upon it. It has the following form: $${\hat{H} = -\frac{\hbar^2}{2m}\left(\frac{\partial^2 }{\partial{x^2}}+\frac{\partial^2 }{\partial{y^2}}+\frac{\partial^2 }{\partial{z^2}}\right)+U(x,y,x)}$$ The first term in the equation is the kinetic energy operator, and the second term is the potential energy operator.