From a mathematical point of view, the Schrödinger Equation is a LINEAR partial differential equation, as is a partial differential equation that is defined by a linear polynomial in the solution and its derivatives.
For a linear differential equation, if you got two different solutions [tex]\psi[/tex] and [tex]\phi[/tex], then the linear combination [tex]\alpha \psi + \beta \phi[/tex], where [tex]\alpha[/tex] and [tex]\beta[/tex] are scalars, is also a solution.
This also is valid for only one solution (think of the other solution as equal to zero, [tex]\phi = 0[/tex] ). So, as the Schrödinger Equation is a Linear partial differential equation, then if [tex]\psi[/tex] is a solution, then [tex]A \psi[/tex] must also be a solution.
This is extremely important for physicist, as let us know that the superposition principle is valid.
